405:
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711:(red solid curve) with the same span and sag. The catenary represents the profile of a simple suspension bridge, or the cable of a suspended-deck suspension bridge on which its deck and hangers have negligible weight compared to its cable. The parabola represents the profile of the cable of a suspended-deck suspension bridge on which its cable and hangers have negligible weight compared to its deck. The profile of the cable of a real suspension bridge with the same span and sag lies between the two curves. The catenary and parabola equations are respectively,
588:
34:
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8669:{\displaystyle {\begin{alignedat}{3}{\frac {dx}{dp}}&={\frac {T_{0}}{T}}{\frac {ds}{dp}}&&=T_{0}\left({\frac {1}{T}}+{\frac {1}{E}}\right)&&={\frac {a}{\sqrt {a^{2}+p^{2}}}}+{\frac {T_{0}}{E}}\\{\frac {dy}{dp}}&={\frac {w_{0}p}{T}}{\frac {ds}{dp}}&&={\frac {T_{0}p}{a}}\left({\frac {1}{T}}+{\frac {1}{E}}\right)&&={\frac {p}{\sqrt {a^{2}+p^{2}}}}+{\frac {T_{0}p}{Ea}}\,.\end{alignedat}}}
62:
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378:(1638), Galileo wrote that a hanging cord is only an approximate parabola, correctly observing that this approximation improves in accuracy as the curvature gets smaller and is almost exact when the elevation is less than 45°. The fact that the curve followed by a chain is not a parabola was proven by
842:
or mooring device than would be the case if it were nearly straight. This enhances the performance of the anchor and raises the level of force it will resist before dragging. To maintain the catenary shape in the presence of wind, a heavy chain is needed, so that only larger ships in deeper water can
691:
with a suspended roadway, the chains or cables support the weight of the bridge, and so do not hang freely. In most cases the roadway is flat, so when the weight of the cable is negligible compared with the weight being supported, the force exerted is uniform with respect to horizontal distance, and
1469:
and everything is inverted. An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium. Equations which define the shape of the curve and the tension of the chain at each point may be derived by a careful inspection of the various forces acting on a
1398:, independent of the interval selected. The catenary is the only plane curve other than a horizontal line with this property. Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the perpendicular segment connecting the centroid of the curve itself and the
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In a catenary of equal strength, the cable is strengthened according to the magnitude of the tension at each point, so its resistance to breaking is constant along its length. Assuming that the strength of the cable is proportional to its density per unit length, the weight,
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present a special case of marine vehicles moving although moored by the two catenaries each of one or more cables (wire ropes or chains) passing through the vehicle and moved along by motorized sheaves. The catenaries can be evaluated graphically.
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of arches, by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect
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where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for
Building." He did not publish the solution to this anagram in his lifetime, but in 1705 his executor provided it as
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Pu, Mingbo; Guo, Yinghui; Li, Xiong; Ma, Xiaoliang; Luo, Xiangang (2018-07-05). "Revisitation of
Extraordinary Young's Interference: from Catenary Optical Fields to Spin–Orbit Interaction in Metasurfaces".
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8857:{\displaystyle {\begin{aligned}x&=a\operatorname {arsinh} \left({\frac {p}{a}}\right)+{\frac {T_{0}}{E}}p+\alpha \,,\\y&={\sqrt {a^{2}+p^{2}}}+{\frac {T_{0}}{2Ea}}p^{2}+\beta \,.\end{aligned}}}
5927:
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2539:
511:. To create the desired curve, the shape of a hanging chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material.
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Calladine, C. R. (2015-04-13), "An amateur's contribution to the design of
Telford's Menai Suspension Bridge: a commentary on Gilbert (1826) 'On the mathematical theory of suspension bridges'",
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9045:{\displaystyle {\begin{aligned}x&=a\operatorname {arsinh} \left({\frac {p}{a}}\right)+{\frac {T_{0}}{E}}p\,,\\y&={\sqrt {a^{2}+p^{2}}}+{\frac {T_{0}}{2Ea}}p^{2}\end{aligned}}}
7989:
7768:
7719:
5714:
3331:
696:, as discussed below (although the term "catenary" is often still used, in an informal sense). If the cable is heavy then the resulting curve is between a catenary and a parabola.
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2072:
1946:
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5655:{\displaystyle {\begin{aligned}T^{2}(\cos ^{2}\varphi +\sin ^{2}\varphi )&=(wa)^{2}+(ws)^{2}\\T^{2}&=w^{2}(a^{2}+s^{2})\\T&=w{\sqrt {a^{2}+s^{2}}}\,.\end{aligned}}}
740:
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If the density of the chain is variable then the analysis above can be adapted to produce equations for the curve given the density, or given the curve to find the density.
2978:
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2230:
805:
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1192:
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522:, United States is sometimes said to be an (inverted) catenary, but this is incorrect. It is close to a more general curve called a flattened catenary, with equation
11695:
4927:
2756:
3325:
2637:
5287:
3132:
8259:{\displaystyle {\begin{aligned}{\frac {dx}{ds}}&=\cos \varphi ={\frac {T_{0}}{T}}\\{\frac {dy}{ds}}&=\sin \varphi ={\frac {w_{0}p}{T}}\,,\end{aligned}}}
6431:
6217:
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9346:
9404:
These equations can be used as the starting point in the analysis of a flexible chain acting under any external force. In the case of the standard catenary,
7312:
1109:
673:
In free-hanging chains, the force exerted is uniform with respect to length of the chain, and so the chain follows the catenary curve. The same is true of a
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1955:
1465:
exerted by the chain is parallel to the chain. The analysis of the curve for an optimal arch is similar except that the forces of tension become forces of
404:
313:
In optics and electromagnetics, the hyperbolic cosine and sine functions are basic solutions to
Maxwell's equations. The symmetric modes consisting of two
2232:, given above, can be solved to produce equations for the curve. We will solve the equation using the boundary condition that the vertex is positioned at
587:
11476:
3945:
10835:. A different (though ultimately mathematically equivalent) derivation, which does not make use of hyperbolic function notation, can be found in
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2941:
Since the primary interest here is simply the shape of the curve, the placement of the coordinate axes are arbitrary; so make the convenient choice of
561:
544:. While a catenary is the ideal shape for a freestanding arch of constant thickness, the Gateway Arch is narrower near the top. According to the U.S.
12486:
8274:
3885:
306:, a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. In the rail industry it refers to the
11317:– The equations governing the shape (static case) as well as dynamics (dynamic case) of a centenary is derived. Solution to the equations discussed.
6015:
3741:{\displaystyle {\frac {dy}{d\varphi }}={\frac {dy}{ds}}{\frac {ds}{d\varphi }}=\sin \varphi \cdot a\sec ^{2}\varphi =a\tan \varphi \sec \varphi \,.}
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5141:
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9103:. The chain is flexible so it can only exert a force parallel to itself. Since tension is defined as the force that the chain exerts on itself,
6764:{\displaystyle w=T_{0}{\frac {d}{ds}}{\frac {dy}{dx}}={\frac {T_{0}{\dfrac {d^{2}y}{dx^{2}}}}{\sqrt {1+\left({\dfrac {dy}{dx}}\right)^{2}}}}\,.}
1539:
12481:
11655:
11356:
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and the weight of the cable and the wire supporting the bridge is negligible in comparison, then the weight on the cable (see the figure in
5936:
4741:{\displaystyle L^{2}-v^{2}=2a^{2}\left(\cosh \left({\frac {x_{2}-x_{1}}{a}}\right)-1\right)=4a^{2}\sinh ^{2}\left({\frac {H}{2a}}\right)\,,}
11635:
10257:
National
Register of Historic Places Inventory-Nomination: Jefferson National Expansion Memorial Gateway Arch / Gateway Arch; or "The Arch"
10195:
10019:
The
Rotational Mechanics of Flexible Or Elastic Bodies 1638–1788: Introduction to Leonhardi Euleri Opera Omnia Vol. X et XI Seriei Secundae
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and so the cable follows a parabola. If the weight of the cable and supporting wires is not negligible then the analysis is more complex.
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3599:{\displaystyle {\frac {dx}{d\varphi }}={\frac {dx}{ds}}{\frac {ds}{d\varphi }}=\cos \varphi \cdot a\sec ^{2}\varphi =a\sec \varphi }
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9114:
1032:
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can roll perfectly smoothly on a road made of a series of bumps in the shape of an inverted catenary curve. The wheels can be any
11263:
8107:{\displaystyle {\frac {dy}{dx}}=\tan \varphi ={\frac {p}{a}}\quad {\text{and}}\quad T={\frac {T_{0}}{a}}{\sqrt {a^{2}+p^{2}}}\,.}
5090:
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1289:
831:. An anchor rode (or anchor line) usually consists of chain or cable or both. Anchor rodes are used by ships, oil rigs, docks,
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7724:
310:
that transfers power to trains. (This often supports a contact wire, in which case it does not follow a true catenary curve.)
11311:– Calculates the deviation from a straight line of a catenary curve and provides derivation of the calculator and references.
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2456:{\displaystyle {\frac {ds}{dx}}={\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}={\sqrt {1+\left({\frac {s}{a}}\right)^{2}}}\,,}
470:
wrote a treatise on the catenary in 1697 in which he provided an incorrect derivation of the correct differential equation.
12266:
9539:
7197:{\displaystyle {\begin{aligned}T\cos \varphi &=T_{0}\,,\\T\sin \varphi &={\frac {1}{c}}\int T\,ds\,.\end{aligned}}}
1371:
from the vertex, that is the roulette traced by a point starting at the vertex when a line is rolled on a catenary, is the
10572:
The parametric equations for the catenary are given by x(t) = t, y(t) = a cosh(t/a), where t=0 corresponds to the vertex
10442:
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the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as a
12440:
12338:
9627:"Herpin effective media resonant underlayers and resonant overlayer designs for ultra-high NA interference lithography"
424:
that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin
7910:{\displaystyle {\frac {dy}{dx}}=\tan \varphi ={\frac {w_{0}p}{T_{0}}}\,,\quad T={\sqrt {T_{0}^{2}+w_{0}^{2}p^{2}}}\,,}
3441:
552:" instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form.
11349:
10027:
1739:
and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written
991:{\displaystyle y=a\cosh \left({\frac {x}{a}}\right)={\frac {a}{2}}\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right),}
176:
127:
10946:
Ira
Freeman investigated the case where only the cable and roadway are significant, see the External links section.
9894:
Jardine, Lisa (2001). "Monuments and
Microscopes: Scientific Thinking on a Grand Scale in the Early Royal Society".
9563:
11790:
7670:
The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.
11314:
1386:
except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels.
12578:
11925:
12501:
12471:
11911:
1364:
1356:
302:
so that forces do not result in bending moments). In the offshore oil and gas industry, "catenary" refers to a
53:
also form a catenary (most prominently visible with high-voltage lines, and with some imperfection near to the
2834:
5669:
467:
262:
9982:
6588:{\displaystyle {\frac {dy}{dx}}=\tan \varphi ={\frac {1}{T_{0}}}\int _{\mathbf {c} }^{\mathbf {r} }w\,ds\,.}
12551:
12364:
12026:
11876:
11845:
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298:
Catenaries and related curves are used in architecture and engineering (e.g., in the design of bridges and
2333:
11289:
7499:
2268:
879:
393:
alluded to a catenary. Some much older arches approximate catenaries, an example of which is the Arch of
9626:
4846:{\displaystyle {\frac {1}{H}}{\sqrt {L^{2}-v^{2}}}={\frac {2a}{H}}\sinh \left({\frac {H}{2a}}\right)\,.}
3083:
3057:
3031:
12588:
12573:
11543:
11528:
7478:{\displaystyle x=c\varphi \,,\quad s=\ln \left(\tan \left({\frac {\pi +2\varphi }{4}}\right)\right)\,.}
1531:
714:
545:
11298:
6298:. Its solution is the usual hyperbolic cosine where the parameters are obtained from the constraints.
11883:
11444:
2902:
409:
17:
9317:{\displaystyle \mathbf {T} (s+\Delta s)-\mathbf {T} (s)+\mathbf {G} \Delta s\approx \mathbf {0} \,.}
6863:
cables follow a parabolic, not a catenary curve, because the roadway is much heavier than the cable.
12476:
11897:
11268:
9725:
7492:
3239:{\displaystyle \cosh ^{-1}{\frac {y-\delta }{a}}={\frac {x-x_{0}}{a}}=\sinh ^{-1}{\frac {s}{a}}\,,}
674:
647:
11104:
10757:
9880:
9840:
4553:{\displaystyle L=a\sinh \left({\frac {x_{2}}{a}}\right)-a\sinh \left({\frac {x_{1}}{a}}\right)\,.}
4433:{\displaystyle v=a\cosh \left({\frac {x_{2}}{a}}\right)-a\cosh \left({\frac {x_{1}}{a}}\right)\,.}
12200:
11890:
11533:
11414:
10880:
6295:
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2640:
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1703:
since the other case is implied by symmetry. The forces acting on the section of the chain from
12491:
12332:
12228:
11609:
10447:
10263:
9809:
9803:
6009:
4221:
3422:{\displaystyle y-\delta =a{\sqrt {1+\left({\frac {s}{a}}\right)^{2}}}={\sqrt {a^{2}+s^{2}}}\,.}
2187:
1360:
745:
681:
658:
390:
360:
12390:
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which can stretch in response to tension. The spring is assumed to stretch in accordance with
6085:
5397:
1436:
with fixed radii at either end that has minimum surface area is a catenary revolved about the
1177:
1074:
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12015:
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11174:
10712:
10255:
10222:
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5719:
2235:
1654:
1466:
1433:
1378:
Another roulette, formed by rolling a line on a catenary, is another line. This implies that
1258:
1011:
832:
250:
9184:
be the external force per unit length acting on a small segment of a chain as a function of
4906:
1394:
Over any horizontal interval, the ratio of the area under the catenary to its length equals
843:
rely on this effect. Smaller boats also rely on catenary to maintain maximum holding power.
12297:
12168:
12049:
11662:
10986:
10054:
9737:
9638:
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denote the weight per unit length of the chain, then the weight of the chain has magnitude
2713:
2615:
1478:
303:
9726:"Theory of microscopic meta-surface waves based on catenary optical fields and dispersion"
6294:
This is an ordinary first order differential equation that can be solved by the method of
437:, meaning "As hangs a flexible cable so, inverted, stand the touching pieces of an arch."
8:
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12447:
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9939:
Super
Structures: The Science of Bridges, Buildings, Dams, and Other Feats of Engineering
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is the natural length of a section of spring, then the length of the spring with tension
6103:
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618:
519:
242:
70:
50:
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9741:
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11007:
10974:
10693:
10658:
10389:
Trinks, W.; Mawhinney, M. H.; Shannon, R. A.; Reed, R. J.; Garvey, J. R. (2003-12-05).
10075:
9919:
9911:
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6109:
5837:
5817:
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1471:
1454:
666:
445:
288:
11720:
10950:
gives the case where only the supporting wires have significant weight as an exercise.
9229:
at the other end, and the external force acting on the segment which is approximately
5284:- coordinate. Begin by combining the squares of the vector components of the tension:
1195:
12248:
12222:
12104:
11765:
11219:
11187:
11012:
10718:
10697:
10554:
10421:
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9611:
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is the position of the axis. The equation can be determined in this case as follows:
1462:
688:
549:
441:
265:. A hanging chain will assume a shape of least potential energy which is a catenary.
246:
33:
12236:
11779:
11085:
6867:
A similar analysis can be done to find the curve followed by the cable supporting a
4340:
respectively. The curve passes through these points, so the difference of height is
389:, whose "true mathematical and mechanical form" in the context of the rebuilding of
12398:
12089:
12039:
11985:
11940:
11869:
11669:
11449:
11002:
10994:
10685:
10650:
10332:
10106:
10070:
10062:
9903:
9745:
9690:
9646:
9589:
9445:
7498:
Recently, it was shown that this type of catenary could act as a building block of
5276:
There is a simple relationship between the tension in the cable at a point and its
1822:
1352:
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1026:
449:
374:
348:
314:
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144:
98:
92:
54:
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which is the length of chain whose weight is equal in magnitude to the tension at
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12149:
12064:
12044:
11995:
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11491:
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11303:
11170:
11154:
10831:
A minor variation of the derivation presented here can be found on page 107 of
9694:
7515:
6192:{\displaystyle {\mathcal {L}}-y'{\frac {\partial {\mathcal {L}}}{\partial y'}}=C}
1527:
1411:
1383:
1019:
838:
When the rope is slack, the catenary curve presents a lower angle of pull on the
598:
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462:
457:
413:
379:
369:
266:
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66:
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10190:
9781:
9724:
Pu, Mingbo; Ma, XiaoLiang; Guo, Yinghui; Li, Xiong; Luo, Xiangang (2018-07-23).
4158:
Three catenaries through the same two points, depending on the horizontal force
329:
204:
The catenary curve has a U-like shape, superficially similar in appearance to a
12099:
12074:
12020:
12005:
11961:
11832:
11824:
11705:
11424:
10975:"Catenary Optics for Achromatic Generation of Perfect Optical Angular Momentum"
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be the lowest point on the chain, called the vertex of the catenary. The slope
1470:
segment using the fact that these forces must be in balance if the chain is in
1423:
1415:
708:
699:
385:
The application of the catenary to the construction of arches is attributed to
11308:
11190:
10889:
9805:
100 Essential Things You Didn't Know You Didn't Know: Math
Explains Your World
9593:
1010:
is the distance of the lowest point above the x axis. All catenary curves are
12567:
12517:
12496:
12433:
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12079:
11990:
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The straight line, the catenary, the brachistochrone, the circle, and Fermat
9650:
9536:
Department of Mechanical & Aerospace Engineering - University of Florida
9528:
7381:
In this case, the curve has vertical asymptotes and this limits the span to
5387:{\displaystyle (T\cos \varphi )^{2}+(T\sin \varphi )^{2}=T_{0}^{2}+(ws)^{2}}
476:
proved in 1744 that the catenary is the curve which, when rotated about the
238:
describes catenaries in a classic statics problem involving a hanging rope.
23:"Chainette" redirects here. For the wine grape also known as Chainette, see
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with a horizontal roadway. If the weight of the roadway per unit length is
1379:
614:
515:
489:
481:
386:
352:
280:
6486:{\displaystyle T\sin \varphi =\int _{\mathbf {c} }^{\mathbf {r} }w\,ds\,,}
3317:{\displaystyle y-\delta =a\cosh \left(\sinh ^{-1}{\frac {s}{a}}\right)\,,}
573:
408:
Analogy between an arch and a hanging chain and comparison to the dome of
333:
219:—the shape assumed by a soap film bounded by two parallel circular rings.
12358:
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is large, the shape of the curve reduces to that of a non-elastic chain.
6852:
4232:
is adjusted so the catenary satisfies the standard equation of the curve
3436:
The differential equation can be solved using a different approach. From
1332:
828:
654:
394:
9395:{\displaystyle {\frac {d\mathbf {T} }{ds}}+\mathbf {G} =\mathbf {0} \,.}
12179:
12084:
12034:
11614:
11409:
11278:
10662:
10638:
9451:
7372:{\displaystyle y=c\ln \left(\sec \left({\frac {x}{c}}\right)\right)\,.}
1598:
1517:
1167:{\displaystyle {\frac {d\varphi }{ds}}={\frac {\cos ^{2}\varphi }{a}},}
1100:
850:
11334:
10530:
9915:
4140:{\displaystyle s=a\tan \varphi =a\sinh \left({\frac {x}{a}}\right)\,.}
4071:{\displaystyle y=a\sec \varphi =a\cosh \left({\frac {x}{a}}\right)\,,}
4009:
Adding and subtracting the last two equations then gives the solution
2938:
being the constant of integration satisfying the boundary condition).
12143:
11730:
11710:
11379:
11195:
11159:"XI Flexible Strings. Inextensible, XII Flexible Strings. Extensible"
7603:
6982:{\displaystyle {\frac {dy}{dx}}=\tan \varphi ={\frac {w}{T_{0}}}x\,.}
2023:{\displaystyle {\frac {dy}{dx}}=\tan \varphi ={\frac {ws}{T_{0}}}\,.}
1276:
669:, also follow a catenary curve, with cables embedded in a rigid deck.
578:
398:
325:
194:
10654:
10223:"The Incredible Gateway Arch: America's Mightiest National Monument"
7091:
is constant, and the analysis for nonuniform chains can be applied.
372:
thought the curve of a hanging chain was parabolic. However, in his
279:. The mathematical properties of the catenary curve were studied by
45:
11956:
11558:
11456:
7944:
is the weight per unit length of the spring with no tension. Write
4154:
1419:
1372:
1368:
1348:
835:, and other marine equipment which must be anchored to the seabed.
693:
643:
485:
453:
276:
254:
216:
205:
82:
11320:
11274:
4000:{\displaystyle \sec \varphi -\tan \varphi =e^{-{\frac {x}{a}}}\,.}
2543:
A reasonably straightforward approach to integrate this is to use
1649:. Since the chain is at rest the sum of these forces must be zero.
568:
435:
ut pendet continuum flexile, sic stabit contiguum rigidum inversum
201:
when supported only at its ends in a uniform gravitational field.
16:
This article is about the mathematical curve. For other uses, see
11858:
11597:
11511:
11389:
10894:
10676:
Parker, Edward (2010). "A Property Characterizing the Catenary".
5933:
is the weight per unit length) and is subject to the constraint
425:
284:
78:
61:
24:
12455:
Viewpoints: Mathematical Perspective and Fractal Geometry in Art
6219:
is an integration constant, in order to obtain a first integral
827:
The catenary produced by gravity provides an advantage to heavy
684:
is a more sophisticated structure with the same catenary shape.
11563:
11327:
Ira Freeman "A General Form of the Suspension Bridge Catenary"
3935:{\displaystyle \sec \varphi +\tan \varphi =e^{\frac {x}{a}}\,,}
839:
823:
chain forms a catenary, with a low angle of pull on the anchor.
820:
492:
gave equations describing the equilibrium of a chain under any
198:
9836:
Memoirs, Correspondence and Private Papers of Thomas Jefferson
6075:{\displaystyle {\mathcal {L}}=(wy-\lambda ){\sqrt {1+y'^{2}}}}
2826:{\displaystyle {\frac {dy}{dx}}=\sinh {\frac {x-x_{0}}{a}}\,,}
867:
650:
are essentially thickened cables, and follow a catenary curve.
230:, or, particularly in the materials sciences, an example of a
165:
159:
116:
113:
11548:
5185:{\displaystyle H=2a\operatorname {arcosh} {\frac {h+a}{a}}\,}
1458:
493:
473:
382:(1587–1657); this result was published posthumously in 1669.
344:
190:
186:
38:
10973:
Pu, Mingbo; Li, Xiong; Ma, Xiaoliang; Luo, Xiangang (2015).
10507:"Chain, Rope, and Catenary – Anchor Systems For Small Boats"
6838:{\displaystyle w={\frac {T_{0}}{\rho \cos ^{2}\varphi }}\,.}
6285:{\displaystyle {\frac {(wy-\lambda )}{\sqrt {1+y'^{2}}}}=-C}
2701:{\displaystyle {\frac {s}{a}}=\sinh {\frac {x-x_{0}}{a}}\,.}
11570:
11553:
6363:{\displaystyle \int _{\mathbf {c} }^{\mathbf {r} }w\,ds\,,}
5037:{\displaystyle a={\frac {{\frac {1}{4}}L^{2}-h^{2}}{2h}}\,}
3810:{\displaystyle x=a\ln(\sec \varphi +\tan \varphi )+\alpha }
815:
677:
or "catenary bridge," where the roadway follows the cable.
508:
168:
150:
119:
11315:
Dynamic as well as static cetenary curve equations derived
10388:
7610:
is variable, but ratio remains valid at a local level, so
5834:
suspended from two points of equal height and at distance
5663:
1856:
The chain is in equilibrium so the sum of three forces is
10004:
abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux
9961:
9081:
acting on the chain, the following analysis can be made.
1602:
Diagram of forces acting on a segment of a catenary from
633:
Catenary arch kiln under construction over temporary form
107:
9188:. The forces acting on the segment of the chain between
4224:
the axes so that the vertex of the catenary lies on the
2179:
1576:{\displaystyle {\frac {d\mathbf {r} }{ds}}=\mathbf {u} }
347:". The English word "catenary" is usually attributed to
11299:"Catenary" at Visual Dictionary of Special Plane Curves
10165:
The Craft and Art of Clay: A Complete Potter's Handbook
6911:. Proceeding as before gives the differential equation
2534:{\displaystyle {\frac {ds}{\sqrt {1+(s/a)^{2}}}}=dx\,.}
2156:
is constant and the vertical component of the tension,
6301:
5999:{\displaystyle \int _{0}^{D}{\sqrt {1+y'^{2}}}dx=L\,.}
2947:
2716:
1825:). Finally, the weight of the chain is represented by
1777:
can be split into two components so it may be written
9349:
9247:
9217:
at one end of the segment, the nearly opposite force
9117:
9054:
are parametric equations for the curve. At the rigid
8890:
8686:
8272:
8122:
7997:
7950:
7780:
7727:
7681:
7616:
7538:
7502:
and was known as "catenary of equal phase gradient".
7396:
7315:
7269:
7252:{\displaystyle c\tan \varphi =\int \sec \varphi \,ds}
7214:
7102:
6999:
6919:
6789:
6723:
6674:
6605:
6499:
6434:
6392:
6326:
6225:
6205:
6135:
6112:
6088:
6018:
5939:
5922:{\displaystyle U=\int _{0}^{D}wy{\sqrt {1+y'^{2}}}dx}
5860:
5840:
5820:
5755:
5722:
5672:
5436:
5400:
5290:
5268:
is the weight per unit length of the chain or cable.
5144:
5093:
4979:
4936:
and so there is at most one position of equilibrium.
4909:
4869:
4754:
4580:
4468:
4348:
4240:
4084:
4015:
3948:
3888:
3823:
3758:
3612:
3483:
3444:
3334:
3252:
3140:
3112:
3086:
3060:
3034:
3019:{\displaystyle y=a\cosh {\frac {x}{a}}\quad \square }
2986:
2905:
2892:{\displaystyle y=a\cosh {\frac {x-x_{0}}{a}}+\delta }
2842:
2764:
2649:
2618:
2553:
2473:
2342:
2271:
2238:
2190:
2090:
2040:
1958:
1912:
1870:
1542:
1292:
1261:
1203:
1180:
1112:
1077:
1035:
890:
748:
717:
177:
162:
147:
128:
104:
11143:"Art. 134 Transcendental Curves; Catenary, Tractrix"
10330:
7663:{\displaystyle {\frac {ds}{dp}}=1+{\frac {T}{E}}\,.}
6384:. Balancing forces as in the uniform chain produces
4277:{\displaystyle y=a\cosh \left({\frac {x}{a}}\right)}
359:
I have lately received from Italy a treatise on the
211:
The curve appears in the design of certain types of
156:
110:
9075:With no assumptions being made regarding the force
7581:{\displaystyle s=\left(1+{\frac {T}{E}}\right)p\,,}
1771:and pulls the section to the right. The tension at
339:The word "catenary" is derived from the Latin word
153:
101:
12352:The Drawing of Geometric Patterns in Saracenic Art
11241:. Trans. Daniel Cresswell. J. Nicholson & Son.
10839:(Article 443, starting in particular at page 317).
9442:– power lines suspended over rail or tram vehicles
9394:
9316:
9140:
9044:
8856:
8668:
8258:
8106:
7983:
7909:
7762:
7713:
7662:
7580:
7477:
7371:
7290:
7251:
7196:
7046:
6981:
6837:
6763:
6587:
6485:
6420:
6362:
6284:
6211:
6191:
6126:does not appear in the Lagrangian, we can use the
6118:
6094:
6074:
5998:
5921:
5846:
5826:
5799:
5741:
5708:
5654:
5422:
5386:
5184:
5130:
5036:
4921:
4895:
4845:
4740:
4572:is expanded using these expressions the result is
4552:
4432:
4276:
4139:
4070:
3999:
3934:
3854:
3809:
3740:
3598:
3465:
3421:
3316:
3238:
3126:
3098:
3072:
3046:
3018:
2972:
2930:
2891:
2825:
2750:
2700:
2631:
2605:{\displaystyle a\sinh ^{-1}{\frac {s}{a}}+x_{0}=x}
2604:
2533:
2455:
2321:
2257:
2224:
2126:
2066:
2022:
1940:
1898:
1575:
1323:
1267:
1247:
1186:
1166:
1083:
1063:
990:
799:
734:
12487:Goudreau Museum of Mathematics in Art and Science
11252:
11185:
10361:
10047:Philosophical Transactions of the Royal Society A
7047:{\displaystyle y={\frac {w}{2T_{0}}}x^{2}+\beta }
5854:. The curve has to minimize its potential energy
5044:where L is the total length of the curve between
1248:{\displaystyle \kappa ={\frac {a}{s^{2}+a^{2}}},}
12565:
10161:
10136:
10006:: the letters of the Latin phrase, alphabetized.
9896:Notes and Records of the Royal Society of London
9625:Bourke, Levi; Blaikie, Richard J. (2017-12-01).
11175:"Chapter V: The Equilibrium of a Flexible Body"
9109:must be parallel to the chain. In other words,
6876:
3054:relation can be derived by solving each of the
2170:is proportional to the length of chain between
2127:{\displaystyle {\frac {dy}{dx}}={\frac {s}{a}}}
11304:The Catenary - Chains, Arches, and Soap Films.
11147:An Elementary Course of Infinitesimal Calculus
10466:
10464:
10462:
10283:Encyclopedia of Twentieth Century Architecture
10097:Gregorii, Davidis (August 1697), "Catenaria",
9856:
9854:
9852:
9850:
9070:
7069:, per unit length of the chain can be written
6106:to be determined. As the independent variable
5800:{\displaystyle T=wy=wa\cosh {\frac {x}{a}}\,.}
1841:is the length of the segment of chain between
1759:is the magnitude of the force. The tension at
1727:, and the weight of the chain. The tension at
810:
283:in the 1670s, and its equation was derived by
12482:European Society for Mathematics and the Arts
11656:Mathematica: A World of Numbers... and Beyond
11350:
10443:"Cranked stress ribbon design to span Medway"
9839:. Henry Colbura and Richard Bertley. p.
9624:
9141:{\displaystyle \mathbf {T} =T\mathbf {u} \,,}
7059:
1657:for the curve may be derived as follows. Let
1448:
1422:if the charge velocity is much less than the
1064:{\displaystyle \tan \varphi ={\frac {s}{a}},}
355:on the construction of an arch for a bridge:
11636:List of works designed with the golden ratio
10196:Notices of the American Mathematical Society
10040:
10038:
7673:The equations for tension of the spring are
7606:of the spring. In the catenary the value of
6991:This is solved by simple integration to get
1477:Let the path followed by the chain be given
1014:to each other, since changing the parameter
10608:
10606:
10604:
10459:
10139:Wood-fired Ceramics: Contemporary Practices
9981:. Lindahall.org. 2002-10-28. Archived from
9847:
9499:(5th ed.). Prentice Hall. p. 22.
9475:
9473:
8678:Integrating gives the parametric equations
7094:In this case the equations for tension are
5192:where H is the horizontal distance between
5131:{\displaystyle L=2a\sinh {\frac {H}{2a}}\,}
1418:travels along a catenary (which tends to a
1342:
857:
269:in 1638 discussed the catenary in the book
11357:
11343:
10141:. University of Pennsylvania. p. 42.
7923:is the natural length of the segment from
6847:
5809:
3855:{\displaystyle y=a\sec \varphi +\beta \,.}
241:Mathematically, the catenary curve is the
11696:Cathedral of Saint Mary of the Assumption
11232:
11153:
11006:
10972:
10636:
10265:Accompanying one photo, aerial, from 1975
10137:Minogue, Coll; Sanderson, Robert (2000).
10110:
10074:
10044:
10035:
10015:
9832:
9749:
9723:
9679:
9388:
9310:
9134:
9099:be the force of tension as a function of
8955:
8846:
8757:
8658:
8248:
8100:
7903:
7845:
7756:
7707:
7656:
7574:
7471:
7409:
7365:
7242:
7186:
7179:
7136:
6975:
6831:
6757:
6581:
6574:
6479:
6472:
6356:
6349:
5992:
5793:
5644:
5181:
5127:
5033:
4839:
4734:
4546:
4426:
4149:
4133:
4064:
3993:
3928:
3848:
3734:
3431:
3415:
3310:
3232:
2819:
2694:
2527:
2449:
2139:The horizontal component of the tension,
2016:
1934:
1389:
1324:{\displaystyle \rho =a\sec ^{2}\varphi ,}
499:
275:recognizing that it was different from a
11083:
10601:
10365:Bridge Engineering: A Global Perspective
10253:
10247:
10188:
10096:
9470:
9065:
7984:{\displaystyle a={\frac {T_{0}}{w_{0}}}}
7763:{\displaystyle T\sin \varphi =w_{0}p\,,}
6851:
4153:
3942:and taking the reciprocal of both sides
1597:
1461:and that it is so flexible any force of
1367:of the parabola is also a catenary. The
866:
814:
698:
653:
642:
460:; their solutions were published in the
403:
324:
60:
44:
32:
12242:Vier Bücher von Menschlicher Proportion
11364:
11264:MacTutor History of Mathematics Archive
11169:
11086:"Chapter 13: The Tractrix and Catenary"
10280:
10162:Peterson, Susan; Peterson, Jan (2003).
9968:, which appeared in the next paragraph.
9893:
7714:{\displaystyle T\cos \varphi =T_{0}\,,}
5709:{\displaystyle y={\sqrt {a^{2}+s^{2}}}}
12566:
11127:
11103:. Hodges, Foster and Figgis. pp.
11096:
10890:"Chaînette - partie 3 : longueur"
10749:
10710:
10675:
10531:"Efficiency of Cable Ferries - Part 2"
10482:
10305:
9808:. W. W. Norton & Company. p.
9801:
9494:
5210:which are located at the same level (
5066:is the sag (vertical distance between
507:are often used in the construction of
11338:
11211:
11186:
11181:. J. & J.J. Deighton. p. 65.
11111:
10717:. Butterworth-Heinemann. p. 56.
10612:
10553:
10440:
10415:
10285:. Taylor & Francis. p. 224.
10220:
9936:
9872:
7514:catenary, the chain is replaced by a
4929:, there is at most one solution with
4855:This is a transcendental equation in
2180:Derivation of equations for the curve
1351:is rolled along a straight line, the
41:hanging from points forms a catenary.
11140:
10362:Fernández Troyano, Leonardo (2003).
10308:Planning and Building a Conservatory
10221:Hicks, Clifford B. (December 1963).
10022:, Zürich: Orell Füssli, p. 66,
7291:{\displaystyle c=\rho \cos \varphi }
6421:{\displaystyle T\cos \varphi =T_{0}}
6372:where the limits of integration are
6306:
5271:
4939:However, if both ends of the curve (
2067:{\displaystyle a={\frac {T_{0}}{w}}}
1941:{\displaystyle T\sin \varphi =ws\,,}
1899:{\displaystyle T\cos \varphi =T_{0}}
1691:since it is a minimum point. Assume
480:-axis, gives the surface of minimum
12427:Journal of Mathematics and the Arts
11323:Unified approach to some geodesics.
11239:Elements of the Theory of Mechanics
9583:
7505:
6899:is the horizontal distance between
6877:Catenary#Model of chains and arches
6302:Generalizations with vertical force
2322:{\displaystyle (x,y)=(x_{0},y_{0})}
2136:is an equation defining the curve.
871:Catenaries for different values of
638:
13:
12441:Making Mathematics with Needlework
11205:
10254:Harrison, Laura Soullière (1985),
9296:
9262:
7491:and, apparently independently, by
6169:
6162:
6157:
6138:
6021:
5749:), so we get the simple relations
3099:{\displaystyle x\leftrightarrow s}
3073:{\displaystyle x\leftrightarrow y}
3047:{\displaystyle y\leftrightarrow s}
1837:is the weight per unit length and
1811:is the angle between the curve at
1807:is the magnitude of the force and
567:Catenary arches under the roof of
548:nomination for the arch, it is a "
488:) for the given bounding circles.
14:
12600:
12267:I quattro libri dell'architettura
11246:
10562:MathWorld--A Wolfram Web Resource
10191:"Mathematics of the Gateway Arch"
9448:– an elliptic/hyperbolic catenary
5233:The horizontal traction force at
4442:and the length of the curve from
735:{\displaystyle y={\text{cosh }}x}
12546:
12545:
11791:Self-portrait in a Convex Mirror
11475:
11235:"Chapter XXIII: On the Catenary"
11163:A Treatise on Analytical Statics
11121:A Treatise on Analytical Statics
9384:
9376:
9357:
9306:
9292:
9275:
9249:
9130:
9119:
6565:
6558:
6463:
6456:
6340:
6333:
3466:{\displaystyle s=a\tan \varphi }
1715:are the tension of the chain at
1569:
1550:
626:
606:
586:
560:
420:In 1671, Hooke announced to the
222:The catenary is also called the
143:
97:
11077:
11060:
11047:
11035:
11023:
10966:
10953:
10940:
10927:
10914:
10902:
10873:
10860:
10851:
10842:
10825:
10813:
10801:
10789:
10776:
10764:
10743:
10731:
10711:Landau, Lev Davidovich (1975).
10704:
10669:
10630:
10621:
10589:
10577:
10547:
10523:
10499:
10476:
10434:
10418:Dictionary Of Civil Engineering
10409:
10382:
10368:. Thomas Telford. p. 514.
10355:
10324:
10299:
10274:
10214:
10182:
10155:
10130:
10118:
10090:
10009:
9996:
9971:
9955:
9941:. JHU Press. pp. 112–113.
9930:
9887:
9866:
9826:
9795:
9545:from the original on 2018-09-20
9239:. These forces must balance so
8049:
8043:
7849:
7413:
3012:
2931:{\displaystyle \delta =y_{0}-a}
215:and as a cross section of the
30:Curve formed by a hanging chain
12502:National Museum of Mathematics
12254:Regole generali d'architettura
10750:Church, Irving Porter (1890).
10714:The Classical Theory of Fields
10583:
10483:Kunkel, Paul (June 30, 2006).
10338:Methods for Euclidean Geometry
10168:. Laurence King. p. 224.
9774:
9717:
9673:
9618:
9577:
9556:
9521:
9484:
9479:
9285:
9279:
9268:
9253:
7487:The curve was studied 1826 by
6244:
6229:
6044:
6029:
5597:
5571:
5531:
5521:
5509:
5499:
5489:
5451:
5375:
5365:
5335:
5319:
5307:
5291:
5084:and the vertex of the curve).
4882:
4876:
4202:be the vertical distance from
3798:
3774:
3090:
3064:
3038:
2506:
2491:
2332:First, invoke the formula for
2316:
2290:
2284:
2272:
1721:, the tension of the chain at
1641:, and the weight of the chain
878:The equation of a catenary in
788:
779:
768:
765:
456:in response to a challenge by
1:
10470:
10441:Finch, Paul (19 March 1998).
10416:Scott, John S. (1992-10-31).
10331:Byer, Owen; Lazebnik, Felix;
9860:
4176:Relabel if necessary so that
1614:. The forces are the tension
317:would form a catenary shape.
263:minimal surface of revolution
12027:Garden of Cosmic Speculation
11233:Venturoli, Giuseppe (1822).
11128:Maurer, Edward Rose (1914).
10857:Following Todhunter Art. 186
10807:
10770:
10641:(1992). "Roads and Wheels".
10487:. Whistler Alley Mathematics
9695:10.1021/acsphotonics.8b00437
9564:"The Calculus of Variations"
9166:is the unit tangent vector.
7304:is the radius of curvature.
6777:and the radius of curvature
2973:{\textstyle x_{0}=0=\delta }
1765:is parallel to the curve at
7:
10832:
10615:Curves and their Properties
10310:. New Holland. p. 36.
9873:Fahie, John Joseph (1903).
9422:
9415:where the chain has weight
9071:Chain under a general force
7500:electromagnetic metasurface
6597:Differentiation then gives
5814:Consider a chain of length
4286:and let the coordinates of
3880:can be taken to be 0. Then
1733:is tangent to the curve at
1443:
1331:which is the length of the
862:
811:Anchoring of marine objects
707:(black dotted curve) and a
597:is enclosed by a series of
430:Description of Helioscopes,
351:, who wrote in a letter to
253:of the catenary curve, the
10:
12607:
11544:Islamic geometric patterns
11309:Cable Sag Error Calculator
11067:
11054:
11041:
11029:
10960:
10947:
10934:
10921:
10908:
10867:
10836:
10819:
10795:
10783:
10737:
10595:
10124:
10099:Philosophical Transactions
9876:Galileo, His Life and Work
9833:Jefferson, Thomas (1829).
9782:""Catenary" at Math Words"
9495:Shodek, Daniel L. (2004).
9454:– the shape of a spun rope
7060:Catenary of equal strength
5087:It can also be shown that
4896:{\displaystyle \sinh(x)/x}
2980:to simplify the result to
2184:The differential equation
2032:It is convenient to write
1534:and has the property that
1449:Model of chains and arches
1405:
1335:between the curve and the
1004:hyperbolic cosine function
621:) is a flattened catenary.
546:National Historic Landmark
320:
189:that an idealized hanging
22:
15:
12541:
12510:
12464:
12418:
12365:A Mathematician's Apology
12325:
12278:
12161:
12124:
12117:
11949:
11802:
11748:
11739:
11686:
11628:
11484:
11473:
11372:
11285:Catenary curve calculator
11141:Lamb, Sir Horace (1897).
10613:Yates, Robert C. (1952).
10420:. Springer. p. 433.
10281:Sennott, Stephen (2004).
10189:Osserman, Robert (2010),
10002:The original anagram was
9784:. Pballew.net. 1995-11-21
9594:10.1007/978-981-13-4818-1
9529:"Shape of a hanging rope"
9202:are the force of tension
8882:can be taken to be 0. So
4957:) are at the same level (
4903:is strictly monotonic on
4169:In general the parameter
2225:{\displaystyle dy/dx=s/a}
1950:and dividing these gives
800:{\displaystyle y=x^{2}+1}
648:Simple suspension bridges
537:, which is a catenary if
18:Catenary (disambiguation)
12477:The Bridges Organization
11269:University of St Andrews
11130:"Art. 26 Catenary Cable"
10753:Mechanics of Engineering
10690:10.4169/002557010X485120
9802:Barrow, John D. (2010).
9463:
8874:-axes can be shifted so
7493:Gaspard-Gustave Coriolis
7307:The solution to this is
6095:{\displaystyle \lambda }
5423:{\displaystyle T_{0}=wa}
4973:), it can be shown that
3872:-axes can be shifted so
1685:of the curve is zero at
1532:natural parameterization
1343:Relation to other curves
1187:{\displaystyle \varphi }
1084:{\displaystyle \varphi }
858:Mathematical description
675:simple suspension bridge
12339:The Grammar of Ornament
12291:Nature's Harmonic Unity
12201:De prospectiva pingendi
11218:. MAA. pp. 128–9.
11117:"Chapter X: On Strings"
11097:Salmon, George (1879).
11084:Lockwood, E.H. (1961).
10260:, National Park Service
9651:10.1364/JOSAA.34.002243
9588:. Singapore: Springer.
9436:or self-siphoning beads
7594:is a constant equal to
7261:and by differentiation
6848:Suspension bridge curve
6296:separation of variables
5810:Variational formulation
5742:{\displaystyle y_{0}=a}
2641:constant of integration
2545:hyperbolic substitution
2258:{\displaystyle s_{0}=0}
1268:{\displaystyle \kappa }
659:Stressed ribbon bridges
595:Sheffield Winter Garden
12579:Differential equations
12492:Institute For Figuring
12404:The 'Life' of a Carpet
12229:A Treatise on Painting
11215:Learn from the Masters
11136:. J. Wiley & Sons.
10999:10.1126/sciadv.1500396
10485:"Hanging With Galileo"
10395:. Wiley. p. 132.
10112:10.1098/rstl.1695.0114
10067:10.1098/rsta.2014.0346
10016:Truesdell, C. (1960),
9908:10.1098/rsnr.2001.0145
9879:. J. Murray. pp.
9584:Luo, Xiangang (2019).
9396:
9333:and take the limit as
9318:
9142:
9046:
8858:
8670:
8260:
8108:
7985:
7911:
7764:
7715:
7664:
7582:
7479:
7388:. Other relations are
7373:
7292:
7253:
7198:
7048:
6983:
6864:
6839:
6765:
6589:
6487:
6422:
6364:
6286:
6213:
6193:
6120:
6096:
6076:
6000:
5923:
5848:
5828:
5801:
5743:
5710:
5656:
5430:) can be rewritten as
5424:
5394:which (recalling that
5388:
5186:
5132:
5038:
4923:
4922:{\displaystyle x>0}
4897:
4847:
4742:
4554:
4434:
4278:
4198:be the horizontal and
4166:
4150:Determining parameters
4141:
4072:
4001:
3936:
3856:
3811:
3742:
3600:
3467:
3432:Alternative derivation
3423:
3318:
3240:
3128:
3100:
3074:
3048:
3028:For completeness, the
3020:
2974:
2932:
2893:
2827:
2752:
2751:{\textstyle s/a=dy/dx}
2702:
2633:
2606:
2535:
2457:
2323:
2259:
2226:
2128:
2068:
2024:
1942:
1900:
1650:
1577:
1390:Geometrical properties
1325:
1269:
1249:
1188:
1168:
1106:Differentiating gives
1085:
1065:
992:
875:
833:floating wind turbines
824:
807:
801:
736:
682:stressed ribbon bridge
670:
651:
500:Inverted catenary arch
428:in an appendix to his
417:
410:Saint Peter's Basilica
368:It is often said that
366:
336:
197:assumes under its own
74:
58:
42:
12373:George David Birkhoff
12347:Ernest Hanbury Hankin
12215:De divina proportione
12195:Piero della Francesca
12174:Leon Battista Alberti
11761:Piero della Francesca
11400:Hyperboloid structure
11212:Swetz, Frank (1995).
10848:Following Lamb p. 342
10306:Hymers, Paul (2005).
9397:
9319:
9143:
9066:Other generalizations
9047:
8859:
8671:
8261:
8109:
7986:
7912:
7765:
7716:
7665:
7583:
7480:
7374:
7293:
7254:
7199:
7049:
6984:
6855:
6840:
6766:
6590:
6488:
6423:
6365:
6287:
6214:
6194:
6121:
6097:
6077:
6001:
5924:
5849:
5829:
5802:
5744:
5711:
5657:
5425:
5389:
5187:
5133:
5039:
4924:
4898:
4848:
4743:
4555:
4435:
4279:
4228:-axis and its height
4157:
4142:
4073:
4002:
3937:
3857:
3812:
3743:
3601:
3468:
3424:
3319:
3241:
3129:
3101:
3075:
3049:
3021:
2975:
2933:
2894:
2828:
2753:
2703:
2634:
2632:{\displaystyle x_{0}}
2607:
2536:
2458:
2324:
2260:
2227:
2129:
2069:
2025:
1943:
1901:
1655:differential equation
1601:
1578:
1434:surface of revolution
1326:
1270:
1250:
1189:
1169:
1086:
1066:
993:
880:Cartesian coordinates
870:
818:
802:
737:
702:
657:
646:
407:
357:
332:'s catenary model at
328:
251:surface of revolution
64:
48:
36:
12298:Frederik Macody Lund
12169:Filippo Brunelleschi
12050:Hamid Naderi Yeganeh
11912:La condition humaine
11255:Robertson, Edmund F.
10678:Mathematics Magazine
10643:Mathematics Magazine
10535:Human Power eJournal
10341:. MAA. p. 210.
10333:Smeltzer, Deirdre L.
9937:Denny, Mark (2010).
9751:10.1364/OE.26.019555
9347:
9245:
9154:is the magnitude of
9115:
8888:
8684:
8270:
8120:
7995:
7948:
7778:
7725:
7679:
7614:
7536:
7394:
7313:
7267:
7212:
7100:
6997:
6917:
6787:
6603:
6497:
6432:
6390:
6324:
6223:
6203:
6133:
6110:
6086:
6016:
5937:
5858:
5838:
5818:
5753:
5720:
5670:
5434:
5398:
5288:
5142:
5091:
4977:
4907:
4867:
4752:
4578:
4466:
4346:
4238:
4082:
4013:
3946:
3886:
3821:
3756:
3610:
3481:
3442:
3332:
3250:
3138:
3110:
3084:
3058:
3032:
2984:
2945:
2903:
2840:
2762:
2714:
2647:
2616:
2551:
2471:
2340:
2269:
2236:
2188:
2088:
2038:
1956:
1910:
1868:
1540:
1355:curve traced by its
1290:
1259:
1201:
1178:
1110:
1075:
1033:
1029:for the catenary is
888:
746:
715:
304:steel catenary riser
51:overhead power lines
12523:Mathematical beauty
12448:Rhythm of Structure
12391:Gödel, Escher, Bach
12187:De re aedificatoria
11818:The Ancient of Days
11437:Projective geometry
11366:Mathematics and art
11329:Bulletin of the AMS
11294:The Geometry Center
11253:O'Connor, John J.;
11149:. University Press.
11134:Technical Mechanics
11123:. University Press.
11100:Higher Plane Curves
10991:2015SciA....1E0396P
10617:. NCTM. p. 13.
10448:Architects' Journal
10392:Industrial Furnaces
10059:2015RSPTA.37340346C
9742:2018OExpr..2619555P
9736:(15): 19555–19562.
9643:2017JOSAA..34.2243B
7890:
7872:
7530:applied has length
7522:. Specifically, if
6570:
6468:
6345:
6104:Lagrange multiplier
5954:
5881:
5361:
4859:and must be solved
3750:Integrating gives,
3127:{\displaystyle x/a}
1697:is to the right of
1593:unit tangent vector
1359:is a catenary. The
1284:radius of curvature
1018:is equivalent to a
663:Leonel Viera Bridge
619:St. Louis, Missouri
520:St. Louis, Missouri
391:St Paul's Cathedral
208:, which it is not.
12528:Patterns in nature
12385:Douglas Hofstadter
12011:Desmond Paul Henry
12001:Bathsheba Grossman
11933:The Swallow's Tail
11854:Giorgio de Chirico
11726:Sydney Opera House
11581:Croatian interlace
11188:Weisstein, Eric W.
11179:Analytical Statics
11113:Routh, Edward John
10555:Weisstein, Eric W.
10127:Art. 455, footnote
10053:(2039): 20140346,
9392:
9314:
9138:
9042:
9040:
8854:
8852:
8666:
8664:
8256:
8254:
8104:
7981:
7907:
7876:
7858:
7760:
7711:
7660:
7578:
7475:
7369:
7288:
7249:
7194:
7192:
7044:
6979:
6865:
6857:Golden Gate Bridge
6835:
6761:
6742:
6707:
6585:
6552:
6483:
6450:
6418:
6360:
6327:
6282:
6209:
6189:
6116:
6092:
6072:
5996:
5940:
5919:
5867:
5844:
5824:
5797:
5739:
5706:
5652:
5650:
5420:
5384:
5347:
5182:
5128:
5034:
4919:
4893:
4843:
4738:
4550:
4430:
4274:
4185:is to the left of
4167:
4137:
4068:
3997:
3932:
3852:
3807:
3738:
3596:
3463:
3419:
3314:
3236:
3124:
3096:
3070:
3044:
3016:
2970:
2928:
2889:
2823:
2748:
2698:
2629:
2602:
2531:
2465:separate variables
2453:
2319:
2255:
2222:
2124:
2064:
2020:
1938:
1896:
1651:
1573:
1472:static equilibrium
1455:mathematical model
1321:
1265:
1245:
1184:
1164:
1081:
1061:
988:
876:
825:
808:
797:
732:
671:
667:Maldonado, Uruguay
652:
446:Christiaan Huygens
418:
337:
75:
69:forming multiple
59:
43:
12589:Analytic geometry
12574:Roulettes (curve)
12561:
12560:
12414:
12413:
12378:Aesthetic Measure
12249:Sebastiano Serlio
12223:Leonardo da Vinci
12113:
12112:
12105:Margaret Wertheim
11766:Leonardo da Vinci
11225:978-0-88385-703-8
10756:. Wiley. p.
10724:978-0-7506-2768-9
10511:Petersmith.net.nz
10427:978-0-412-98421-1
10402:978-0-471-38706-0
10375:978-0-7277-3215-6
10348:978-0-88385-763-2
10317:978-1-84330-910-9
10292:978-1-57958-433-7
10228:Popular Mechanics
10175:978-1-85669-354-7
10148:978-0-8122-3514-2
9948:978-0-8018-9437-4
9819:978-0-393-33867-6
9637:(12): 2243–2249.
9603:978-981-13-4818-1
9506:978-0-13-048879-4
9458:Weighted catenary
9440:Overhead catenary
9419:per unit length.
9370:
9026:
8998:
8950:
8926:
8828:
8800:
8746:
8722:
8656:
8626:
8625:
8581:
8568:
8553:
8523:
8503:
8474:
8450:
8430:
8429:
8385:
8372:
8339:
8319:
8295:
8246:
8205:
8181:
8145:
8098:
8071:
8047:
8041:
8016:
7979:
7901:
7843:
7799:
7654:
7635:
7564:
7460:
7354:
7171:
7026:
6970:
6938:
6869:suspension bridge
6861:suspension bridge
6829:
6755:
6754:
6741:
6706:
6655:
6635:
6550:
6518:
6307:Nonuniform chains
6271:
6270:
6212:{\displaystyle C}
6181:
6128:Beltrami identity
6119:{\displaystyle x}
6070:
5978:
5911:
5847:{\displaystyle D}
5827:{\displaystyle L}
5791:
5704:
5642:
5272:Tension relations
5179:
5125:
5031:
4997:
4833:
4808:
4790:
4763:
4728:
4665:
4540:
4503:
4420:
4383:
4268:
4127:
4058:
3989:
3925:
3674:
3654:
3631:
3545:
3525:
3502:
3413:
3383:
3371:
3303:
3230:
3201:
3173:
3010:
2881:
2817:
2783:
2692:
2658:
2581:
2516:
2515:
2447:
2435:
2409:
2397:
2361:
2122:
2109:
2062:
2014:
1977:
1563:
1240:
1159:
1131:
1056:
976:
954:
935:
918:
774:
727:
689:suspension bridge
550:weighted catenary
442:Gottfried Leibniz
261:, specifically a
247:hyperbolic cosine
12596:
12549:
12548:
12399:Nikos Salingaros
12122:
12121:
12090:Hiroshi Sugimoto
12040:Robert Longhurst
11986:Helaman Ferguson
11941:Crockett Johnson
11870:Circle Limit III
11839:Danseuse au café
11746:
11745:
11716:Pyramid of Khufu
11479:
11359:
11352:
11345:
11336:
11335:
11271:
11242:
11229:
11201:
11200:
11182:
11171:Whewell, William
11166:
11155:Todhunter, Isaac
11150:
11137:
11124:
11108:
11093:
11090:A Book of Curves
11071:
11064:
11058:
11051:
11045:
11039:
11033:
11027:
11021:
11020:
11010:
10979:Science Advances
10970:
10964:
10957:
10951:
10944:
10938:
10931:
10925:
10918:
10912:
10906:
10900:
10899:
10877:
10871:
10864:
10858:
10855:
10849:
10846:
10840:
10829:
10823:
10817:
10811:
10805:
10799:
10793:
10787:
10780:
10774:
10768:
10762:
10761:
10747:
10741:
10740:Art. 442, p. 316
10735:
10729:
10728:
10708:
10702:
10701:
10673:
10667:
10666:
10634:
10628:
10625:
10619:
10618:
10610:
10599:
10593:
10587:
10581:
10575:
10574:
10569:
10568:
10551:
10545:
10544:
10542:
10541:
10527:
10521:
10520:
10518:
10517:
10503:
10497:
10496:
10494:
10492:
10480:
10474:
10468:
10457:
10456:
10438:
10432:
10431:
10413:
10407:
10406:
10386:
10380:
10379:
10359:
10353:
10352:
10328:
10322:
10321:
10303:
10297:
10296:
10278:
10272:
10271:
10261:
10251:
10245:
10244:
10218:
10212:
10211:
10186:
10180:
10179:
10159:
10153:
10152:
10134:
10128:
10122:
10116:
10115:
10114:
10105:(231): 637–652,
10094:
10088:
10087:
10078:
10042:
10033:
10032:
10013:
10007:
10000:
9994:
9993:
9991:
9990:
9975:
9969:
9964:the anagram for
9959:
9953:
9952:
9934:
9928:
9927:
9891:
9885:
9884:
9870:
9864:
9858:
9845:
9844:
9830:
9824:
9823:
9799:
9793:
9792:
9790:
9789:
9778:
9772:
9771:
9753:
9721:
9715:
9714:
9689:(8): 3198–3204.
9677:
9671:
9670:
9622:
9616:
9615:
9581:
9575:
9574:
9572:
9571:
9560:
9554:
9553:
9551:
9550:
9544:
9533:
9525:
9519:
9518:
9488:
9482:
9477:
9446:Roulette (curve)
9418:
9414:
9401:
9399:
9398:
9393:
9387:
9379:
9371:
9369:
9361:
9360:
9351:
9340:
9332:
9323:
9321:
9320:
9315:
9309:
9295:
9278:
9252:
9238:
9228:
9216:
9201:
9191:
9187:
9183:
9165:
9159:
9153:
9147:
9145:
9144:
9139:
9133:
9122:
9108:
9102:
9098:
9080:
9061:
9051:
9049:
9048:
9043:
9041:
9037:
9036:
9027:
9025:
9014:
9013:
9004:
8999:
8997:
8996:
8984:
8983:
8974:
8951:
8946:
8945:
8936:
8931:
8927:
8919:
8881:
8877:
8873:
8869:
8863:
8861:
8860:
8855:
8853:
8839:
8838:
8829:
8827:
8816:
8815:
8806:
8801:
8799:
8798:
8786:
8785:
8776:
8747:
8742:
8741:
8732:
8727:
8723:
8715:
8675:
8673:
8672:
8667:
8665:
8657:
8655:
8647:
8643:
8642:
8632:
8627:
8624:
8623:
8611:
8610:
8601:
8597:
8589:
8587:
8583:
8582:
8574:
8569:
8561:
8554:
8549:
8545:
8544:
8534:
8526:
8524:
8522:
8514:
8506:
8504:
8499:
8495:
8494:
8484:
8475:
8473:
8465:
8457:
8451:
8446:
8445:
8436:
8431:
8428:
8427:
8415:
8414:
8405:
8401:
8393:
8391:
8387:
8386:
8378:
8373:
8365:
8358:
8357:
8342:
8340:
8338:
8330:
8322:
8320:
8315:
8314:
8305:
8296:
8294:
8286:
8278:
8265:
8263:
8262:
8257:
8255:
8247:
8242:
8238:
8237:
8227:
8206:
8204:
8196:
8188:
8182:
8177:
8176:
8167:
8146:
8144:
8136:
8128:
8113:
8111:
8110:
8105:
8099:
8097:
8096:
8084:
8083:
8074:
8072:
8067:
8066:
8057:
8048:
8045:
8042:
8034:
8017:
8015:
8007:
7999:
7990:
7988:
7987:
7982:
7980:
7978:
7977:
7968:
7967:
7958:
7943:
7934:
7928:
7922:
7916:
7914:
7913:
7908:
7902:
7900:
7899:
7889:
7884:
7871:
7866:
7857:
7844:
7842:
7841:
7832:
7828:
7827:
7817:
7800:
7798:
7790:
7782:
7769:
7767:
7766:
7761:
7752:
7751:
7720:
7718:
7717:
7712:
7706:
7705:
7669:
7667:
7666:
7661:
7655:
7647:
7636:
7634:
7626:
7618:
7609:
7601:
7597:
7593:
7587:
7585:
7584:
7579:
7570:
7566:
7565:
7557:
7529:
7525:
7506:Elastic catenary
7484:
7482:
7481:
7476:
7470:
7466:
7465:
7461:
7456:
7442:
7387:
7378:
7376:
7375:
7370:
7364:
7360:
7359:
7355:
7347:
7303:
7297:
7295:
7294:
7289:
7258:
7256:
7255:
7250:
7206:Combining gives
7203:
7201:
7200:
7195:
7193:
7172:
7164:
7135:
7134:
7090:
7086:
7085:
7083:
7082:
7079:
7076:
7068:
7053:
7051:
7050:
7045:
7037:
7036:
7027:
7025:
7024:
7023:
7007:
6988:
6986:
6985:
6980:
6971:
6969:
6968:
6956:
6939:
6937:
6929:
6921:
6910:
6904:
6898:
6894:
6890:
6884:
6874:
6844:
6842:
6841:
6836:
6830:
6828:
6821:
6820:
6807:
6806:
6797:
6780:
6776:
6770:
6768:
6767:
6762:
6756:
6753:
6752:
6747:
6743:
6740:
6732:
6724:
6710:
6709:
6708:
6705:
6704:
6703:
6690:
6686:
6685:
6675:
6672:
6671:
6661:
6656:
6654:
6646:
6638:
6636:
6634:
6623:
6621:
6620:
6594:
6592:
6591:
6586:
6569:
6568:
6562:
6561:
6551:
6549:
6548:
6536:
6519:
6517:
6509:
6501:
6492:
6490:
6489:
6484:
6467:
6466:
6460:
6459:
6427:
6425:
6424:
6419:
6417:
6416:
6383:
6377:
6369:
6367:
6366:
6361:
6344:
6343:
6337:
6336:
6317:
6291:
6289:
6288:
6283:
6272:
6269:
6268:
6267:
6248:
6247:
6227:
6218:
6216:
6215:
6210:
6198:
6196:
6195:
6190:
6182:
6180:
6179:
6167:
6166:
6165:
6155:
6153:
6142:
6141:
6125:
6123:
6122:
6117:
6101:
6099:
6098:
6093:
6081:
6079:
6078:
6073:
6071:
6069:
6068:
6067:
6048:
6025:
6024:
6005:
6003:
6002:
5997:
5979:
5977:
5976:
5975:
5956:
5953:
5948:
5932:
5928:
5926:
5925:
5920:
5912:
5910:
5909:
5908:
5889:
5880:
5875:
5853:
5851:
5850:
5845:
5833:
5831:
5830:
5825:
5806:
5804:
5803:
5798:
5792:
5784:
5748:
5746:
5745:
5740:
5732:
5731:
5715:
5713:
5712:
5707:
5705:
5703:
5702:
5690:
5689:
5680:
5661:
5659:
5658:
5653:
5651:
5643:
5641:
5640:
5628:
5627:
5618:
5596:
5595:
5583:
5582:
5570:
5569:
5553:
5552:
5539:
5538:
5517:
5516:
5482:
5481:
5463:
5462:
5450:
5449:
5429:
5427:
5426:
5421:
5410:
5409:
5393:
5391:
5390:
5385:
5383:
5382:
5360:
5355:
5343:
5342:
5315:
5314:
5283:
5279:
5267:
5263:
5250:
5241:
5229:
5209:
5200:
5191:
5189:
5188:
5183:
5180:
5175:
5164:
5137:
5135:
5134:
5129:
5126:
5124:
5113:
5083:
5074:
5065:
5061:
5052:
5043:
5041:
5040:
5035:
5032:
5030:
5022:
5021:
5020:
5008:
5007:
4998:
4990:
4987:
4972:
4956:
4947:
4935:
4928:
4926:
4925:
4920:
4902:
4900:
4899:
4894:
4889:
4858:
4852:
4850:
4849:
4844:
4838:
4834:
4832:
4821:
4809:
4804:
4796:
4791:
4789:
4788:
4776:
4775:
4766:
4764:
4756:
4747:
4745:
4744:
4739:
4733:
4729:
4727:
4716:
4707:
4706:
4697:
4696:
4681:
4677:
4670:
4666:
4661:
4660:
4659:
4647:
4646:
4636:
4619:
4618:
4603:
4602:
4590:
4589:
4571:
4559:
4557:
4556:
4551:
4545:
4541:
4536:
4535:
4526:
4508:
4504:
4499:
4498:
4489:
4459:
4450:
4439:
4437:
4436:
4431:
4425:
4421:
4416:
4415:
4406:
4388:
4384:
4379:
4378:
4369:
4339:
4321:
4303:
4294:
4283:
4281:
4280:
4275:
4273:
4269:
4261:
4231:
4227:
4219:
4210:
4201:
4197:
4193:
4184:
4172:
4164:
4146:
4144:
4143:
4138:
4132:
4128:
4120:
4077:
4075:
4074:
4069:
4063:
4059:
4051:
4006:
4004:
4003:
3998:
3992:
3991:
3990:
3982:
3941:
3939:
3938:
3933:
3927:
3926:
3918:
3879:
3875:
3871:
3867:
3861:
3859:
3858:
3853:
3816:
3814:
3813:
3808:
3747:
3745:
3744:
3739:
3703:
3702:
3675:
3673:
3665:
3657:
3655:
3653:
3645:
3637:
3632:
3630:
3622:
3614:
3605:
3603:
3602:
3597:
3574:
3573:
3546:
3544:
3536:
3528:
3526:
3524:
3516:
3508:
3503:
3501:
3493:
3485:
3475:it follows that
3472:
3470:
3469:
3464:
3428:
3426:
3425:
3420:
3414:
3412:
3411:
3399:
3398:
3389:
3384:
3382:
3381:
3376:
3372:
3364:
3351:
3326:can be rewritten
3323:
3321:
3320:
3315:
3309:
3305:
3304:
3296:
3291:
3290:
3245:
3243:
3242:
3237:
3231:
3223:
3218:
3217:
3202:
3197:
3196:
3195:
3179:
3174:
3169:
3158:
3153:
3152:
3133:
3131:
3130:
3125:
3120:
3105:
3103:
3102:
3097:
3079:
3077:
3076:
3071:
3053:
3051:
3050:
3045:
3025:
3023:
3022:
3017:
3011:
3003:
2979:
2977:
2976:
2971:
2957:
2956:
2937:
2935:
2934:
2929:
2921:
2920:
2898:
2896:
2895:
2890:
2882:
2877:
2876:
2875:
2859:
2832:
2830:
2829:
2824:
2818:
2813:
2812:
2811:
2795:
2784:
2782:
2774:
2766:
2757:
2755:
2754:
2749:
2741:
2724:
2707:
2705:
2704:
2699:
2693:
2688:
2687:
2686:
2670:
2659:
2651:
2638:
2636:
2635:
2630:
2628:
2627:
2611:
2609:
2608:
2603:
2595:
2594:
2582:
2574:
2569:
2568:
2540:
2538:
2537:
2532:
2517:
2514:
2513:
2501:
2484:
2483:
2475:
2462:
2460:
2459:
2454:
2448:
2446:
2445:
2440:
2436:
2428:
2415:
2410:
2408:
2407:
2402:
2398:
2396:
2388:
2380:
2367:
2362:
2360:
2352:
2344:
2328:
2326:
2325:
2320:
2315:
2314:
2302:
2301:
2264:
2262:
2261:
2256:
2248:
2247:
2231:
2229:
2228:
2223:
2218:
2201:
2176:and the vertex.
2175:
2169:
2155:
2133:
2131:
2130:
2125:
2123:
2115:
2110:
2108:
2100:
2092:
2081:
2073:
2071:
2070:
2065:
2063:
2058:
2057:
2048:
2029:
2027:
2026:
2021:
2015:
2013:
2012:
2003:
1995:
1978:
1976:
1968:
1960:
1947:
1945:
1944:
1939:
1905:
1903:
1902:
1897:
1895:
1894:
1861:
1852:
1846:
1840:
1836:
1832:
1823:tangential angle
1820:
1816:
1810:
1806:
1802:
1776:
1770:
1764:
1758:
1749:
1738:
1732:
1726:
1720:
1714:
1708:
1702:
1696:
1690:
1684:
1683:
1681:
1680:
1675:
1672:
1662:
1648:
1640:
1634:
1628:
1622:
1613:
1607:
1590:
1582:
1580:
1579:
1574:
1572:
1564:
1562:
1554:
1553:
1544:
1525:
1515:
1511:
1439:
1428:
1401:
1397:
1338:
1330:
1328:
1327:
1322:
1311:
1310:
1274:
1272:
1271:
1266:
1254:
1252:
1251:
1246:
1241:
1239:
1238:
1237:
1225:
1224:
1211:
1193:
1191:
1190:
1185:
1174:and eliminating
1173:
1171:
1170:
1165:
1160:
1155:
1148:
1147:
1137:
1132:
1130:
1122:
1114:
1098:
1093:tangential angle
1090:
1088:
1087:
1082:
1070:
1068:
1067:
1062:
1057:
1049:
1027:Whewell equation
1017:
1009:
1001:
997:
995:
994:
989:
984:
980:
979:
978:
977:
969:
956:
955:
947:
936:
928:
923:
919:
911:
874:
806:
804:
803:
798:
775:
772:
764:
763:
741:
739:
738:
733:
728:
725:
703:Comparison of a
692:the result is a
639:Catenary bridges
630:
610:
590:
564:
543:
536:
479:
450:Johann Bernoulli
375:Two New Sciences
349:Thomas Jefferson
315:evanescent waves
293:Johann Bernoulli
272:Two New Sciences
181:
175:
174:
171:
170:
167:
164:
161:
158:
155:
152:
149:
142:
131:
126:
125:
122:
121:
118:
115:
112:
109:
106:
103:
96:
12606:
12605:
12599:
12598:
12597:
12595:
12594:
12593:
12564:
12563:
12562:
12557:
12537:
12533:Sacred geometry
12506:
12472:Ars Mathematica
12460:
12410:
12321:
12274:
12261:Andrea Palladio
12157:
12150:De architectura
12109:
12065:Antoine Pevsner
12045:Jeanette McLeod
11996:Susan Goldstine
11945:
11804:
11798:
11735:
11721:Sagrada Família
11682:
11624:
11492:Algorithmic art
11480:
11471:
11467:Wallpaper group
11405:Minimal surface
11368:
11363:
11249:
11226:
11208:
11206:Further reading
11080:
11075:
11074:
11065:
11061:
11052:
11048:
11040:
11036:
11028:
11024:
10985:(9): e1500396.
10971:
10967:
10958:
10954:
10945:
10941:
10932:
10928:
10919:
10915:
10911:Art 443, p. 318
10907:
10903:
10888:
10885:Wayback Machine
10878:
10874:
10865:
10861:
10856:
10852:
10847:
10843:
10830:
10826:
10822:Art 443, p. 318
10818:
10814:
10806:
10802:
10798:Art. 443 p. 317
10794:
10790:
10786:Art. 443 p. 316
10781:
10777:
10769:
10765:
10748:
10744:
10736:
10732:
10725:
10709:
10705:
10674:
10670:
10655:10.2307/2691240
10635:
10631:
10626:
10622:
10611:
10602:
10594:
10590:
10582:
10578:
10566:
10564:
10552:
10548:
10539:
10537:
10529:
10528:
10524:
10515:
10513:
10505:
10504:
10500:
10490:
10488:
10481:
10477:
10469:
10460:
10439:
10435:
10428:
10414:
10410:
10403:
10387:
10383:
10376:
10360:
10356:
10349:
10329:
10325:
10318:
10304:
10300:
10293:
10279:
10275:
10269:
10252:
10248:
10219:
10215:
10187:
10183:
10176:
10160:
10156:
10149:
10135:
10131:
10123:
10119:
10095:
10091:
10043:
10036:
10030:
10014:
10010:
10001:
9997:
9988:
9986:
9977:
9976:
9972:
9960:
9956:
9949:
9935:
9931:
9892:
9888:
9871:
9867:
9859:
9848:
9831:
9827:
9820:
9800:
9796:
9787:
9785:
9780:
9779:
9775:
9722:
9718:
9678:
9674:
9623:
9619:
9604:
9586:Catenary optics
9582:
9578:
9569:
9567:
9562:
9561:
9557:
9548:
9546:
9542:
9531:
9527:
9526:
9522:
9507:
9489:
9485:
9478:
9471:
9466:
9425:
9416:
9405:
9383:
9375:
9362:
9356:
9352:
9350:
9348:
9345:
9344:
9334:
9327:
9305:
9291:
9274:
9248:
9246:
9243:
9242:
9230:
9218:
9203:
9193:
9189:
9185:
9170:
9161:
9155:
9151:
9129:
9118:
9116:
9113:
9112:
9104:
9100:
9085:
9076:
9073:
9068:
9059:
9039:
9038:
9032:
9028:
9015:
9009:
9005:
9003:
8992:
8988:
8979:
8975:
8973:
8966:
8960:
8959:
8941:
8937:
8935:
8918:
8914:
8898:
8891:
8889:
8886:
8885:
8879:
8875:
8871:
8867:
8851:
8850:
8834:
8830:
8817:
8811:
8807:
8805:
8794:
8790:
8781:
8777:
8775:
8768:
8762:
8761:
8737:
8733:
8731:
8714:
8710:
8694:
8687:
8685:
8682:
8681:
8663:
8662:
8648:
8638:
8634:
8633:
8631:
8619:
8615:
8606:
8602:
8596:
8588:
8573:
8560:
8559:
8555:
8540:
8536:
8535:
8533:
8525:
8515:
8507:
8505:
8490:
8486:
8485:
8483:
8476:
8466:
8458:
8456:
8453:
8452:
8441:
8437:
8435:
8423:
8419:
8410:
8406:
8400:
8392:
8377:
8364:
8363:
8359:
8353:
8349:
8341:
8331:
8323:
8321:
8310:
8306:
8304:
8297:
8287:
8279:
8277:
8273:
8271:
8268:
8267:
8253:
8252:
8233:
8229:
8228:
8226:
8207:
8197:
8189:
8187:
8184:
8183:
8172:
8168:
8166:
8147:
8137:
8129:
8127:
8123:
8121:
8118:
8117:
8092:
8088:
8079:
8075:
8073:
8062:
8058:
8056:
8044:
8033:
8008:
8000:
7998:
7996:
7993:
7992:
7973:
7969:
7963:
7959:
7957:
7949:
7946:
7945:
7942:
7936:
7930:
7924:
7920:
7895:
7891:
7885:
7880:
7867:
7862:
7856:
7837:
7833:
7823:
7819:
7818:
7816:
7791:
7783:
7781:
7779:
7776:
7775:
7747:
7743:
7726:
7723:
7722:
7701:
7697:
7680:
7677:
7676:
7646:
7627:
7619:
7617:
7615:
7612:
7611:
7607:
7599:
7595:
7591:
7556:
7549:
7545:
7537:
7534:
7533:
7527:
7523:
7508:
7443:
7441:
7437:
7430:
7426:
7395:
7392:
7391:
7382:
7346:
7342:
7335:
7331:
7314:
7311:
7310:
7301:
7268:
7265:
7264:
7213:
7210:
7209:
7191:
7190:
7163:
7156:
7141:
7140:
7130:
7126:
7119:
7103:
7101:
7098:
7097:
7088:
7080:
7077:
7074:
7073:
7071:
7070:
7066:
7062:
7032:
7028:
7019:
7015:
7011:
7006:
6998:
6995:
6994:
6964:
6960:
6955:
6930:
6922:
6920:
6918:
6915:
6914:
6906:
6900:
6896:
6892:
6886:
6880:
6872:
6850:
6816:
6812:
6808:
6802:
6798:
6796:
6788:
6785:
6784:
6778:
6774:
6748:
6733:
6725:
6722:
6718:
6717:
6699:
6695:
6691:
6681:
6677:
6676:
6673:
6667:
6663:
6662:
6660:
6647:
6639:
6637:
6627:
6622:
6616:
6612:
6604:
6601:
6600:
6564:
6563:
6557:
6556:
6544:
6540:
6535:
6510:
6502:
6500:
6498:
6495:
6494:
6462:
6461:
6455:
6454:
6433:
6430:
6429:
6412:
6408:
6391:
6388:
6387:
6379:
6373:
6339:
6338:
6332:
6331:
6325:
6322:
6321:
6315:
6309:
6304:
6263:
6259:
6255:
6228:
6226:
6224:
6221:
6220:
6204:
6201:
6200:
6172:
6168:
6161:
6160:
6156:
6154:
6146:
6137:
6136:
6134:
6131:
6130:
6111:
6108:
6107:
6087:
6084:
6083:
6063:
6059:
6055:
6047:
6020:
6019:
6017:
6014:
6013:
5971:
5967:
5963:
5955:
5949:
5944:
5938:
5935:
5934:
5930:
5904:
5900:
5896:
5888:
5876:
5871:
5859:
5856:
5855:
5839:
5836:
5835:
5819:
5816:
5815:
5812:
5783:
5754:
5751:
5750:
5727:
5723:
5721:
5718:
5717:
5716:(assuming that
5698:
5694:
5685:
5681:
5679:
5671:
5668:
5667:
5649:
5648:
5636:
5632:
5623:
5619:
5617:
5607:
5601:
5600:
5591:
5587:
5578:
5574:
5565:
5561:
5554:
5548:
5544:
5541:
5540:
5534:
5530:
5512:
5508:
5492:
5477:
5473:
5458:
5454:
5445:
5441:
5437:
5435:
5432:
5431:
5405:
5401:
5399:
5396:
5395:
5378:
5374:
5356:
5351:
5338:
5334:
5310:
5306:
5289:
5286:
5285:
5281:
5277:
5274:
5265:
5257:
5252:
5249:
5243:
5240:
5234:
5228:
5221:
5211:
5208:
5202:
5199:
5193:
5165:
5163:
5143:
5140:
5139:
5117:
5112:
5092:
5089:
5088:
5082:
5076:
5073:
5067:
5063:
5060:
5054:
5051:
5045:
5023:
5016:
5012:
5003:
4999:
4989:
4988:
4986:
4978:
4975:
4974:
4971:
4964:
4958:
4955:
4949:
4946:
4940:
4930:
4908:
4905:
4904:
4885:
4868:
4865:
4864:
4856:
4825:
4820:
4816:
4797:
4795:
4784:
4780:
4771:
4767:
4765:
4755:
4753:
4750:
4749:
4720:
4715:
4711:
4702:
4698:
4692:
4688:
4655:
4651:
4642:
4638:
4637:
4635:
4631:
4624:
4620:
4614:
4610:
4598:
4594:
4585:
4581:
4579:
4576:
4575:
4563:
4531:
4527:
4525:
4521:
4494:
4490:
4488:
4484:
4467:
4464:
4463:
4458:
4452:
4449:
4443:
4411:
4407:
4405:
4401:
4374:
4370:
4368:
4364:
4347:
4344:
4343:
4337:
4330:
4323:
4319:
4312:
4305:
4302:
4296:
4293:
4287:
4260:
4256:
4239:
4236:
4235:
4229:
4225:
4218:
4212:
4209:
4203:
4199:
4195:
4192:
4186:
4183:
4177:
4170:
4163:
4159:
4152:
4119:
4115:
4083:
4080:
4079:
4050:
4046:
4014:
4011:
4010:
3981:
3977:
3973:
3947:
3944:
3943:
3917:
3913:
3887:
3884:
3883:
3877:
3873:
3869:
3865:
3864:As before, the
3822:
3819:
3818:
3757:
3754:
3753:
3698:
3694:
3666:
3658:
3656:
3646:
3638:
3636:
3623:
3615:
3613:
3611:
3608:
3607:
3569:
3565:
3537:
3529:
3527:
3517:
3509:
3507:
3494:
3486:
3484:
3482:
3479:
3478:
3443:
3440:
3439:
3434:
3407:
3403:
3394:
3390:
3388:
3377:
3363:
3359:
3358:
3350:
3333:
3330:
3329:
3295:
3283:
3279:
3278:
3274:
3251:
3248:
3247:
3222:
3210:
3206:
3191:
3187:
3180:
3178:
3159:
3157:
3145:
3141:
3139:
3136:
3135:
3116:
3111:
3108:
3107:
3085:
3082:
3081:
3059:
3056:
3055:
3033:
3030:
3029:
3002:
2985:
2982:
2981:
2952:
2948:
2946:
2943:
2942:
2916:
2912:
2904:
2901:
2900:
2871:
2867:
2860:
2858:
2841:
2838:
2837:
2807:
2803:
2796:
2794:
2775:
2767:
2765:
2763:
2760:
2759:
2737:
2720:
2715:
2712:
2711:
2682:
2678:
2671:
2669:
2650:
2648:
2645:
2644:
2623:
2619:
2617:
2614:
2613:
2590:
2586:
2573:
2561:
2557:
2552:
2549:
2548:
2509:
2505:
2497:
2476:
2474:
2472:
2469:
2468:
2441:
2427:
2423:
2422:
2414:
2403:
2389:
2381:
2379:
2375:
2374:
2366:
2353:
2345:
2343:
2341:
2338:
2337:
2310:
2306:
2297:
2293:
2270:
2267:
2266:
2243:
2239:
2237:
2234:
2233:
2214:
2197:
2189:
2186:
2185:
2182:
2171:
2157:
2154:
2140:
2114:
2101:
2093:
2091:
2089:
2086:
2085:
2077:
2053:
2049:
2047:
2039:
2036:
2035:
2008:
2004:
1996:
1994:
1969:
1961:
1959:
1957:
1954:
1953:
1911:
1908:
1907:
1890:
1886:
1869:
1866:
1865:
1857:
1848:
1842:
1838:
1834:
1826:
1818:
1812:
1808:
1804:
1778:
1772:
1766:
1760:
1757:
1751:
1747:
1740:
1734:
1728:
1722:
1716:
1710:
1704:
1698:
1692:
1686:
1676:
1673:
1668:
1667:
1665:
1664:
1658:
1642:
1636:
1630:
1624:
1621:
1615:
1609:
1603:
1586:
1568:
1555:
1549:
1545:
1543:
1541:
1538:
1537:
1528:position vector
1521:
1513:
1482:
1451:
1446:
1437:
1426:
1408:
1399:
1395:
1392:
1384:regular polygon
1345:
1336:
1306:
1302:
1291:
1288:
1287:
1260:
1257:
1256:
1233:
1229:
1220:
1216:
1215:
1210:
1202:
1199:
1198:
1196:Cesàro equation
1179:
1176:
1175:
1143:
1139:
1138:
1136:
1123:
1115:
1113:
1111:
1108:
1107:
1096:
1076:
1073:
1072:
1048:
1034:
1031:
1030:
1020:uniform scaling
1015:
1007:
999:
968:
964:
960:
946:
942:
941:
937:
927:
910:
906:
889:
886:
885:
872:
865:
860:
813:
771:
759:
755:
747:
744:
743:
724:
716:
713:
712:
641:
634:
631:
622:
611:
602:
599:catenary arches
591:
582:
565:
555:
538:
523:
505:Catenary arches
502:
477:
466:for June 1691.
463:Acta Eruditorum
458:Jakob Bernoulli
414:Giovanni Poleni
380:Joachim Jungius
343:, which means "
323:
308:overhead wiring
267:Galileo Galilei
259:minimal surface
179:
146:
137:
136:
129:
100:
91:
90:
65:The silk on a
49:Freely-hanging
31:
28:
21:
12:
11:
5:
12604:
12603:
12592:
12591:
12586:
12581:
12576:
12559:
12558:
12556:
12555:
12542:
12539:
12538:
12536:
12535:
12530:
12525:
12520:
12514:
12512:
12508:
12507:
12505:
12504:
12499:
12494:
12489:
12484:
12479:
12474:
12468:
12466:
12462:
12461:
12459:
12458:
12451:
12444:
12437:
12430:
12422:
12420:
12416:
12415:
12412:
12411:
12409:
12408:
12407:
12406:
12396:
12395:
12394:
12382:
12381:
12380:
12370:
12369:
12368:
12356:
12355:
12354:
12344:
12343:
12342:
12329:
12327:
12323:
12322:
12320:
12319:
12318:
12317:
12315:The Greek Vase
12307:
12306:
12305:
12295:
12294:
12293:
12282:
12280:
12276:
12275:
12273:
12272:
12271:
12270:
12258:
12257:
12256:
12246:
12245:
12244:
12237:Albrecht Dürer
12234:
12233:
12232:
12220:
12219:
12218:
12206:
12205:
12204:
12192:
12191:
12190:
12183:
12171:
12165:
12163:
12159:
12158:
12156:
12155:
12154:
12153:
12141:
12140:
12139:
12128:
12126:
12119:
12115:
12114:
12111:
12110:
12108:
12107:
12102:
12100:Roman Verostko
12097:
12092:
12087:
12082:
12077:
12075:Alba Rojo Cama
12072:
12067:
12062:
12057:
12052:
12047:
12042:
12037:
12032:
12031:
12030:
12021:Charles Jencks
12018:
12013:
12008:
12006:George W. Hart
12003:
11998:
11993:
11988:
11983:
11978:
11973:
11968:
11959:
11953:
11951:
11947:
11946:
11944:
11943:
11938:
11937:
11936:
11929:
11917:
11916:
11915:
11903:
11902:
11901:
11894:
11887:
11880:
11873:
11861:
11856:
11851:
11850:
11849:
11842:
11833:Jean Metzinger
11830:
11829:
11828:
11821:
11808:
11806:
11800:
11799:
11797:
11796:
11795:
11794:
11782:
11780:Albrecht Dürer
11777:
11776:
11775:
11763:
11758:
11752:
11750:
11743:
11737:
11736:
11734:
11733:
11728:
11723:
11718:
11713:
11708:
11703:
11698:
11692:
11690:
11684:
11683:
11681:
11680:
11673:
11666:
11659:
11652:
11645:
11638:
11632:
11630:
11626:
11625:
11623:
11622:
11617:
11612:
11607:
11606:
11605:
11595:
11590:
11589:
11588:
11583:
11578:
11568:
11567:
11566:
11561:
11556:
11551:
11541:
11536:
11531:
11526:
11521:
11520:
11519:
11514:
11509:
11499:
11497:Anamorphic art
11494:
11488:
11486:
11482:
11481:
11474:
11472:
11470:
11469:
11464:
11459:
11454:
11453:
11452:
11447:
11439:
11434:
11429:
11428:
11427:
11425:Camera obscura
11422:
11412:
11407:
11402:
11397:
11392:
11387:
11382:
11376:
11374:
11370:
11369:
11362:
11361:
11354:
11347:
11339:
11333:
11332:
11324:
11318:
11312:
11306:
11301:
11296:
11287:
11282:
11272:
11248:
11247:External links
11245:
11244:
11243:
11230:
11224:
11207:
11204:
11203:
11202:
11183:
11167:
11151:
11138:
11125:
11109:
11094:
11079:
11076:
11073:
11072:
11059:
11046:
11034:
11022:
10965:
10952:
10939:
10926:
10913:
10901:
10872:
10859:
10850:
10841:
10824:
10812:
10800:
10788:
10775:
10763:
10742:
10730:
10723:
10703:
10668:
10649:(5): 283–301.
10629:
10620:
10600:
10588:
10576:
10546:
10522:
10498:
10475:
10458:
10433:
10426:
10408:
10401:
10381:
10374:
10354:
10347:
10335:(2010-09-02).
10323:
10316:
10298:
10291:
10273:
10246:
10213:
10203:(2): 220–229,
10181:
10174:
10154:
10147:
10129:
10117:
10089:
10034:
10028:
10008:
9995:
9970:
9954:
9947:
9929:
9902:(2): 289–308.
9886:
9865:
9846:
9825:
9818:
9794:
9773:
9730:Optics Express
9716:
9672:
9617:
9602:
9576:
9555:
9538:. 2017-05-02.
9520:
9505:
9483:
9468:
9467:
9465:
9462:
9461:
9460:
9455:
9449:
9443:
9437:
9434:Chain fountain
9431:
9424:
9421:
9391:
9386:
9382:
9378:
9374:
9368:
9365:
9359:
9355:
9313:
9308:
9304:
9301:
9298:
9294:
9290:
9287:
9284:
9281:
9277:
9273:
9270:
9267:
9264:
9261:
9258:
9255:
9251:
9137:
9132:
9128:
9125:
9121:
9072:
9069:
9067:
9064:
9035:
9031:
9024:
9021:
9018:
9012:
9008:
9002:
8995:
8991:
8987:
8982:
8978:
8972:
8969:
8967:
8965:
8962:
8961:
8958:
8954:
8949:
8944:
8940:
8934:
8930:
8925:
8922:
8917:
8913:
8910:
8907:
8904:
8901:
8899:
8897:
8894:
8893:
8849:
8845:
8842:
8837:
8833:
8826:
8823:
8820:
8814:
8810:
8804:
8797:
8793:
8789:
8784:
8780:
8774:
8771:
8769:
8767:
8764:
8763:
8760:
8756:
8753:
8750:
8745:
8740:
8736:
8730:
8726:
8721:
8718:
8713:
8709:
8706:
8703:
8700:
8697:
8695:
8693:
8690:
8689:
8661:
8654:
8651:
8646:
8641:
8637:
8630:
8622:
8618:
8614:
8609:
8605:
8600:
8595:
8592:
8590:
8586:
8580:
8577:
8572:
8567:
8564:
8558:
8552:
8548:
8543:
8539:
8532:
8529:
8527:
8521:
8518:
8513:
8510:
8502:
8498:
8493:
8489:
8482:
8479:
8477:
8472:
8469:
8464:
8461:
8455:
8454:
8449:
8444:
8440:
8434:
8426:
8422:
8418:
8413:
8409:
8404:
8399:
8396:
8394:
8390:
8384:
8381:
8376:
8371:
8368:
8362:
8356:
8352:
8348:
8345:
8343:
8337:
8334:
8329:
8326:
8318:
8313:
8309:
8303:
8300:
8298:
8293:
8290:
8285:
8282:
8276:
8275:
8251:
8245:
8241:
8236:
8232:
8225:
8222:
8219:
8216:
8213:
8210:
8208:
8203:
8200:
8195:
8192:
8186:
8185:
8180:
8175:
8171:
8165:
8162:
8159:
8156:
8153:
8150:
8148:
8143:
8140:
8135:
8132:
8126:
8125:
8103:
8095:
8091:
8087:
8082:
8078:
8070:
8065:
8061:
8055:
8052:
8040:
8037:
8032:
8029:
8026:
8023:
8020:
8014:
8011:
8006:
8003:
7976:
7972:
7966:
7962:
7956:
7953:
7940:
7906:
7898:
7894:
7888:
7883:
7879:
7875:
7870:
7865:
7861:
7855:
7852:
7848:
7840:
7836:
7831:
7826:
7822:
7815:
7812:
7809:
7806:
7803:
7797:
7794:
7789:
7786:
7759:
7755:
7750:
7746:
7742:
7739:
7736:
7733:
7730:
7710:
7704:
7700:
7696:
7693:
7690:
7687:
7684:
7659:
7653:
7650:
7645:
7642:
7639:
7633:
7630:
7625:
7622:
7577:
7573:
7569:
7563:
7560:
7555:
7552:
7548:
7544:
7541:
7507:
7504:
7489:Davies Gilbert
7474:
7469:
7464:
7459:
7455:
7452:
7449:
7446:
7440:
7436:
7433:
7429:
7425:
7422:
7419:
7416:
7412:
7408:
7405:
7402:
7399:
7368:
7363:
7358:
7353:
7350:
7345:
7341:
7338:
7334:
7330:
7327:
7324:
7321:
7318:
7287:
7284:
7281:
7278:
7275:
7272:
7248:
7245:
7241:
7238:
7235:
7232:
7229:
7226:
7223:
7220:
7217:
7189:
7185:
7182:
7178:
7175:
7170:
7167:
7162:
7159:
7157:
7155:
7152:
7149:
7146:
7143:
7142:
7139:
7133:
7129:
7125:
7122:
7120:
7118:
7115:
7112:
7109:
7106:
7105:
7061:
7058:
7043:
7040:
7035:
7031:
7022:
7018:
7014:
7010:
7005:
7002:
6978:
6974:
6967:
6963:
6959:
6954:
6951:
6948:
6945:
6942:
6936:
6933:
6928:
6925:
6849:
6846:
6834:
6827:
6824:
6819:
6815:
6811:
6805:
6801:
6795:
6792:
6760:
6751:
6746:
6739:
6736:
6731:
6728:
6721:
6716:
6713:
6702:
6698:
6694:
6689:
6684:
6680:
6670:
6666:
6659:
6653:
6650:
6645:
6642:
6633:
6630:
6626:
6619:
6615:
6611:
6608:
6584:
6580:
6577:
6573:
6567:
6560:
6555:
6547:
6543:
6539:
6534:
6531:
6528:
6525:
6522:
6516:
6513:
6508:
6505:
6493:and therefore
6482:
6478:
6475:
6471:
6465:
6458:
6453:
6449:
6446:
6443:
6440:
6437:
6415:
6411:
6407:
6404:
6401:
6398:
6395:
6359:
6355:
6352:
6348:
6342:
6335:
6330:
6308:
6305:
6303:
6300:
6281:
6278:
6275:
6266:
6262:
6258:
6254:
6251:
6246:
6243:
6240:
6237:
6234:
6231:
6208:
6188:
6185:
6178:
6175:
6171:
6164:
6159:
6152:
6149:
6145:
6140:
6115:
6091:
6066:
6062:
6058:
6054:
6051:
6046:
6043:
6040:
6037:
6034:
6031:
6028:
6023:
6012:is therefore
5995:
5991:
5988:
5985:
5982:
5974:
5970:
5966:
5962:
5959:
5952:
5947:
5943:
5918:
5915:
5907:
5903:
5899:
5895:
5892:
5887:
5884:
5879:
5874:
5870:
5866:
5863:
5843:
5823:
5811:
5808:
5796:
5790:
5787:
5782:
5779:
5776:
5773:
5770:
5767:
5764:
5761:
5758:
5738:
5735:
5730:
5726:
5701:
5697:
5693:
5688:
5684:
5678:
5675:
5664:as shown above
5647:
5639:
5635:
5631:
5626:
5622:
5616:
5613:
5610:
5608:
5606:
5603:
5602:
5599:
5594:
5590:
5586:
5581:
5577:
5573:
5568:
5564:
5560:
5557:
5555:
5551:
5547:
5543:
5542:
5537:
5533:
5529:
5526:
5523:
5520:
5515:
5511:
5507:
5504:
5501:
5498:
5495:
5493:
5491:
5488:
5485:
5480:
5476:
5472:
5469:
5466:
5461:
5457:
5453:
5448:
5444:
5440:
5439:
5419:
5416:
5413:
5408:
5404:
5381:
5377:
5373:
5370:
5367:
5364:
5359:
5354:
5350:
5346:
5341:
5337:
5333:
5330:
5327:
5324:
5321:
5318:
5313:
5309:
5305:
5302:
5299:
5296:
5293:
5273:
5270:
5255:
5247:
5238:
5226:
5219:
5206:
5197:
5178:
5174:
5171:
5168:
5162:
5159:
5156:
5153:
5150:
5147:
5123:
5120:
5116:
5111:
5108:
5105:
5102:
5099:
5096:
5080:
5071:
5058:
5049:
5029:
5026:
5019:
5015:
5011:
5006:
5002:
4996:
4993:
4985:
4982:
4969:
4962:
4953:
4944:
4918:
4915:
4912:
4892:
4888:
4884:
4881:
4878:
4875:
4872:
4842:
4837:
4831:
4828:
4824:
4819:
4815:
4812:
4807:
4803:
4800:
4794:
4787:
4783:
4779:
4774:
4770:
4762:
4759:
4737:
4732:
4726:
4723:
4719:
4714:
4710:
4705:
4701:
4695:
4691:
4687:
4684:
4680:
4676:
4673:
4669:
4664:
4658:
4654:
4650:
4645:
4641:
4634:
4630:
4627:
4623:
4617:
4613:
4609:
4606:
4601:
4597:
4593:
4588:
4584:
4549:
4544:
4539:
4534:
4530:
4524:
4520:
4517:
4514:
4511:
4507:
4502:
4497:
4493:
4487:
4483:
4480:
4477:
4474:
4471:
4456:
4447:
4429:
4424:
4419:
4414:
4410:
4404:
4400:
4397:
4394:
4391:
4387:
4382:
4377:
4373:
4367:
4363:
4360:
4357:
4354:
4351:
4335:
4328:
4317:
4310:
4300:
4291:
4272:
4267:
4264:
4259:
4255:
4252:
4249:
4246:
4243:
4216:
4207:
4190:
4181:
4161:
4151:
4148:
4136:
4131:
4126:
4123:
4118:
4114:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4067:
4062:
4057:
4054:
4049:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
3996:
3988:
3985:
3980:
3976:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3931:
3924:
3921:
3916:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3851:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3737:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3701:
3697:
3693:
3690:
3687:
3684:
3681:
3678:
3672:
3669:
3664:
3661:
3652:
3649:
3644:
3641:
3635:
3629:
3626:
3621:
3618:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3572:
3568:
3564:
3561:
3558:
3555:
3552:
3549:
3543:
3540:
3535:
3532:
3523:
3520:
3515:
3512:
3506:
3500:
3497:
3492:
3489:
3462:
3459:
3456:
3453:
3450:
3447:
3433:
3430:
3418:
3410:
3406:
3402:
3397:
3393:
3387:
3380:
3375:
3370:
3367:
3362:
3357:
3354:
3349:
3346:
3343:
3340:
3337:
3313:
3308:
3302:
3299:
3294:
3289:
3286:
3282:
3277:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3235:
3229:
3226:
3221:
3216:
3213:
3209:
3205:
3200:
3194:
3190:
3186:
3183:
3177:
3172:
3168:
3165:
3162:
3156:
3151:
3148:
3144:
3123:
3119:
3115:
3106:relations for
3095:
3092:
3089:
3069:
3066:
3063:
3043:
3040:
3037:
3015:
3009:
3006:
3001:
2998:
2995:
2992:
2989:
2969:
2966:
2963:
2960:
2955:
2951:
2927:
2924:
2919:
2915:
2911:
2908:
2888:
2885:
2880:
2874:
2870:
2866:
2863:
2857:
2854:
2851:
2848:
2845:
2822:
2816:
2810:
2806:
2802:
2799:
2793:
2790:
2787:
2781:
2778:
2773:
2770:
2747:
2744:
2740:
2736:
2733:
2730:
2727:
2723:
2719:
2697:
2691:
2685:
2681:
2677:
2674:
2668:
2665:
2662:
2657:
2654:
2626:
2622:
2601:
2598:
2593:
2589:
2585:
2580:
2577:
2572:
2567:
2564:
2560:
2556:
2547:, which gives
2530:
2526:
2523:
2520:
2512:
2508:
2504:
2500:
2496:
2493:
2490:
2487:
2482:
2479:
2452:
2444:
2439:
2434:
2431:
2426:
2421:
2418:
2413:
2406:
2401:
2395:
2392:
2387:
2384:
2378:
2373:
2370:
2365:
2359:
2356:
2351:
2348:
2318:
2313:
2309:
2305:
2300:
2296:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2254:
2251:
2246:
2242:
2221:
2217:
2213:
2210:
2207:
2204:
2200:
2196:
2193:
2181:
2178:
2152:
2121:
2118:
2113:
2107:
2104:
2099:
2096:
2061:
2056:
2052:
2046:
2043:
2019:
2011:
2007:
2002:
1999:
1993:
1990:
1987:
1984:
1981:
1975:
1972:
1967:
1964:
1937:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1893:
1889:
1885:
1882:
1879:
1876:
1873:
1755:
1745:
1629:, the tension
1619:
1571:
1567:
1561:
1558:
1552:
1548:
1530:. This is the
1479:parametrically
1450:
1447:
1445:
1442:
1424:speed of light
1416:electric field
1407:
1404:
1391:
1388:
1344:
1341:
1320:
1317:
1314:
1309:
1305:
1301:
1298:
1295:
1264:
1244:
1236:
1232:
1228:
1223:
1219:
1214:
1209:
1206:
1183:
1163:
1158:
1154:
1151:
1146:
1142:
1135:
1129:
1126:
1121:
1118:
1080:
1060:
1055:
1052:
1047:
1044:
1041:
1038:
1022:of the curve.
987:
983:
975:
972:
967:
963:
959:
953:
950:
945:
940:
934:
931:
926:
922:
917:
914:
909:
905:
902:
899:
896:
893:
864:
861:
859:
856:
812:
809:
796:
793:
790:
787:
784:
781:
778:
770:
767:
762:
758:
754:
751:
731:
723:
720:
709:parabolic arch
687:However, in a
640:
637:
636:
635:
632:
625:
623:
612:
605:
603:
592:
585:
583:
566:
559:
501:
498:
322:
319:
249:function. The
29:
9:
6:
4:
3:
2:
12602:
12601:
12590:
12587:
12585:
12582:
12580:
12577:
12575:
12572:
12571:
12569:
12554:
12553:
12544:
12543:
12540:
12534:
12531:
12529:
12526:
12524:
12521:
12519:
12518:Droste effect
12516:
12515:
12513:
12509:
12503:
12500:
12498:
12497:Mathemalchemy
12495:
12493:
12490:
12488:
12485:
12483:
12480:
12478:
12475:
12473:
12470:
12469:
12467:
12465:Organizations
12463:
12457:
12456:
12452:
12450:
12449:
12445:
12443:
12442:
12438:
12436:
12435:
12434:Lumen Naturae
12431:
12429:
12428:
12424:
12423:
12421:
12417:
12405:
12402:
12401:
12400:
12397:
12393:
12392:
12388:
12387:
12386:
12383:
12379:
12376:
12375:
12374:
12371:
12367:
12366:
12362:
12361:
12360:
12357:
12353:
12350:
12349:
12348:
12345:
12341:
12340:
12336:
12335:
12334:
12331:
12330:
12328:
12324:
12316:
12313:
12312:
12311:
12308:
12304:
12301:
12300:
12299:
12296:
12292:
12289:
12288:
12287:
12286:Samuel Colman
12284:
12283:
12281:
12277:
12269:
12268:
12264:
12263:
12262:
12259:
12255:
12252:
12251:
12250:
12247:
12243:
12240:
12239:
12238:
12235:
12231:
12230:
12226:
12225:
12224:
12221:
12217:
12216:
12212:
12211:
12210:
12207:
12203:
12202:
12198:
12197:
12196:
12193:
12189:
12188:
12184:
12182:
12181:
12177:
12176:
12175:
12172:
12170:
12167:
12166:
12164:
12160:
12152:
12151:
12147:
12146:
12145:
12142:
12138:
12135:
12134:
12133:
12130:
12129:
12127:
12123:
12120:
12116:
12106:
12103:
12101:
12098:
12096:
12095:Daina Taimiņa
12093:
12091:
12088:
12086:
12083:
12081:
12080:Reza Sarhangi
12078:
12076:
12073:
12071:
12068:
12066:
12063:
12061:
12058:
12056:
12053:
12051:
12048:
12046:
12043:
12041:
12038:
12036:
12033:
12029:
12028:
12024:
12023:
12022:
12019:
12017:
12014:
12012:
12009:
12007:
12004:
12002:
11999:
11997:
11994:
11992:
11991:Peter Forakis
11989:
11987:
11984:
11982:
11979:
11977:
11974:
11972:
11969:
11967:
11963:
11960:
11958:
11955:
11954:
11952:
11948:
11942:
11939:
11935:
11934:
11930:
11928:
11927:
11923:
11922:
11921:
11920:Salvador Dalí
11918:
11914:
11913:
11909:
11908:
11907:
11906:René Magritte
11904:
11900:
11899:
11895:
11893:
11892:
11888:
11886:
11885:
11881:
11879:
11878:
11877:Print Gallery
11874:
11872:
11871:
11867:
11866:
11865:
11862:
11860:
11857:
11855:
11852:
11848:
11847:
11846:L'Oiseau bleu
11843:
11841:
11840:
11836:
11835:
11834:
11831:
11827:
11826:
11822:
11820:
11819:
11815:
11814:
11813:
11812:William Blake
11810:
11809:
11807:
11801:
11793:
11792:
11788:
11787:
11786:
11783:
11781:
11778:
11774:
11773:
11772:Vitruvian Man
11769:
11768:
11767:
11764:
11762:
11759:
11757:
11756:Paolo Uccello
11754:
11753:
11751:
11747:
11744:
11742:
11738:
11732:
11729:
11727:
11724:
11722:
11719:
11717:
11714:
11712:
11709:
11707:
11704:
11702:
11699:
11697:
11694:
11693:
11691:
11689:
11685:
11679:
11678:
11677:Pi in the Sky
11674:
11672:
11671:
11667:
11665:
11664:
11660:
11658:
11657:
11653:
11651:
11650:
11649:Mathemalchemy
11646:
11644:
11643:
11639:
11637:
11634:
11633:
11631:
11627:
11621:
11618:
11616:
11613:
11611:
11608:
11604:
11601:
11600:
11599:
11596:
11594:
11591:
11587:
11584:
11582:
11579:
11577:
11574:
11573:
11572:
11569:
11565:
11562:
11560:
11557:
11555:
11552:
11550:
11547:
11546:
11545:
11542:
11540:
11537:
11535:
11532:
11530:
11527:
11525:
11522:
11518:
11517:Vastu shastra
11515:
11513:
11510:
11508:
11507:Geodesic dome
11505:
11504:
11503:
11500:
11498:
11495:
11493:
11490:
11489:
11487:
11483:
11478:
11468:
11465:
11463:
11460:
11458:
11455:
11451:
11448:
11446:
11443:
11442:
11440:
11438:
11435:
11433:
11432:Plastic ratio
11430:
11426:
11423:
11421:
11420:Camera lucida
11418:
11417:
11416:
11413:
11411:
11408:
11406:
11403:
11401:
11398:
11396:
11393:
11391:
11388:
11386:
11383:
11381:
11378:
11377:
11375:
11371:
11367:
11360:
11355:
11353:
11348:
11346:
11341:
11340:
11337:
11331:
11330:
11325:
11322:
11319:
11316:
11313:
11310:
11307:
11305:
11302:
11300:
11297:
11295:
11291:
11288:
11286:
11283:
11280:
11276:
11273:
11270:
11266:
11265:
11260:
11256:
11251:
11250:
11240:
11236:
11231:
11227:
11221:
11217:
11216:
11210:
11209:
11198:
11197:
11192:
11189:
11184:
11180:
11176:
11172:
11168:
11164:
11160:
11156:
11152:
11148:
11144:
11139:
11135:
11131:
11126:
11122:
11118:
11114:
11110:
11106:
11102:
11101:
11095:
11091:
11087:
11082:
11081:
11069:
11063:
11056:
11050:
11043:
11038:
11031:
11026:
11018:
11014:
11009:
11004:
11000:
10996:
10992:
10988:
10984:
10980:
10976:
10969:
10962:
10956:
10949:
10943:
10936:
10930:
10923:
10917:
10910:
10905:
10897:
10896:
10891:
10886:
10882:
10876:
10869:
10863:
10854:
10845:
10838:
10834:
10828:
10821:
10816:
10809:
10804:
10797:
10792:
10785:
10779:
10772:
10767:
10759:
10755:
10754:
10746:
10739:
10734:
10726:
10720:
10716:
10715:
10707:
10699:
10695:
10691:
10687:
10683:
10679:
10672:
10664:
10660:
10656:
10652:
10648:
10644:
10640:
10633:
10624:
10616:
10609:
10607:
10605:
10597:
10592:
10585:
10580:
10573:
10563:
10559:
10556:
10550:
10536:
10532:
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10270:(578 KB)
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10041:
10039:
10031:
10029:9783764314415
10025:
10021:
10020:
10012:
10005:
9999:
9985:on 2010-11-13
9984:
9980:
9979:"Arch Design"
9974:
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9731:
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9712:
9708:
9704:
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9688:
9684:
9683:ACS Photonics
9676:
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9469:
9459:
9456:
9453:
9450:
9447:
9444:
9441:
9438:
9435:
9432:
9430:
9429:Catenary arch
9427:
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9420:
9412:
9408:
9402:
9389:
9380:
9372:
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7974:
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7954:
7951:
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7927:
7917:
7904:
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7877:
7873:
7868:
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7850:
7846:
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7834:
7829:
7824:
7820:
7813:
7810:
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7801:
7795:
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7773:
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7753:
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7728:
7708:
7702:
7698:
7694:
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7688:
7685:
7682:
7674:
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7657:
7651:
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7643:
7640:
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7631:
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7623:
7620:
7605:
7588:
7575:
7571:
7567:
7561:
7558:
7553:
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7546:
7542:
7539:
7531:
7521:
7517:
7513:
7503:
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7496:
7494:
7490:
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7467:
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7457:
7453:
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7438:
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7410:
7406:
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7400:
7397:
7389:
7386:
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7351:
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7339:
7336:
7332:
7328:
7325:
7322:
7319:
7316:
7308:
7305:
7298:
7285:
7282:
7279:
7276:
7273:
7270:
7262:
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7246:
7243:
7239:
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7233:
7230:
7227:
7224:
7221:
7218:
7215:
7207:
7204:
7187:
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7180:
7176:
7173:
7168:
7165:
7160:
7158:
7153:
7150:
7147:
7144:
7137:
7131:
7127:
7123:
7121:
7116:
7113:
7110:
7107:
7095:
7092:
7057:
7054:
7041:
7038:
7033:
7029:
7020:
7016:
7012:
7008:
7003:
7000:
6992:
6989:
6976:
6972:
6965:
6961:
6957:
6952:
6949:
6946:
6943:
6940:
6934:
6931:
6926:
6923:
6912:
6909:
6903:
6889:
6883:
6878:
6870:
6862:
6858:
6854:
6845:
6832:
6825:
6822:
6817:
6813:
6809:
6803:
6799:
6793:
6790:
6782:
6781:this becomes
6771:
6758:
6749:
6744:
6737:
6734:
6729:
6726:
6719:
6714:
6711:
6700:
6696:
6692:
6687:
6682:
6678:
6668:
6664:
6657:
6651:
6648:
6643:
6640:
6631:
6628:
6624:
6617:
6613:
6609:
6606:
6598:
6595:
6582:
6578:
6575:
6571:
6553:
6545:
6541:
6537:
6532:
6529:
6526:
6523:
6520:
6514:
6511:
6506:
6503:
6480:
6476:
6473:
6469:
6451:
6447:
6444:
6441:
6438:
6435:
6413:
6409:
6405:
6402:
6399:
6396:
6393:
6385:
6382:
6376:
6370:
6357:
6353:
6350:
6346:
6328:
6319:
6312:
6299:
6297:
6292:
6279:
6276:
6273:
6264:
6260:
6256:
6252:
6249:
6241:
6238:
6235:
6232:
6206:
6186:
6183:
6176:
6173:
6150:
6147:
6143:
6129:
6113:
6105:
6089:
6064:
6060:
6056:
6052:
6049:
6041:
6038:
6035:
6032:
6026:
6011:
6008:The modified
6006:
5993:
5989:
5986:
5983:
5980:
5972:
5968:
5964:
5960:
5957:
5950:
5945:
5941:
5916:
5913:
5905:
5901:
5897:
5893:
5890:
5885:
5882:
5877:
5872:
5868:
5864:
5861:
5841:
5821:
5807:
5794:
5788:
5785:
5780:
5777:
5774:
5771:
5768:
5765:
5762:
5759:
5756:
5736:
5733:
5728:
5724:
5699:
5695:
5691:
5686:
5682:
5676:
5673:
5665:
5645:
5637:
5633:
5629:
5624:
5620:
5614:
5611:
5609:
5604:
5592:
5588:
5584:
5579:
5575:
5566:
5562:
5558:
5556:
5549:
5545:
5535:
5527:
5524:
5518:
5513:
5505:
5502:
5496:
5494:
5486:
5483:
5478:
5474:
5470:
5467:
5464:
5459:
5455:
5446:
5442:
5417:
5414:
5411:
5406:
5402:
5379:
5371:
5368:
5362:
5357:
5352:
5348:
5344:
5339:
5331:
5328:
5325:
5322:
5316:
5311:
5303:
5300:
5297:
5294:
5269:
5262:
5258:
5246:
5237:
5231:
5225:
5218:
5214:
5205:
5196:
5176:
5172:
5169:
5166:
5160:
5157:
5154:
5151:
5148:
5145:
5121:
5118:
5114:
5109:
5106:
5103:
5100:
5097:
5094:
5085:
5079:
5070:
5057:
5048:
5027:
5024:
5017:
5013:
5009:
5004:
5000:
4994:
4991:
4983:
4980:
4968:
4961:
4952:
4943:
4937:
4933:
4916:
4913:
4910:
4890:
4886:
4879:
4873:
4870:
4862:
4853:
4840:
4835:
4829:
4826:
4822:
4817:
4813:
4810:
4805:
4801:
4798:
4792:
4785:
4781:
4777:
4772:
4768:
4760:
4757:
4735:
4730:
4724:
4721:
4717:
4712:
4708:
4703:
4699:
4693:
4689:
4685:
4682:
4678:
4674:
4671:
4667:
4662:
4656:
4652:
4648:
4643:
4639:
4632:
4628:
4625:
4621:
4615:
4611:
4607:
4604:
4599:
4595:
4591:
4586:
4582:
4573:
4570:
4566:
4560:
4547:
4542:
4537:
4532:
4528:
4522:
4518:
4515:
4512:
4509:
4505:
4500:
4495:
4491:
4485:
4481:
4478:
4475:
4472:
4469:
4461:
4455:
4446:
4440:
4427:
4422:
4417:
4412:
4408:
4402:
4398:
4395:
4392:
4389:
4385:
4380:
4375:
4371:
4365:
4361:
4358:
4355:
4352:
4349:
4341:
4334:
4327:
4316:
4309:
4299:
4290:
4284:
4270:
4265:
4262:
4257:
4253:
4250:
4247:
4244:
4241:
4233:
4223:
4215:
4206:
4189:
4180:
4174:
4156:
4147:
4134:
4129:
4124:
4121:
4116:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4065:
4060:
4055:
4052:
4047:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4007:
3994:
3986:
3983:
3978:
3974:
3970:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3929:
3922:
3919:
3914:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3881:
3862:
3849:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3804:
3801:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3771:
3768:
3765:
3762:
3759:
3751:
3748:
3735:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3699:
3695:
3691:
3688:
3685:
3682:
3679:
3676:
3670:
3667:
3662:
3659:
3650:
3647:
3642:
3639:
3633:
3627:
3624:
3619:
3616:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3570:
3566:
3562:
3559:
3556:
3553:
3550:
3547:
3541:
3538:
3533:
3530:
3521:
3518:
3513:
3510:
3504:
3498:
3495:
3490:
3487:
3476:
3473:
3460:
3457:
3454:
3451:
3448:
3445:
3437:
3429:
3416:
3408:
3404:
3400:
3395:
3391:
3385:
3378:
3373:
3368:
3365:
3360:
3355:
3352:
3347:
3344:
3341:
3338:
3335:
3327:
3311:
3306:
3300:
3297:
3292:
3287:
3284:
3280:
3275:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3233:
3227:
3224:
3219:
3214:
3211:
3207:
3203:
3198:
3192:
3188:
3184:
3181:
3175:
3170:
3166:
3163:
3160:
3154:
3149:
3146:
3142:
3121:
3117:
3113:
3093:
3087:
3067:
3061:
3041:
3035:
3026:
3013:
3007:
3004:
2999:
2996:
2993:
2990:
2987:
2967:
2964:
2961:
2958:
2953:
2949:
2939:
2925:
2922:
2917:
2913:
2909:
2906:
2886:
2883:
2878:
2872:
2868:
2864:
2861:
2855:
2852:
2849:
2846:
2843:
2836:
2835:integrates as
2820:
2814:
2808:
2804:
2800:
2797:
2791:
2788:
2785:
2779:
2776:
2771:
2768:
2745:
2742:
2738:
2734:
2731:
2728:
2725:
2721:
2717:
2708:
2695:
2689:
2683:
2679:
2675:
2672:
2666:
2663:
2660:
2655:
2652:
2643:), and hence
2642:
2624:
2620:
2599:
2596:
2591:
2587:
2583:
2578:
2575:
2570:
2565:
2562:
2558:
2554:
2546:
2541:
2528:
2524:
2521:
2518:
2510:
2502:
2498:
2494:
2488:
2485:
2480:
2477:
2466:
2450:
2442:
2437:
2432:
2429:
2424:
2419:
2416:
2411:
2404:
2399:
2393:
2390:
2385:
2382:
2376:
2371:
2368:
2363:
2357:
2354:
2349:
2346:
2335:
2330:
2311:
2307:
2303:
2298:
2294:
2287:
2281:
2278:
2275:
2252:
2249:
2244:
2240:
2219:
2215:
2211:
2208:
2205:
2202:
2198:
2194:
2191:
2177:
2174:
2168:
2164:
2160:
2151:
2147:
2143:
2137:
2134:
2119:
2116:
2111:
2105:
2102:
2097:
2094:
2083:
2080:
2074:
2059:
2054:
2050:
2044:
2041:
2033:
2030:
2017:
2009:
2005:
2000:
1997:
1991:
1988:
1985:
1982:
1979:
1973:
1970:
1965:
1962:
1951:
1948:
1935:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1891:
1887:
1883:
1880:
1877:
1874:
1871:
1863:
1860:
1854:
1851:
1845:
1830:
1824:
1815:
1800:
1796:
1792:
1788:
1784:
1781:
1775:
1769:
1763:
1754:
1744:
1737:
1731:
1725:
1719:
1713:
1707:
1701:
1695:
1689:
1679:
1671:
1661:
1656:
1646:
1639:
1633:
1627:
1618:
1612:
1606:
1600:
1596:
1594:
1589:
1583:
1565:
1559:
1556:
1546:
1535:
1533:
1529:
1524:
1519:
1509:
1505:
1501:
1497:
1493:
1489:
1485:
1480:
1475:
1473:
1468:
1464:
1460:
1456:
1441:
1435:
1430:
1425:
1421:
1417:
1414:in a uniform
1413:
1403:
1387:
1385:
1381:
1380:square wheels
1376:
1374:
1370:
1366:
1362:
1358:
1354:
1350:
1340:
1334:
1318:
1315:
1312:
1307:
1303:
1299:
1296:
1293:
1285:
1280:
1278:
1262:
1242:
1234:
1230:
1226:
1221:
1217:
1212:
1207:
1204:
1197:
1181:
1161:
1156:
1152:
1149:
1144:
1140:
1133:
1127:
1124:
1119:
1116:
1104:
1102:
1094:
1078:
1058:
1053:
1050:
1045:
1042:
1039:
1036:
1028:
1023:
1021:
1013:
1005:
985:
981:
973:
970:
965:
961:
957:
951:
948:
943:
938:
932:
929:
924:
920:
915:
912:
907:
903:
900:
897:
894:
891:
883:
882:has the form
881:
869:
855:
852:
848:
847:Cable ferries
844:
841:
836:
834:
830:
822:
817:
794:
791:
785:
782:
776:
760:
756:
752:
749:
729:
721:
718:
710:
706:
705:catenary arch
701:
697:
695:
690:
685:
683:
678:
676:
668:
664:
660:
656:
649:
645:
629:
624:
620:
616:
609:
604:
600:
596:
589:
584:
580:
576:
575:
570:
563:
558:
557:
556:
553:
551:
547:
541:
534:
530:
526:
521:
517:
512:
510:
506:
497:
495:
491:
487:
483:
475:
471:
469:
468:David Gregory
465:
464:
459:
455:
451:
447:
443:
438:
436:
431:
427:
423:
422:Royal Society
415:
411:
406:
402:
400:
396:
392:
388:
383:
381:
377:
376:
371:
365:
362:
356:
354:
350:
346:
342:
335:
331:
327:
318:
316:
311:
309:
305:
301:
296:
294:
290:
286:
282:
278:
274:
273:
268:
264:
260:
256:
252:
248:
244:
239:
237:
233:
229:
225:
220:
218:
214:
209:
207:
202:
200:
196:
192:
188:
184:
183:
173:
140:
134:
133:
124:
94:
88:
84:
80:
72:
68:
63:
56:
52:
47:
40:
35:
26:
19:
12584:Exponentials
12550:
12453:
12446:
12439:
12432:
12425:
12419:Publications
12403:
12389:
12377:
12363:
12351:
12337:
12314:
12310:Jay Hambidge
12303:Ad Quadratum
12302:
12290:
12265:
12253:
12241:
12227:
12213:
12209:Luca Pacioli
12199:
12185:
12178:
12148:
12136:
12060:Hinke Osinga
12055:István Orosz
12025:
12016:Anthony Hill
11971:Scott Draves
11966:Erik Demaine
11950:Contemporary
11931:
11924:
11910:
11896:
11889:
11882:
11875:
11868:
11864:M. C. Escher
11844:
11837:
11823:
11816:
11789:
11785:Parmigianino
11770:
11701:Hagia Sophia
11675:
11668:
11661:
11654:
11647:
11640:
11524:Computer art
11502:Architecture
11462:Tessellation
11445:Architecture
11395:Golden ratio
11384:
11328:
11262:
11238:
11214:
11194:
11178:
11165:. Macmillan.
11162:
11146:
11133:
11120:
11099:
11092:. Cambridge.
11089:
11078:Bibliography
11062:
11049:
11037:
11025:
10982:
10978:
10968:
10955:
10942:
10929:
10916:
10904:
10893:
10881:Ghostarchive
10879:Archived at
10875:
10862:
10853:
10844:
10827:
10815:
10803:
10791:
10778:
10766:
10752:
10745:
10733:
10713:
10706:
10681:
10677:
10671:
10646:
10642:
10637:Hall, Leon;
10632:
10623:
10614:
10591:
10579:
10571:
10565:. Retrieved
10561:
10549:
10538:. Retrieved
10534:
10525:
10514:. Retrieved
10510:
10501:
10489:. Retrieved
10478:
10452:
10446:
10436:
10417:
10411:
10391:
10384:
10364:
10357:
10337:
10326:
10307:
10301:
10282:
10276:
10264:
10256:
10249:
10232:
10226:
10216:
10200:
10194:
10184:
10164:
10157:
10138:
10132:
10120:
10102:
10098:
10092:
10050:
10046:
10018:
10011:
10003:
9998:
9987:. Retrieved
9983:the original
9973:
9957:
9938:
9932:
9899:
9895:
9889:
9875:
9868:
9835:
9828:
9804:
9797:
9786:. Retrieved
9776:
9733:
9729:
9719:
9686:
9682:
9675:
9634:
9630:
9620:
9585:
9579:
9568:. Retrieved
9558:
9547:. Retrieved
9535:
9523:
9496:
9490:
9486:
9410:
9406:
9403:
9343:
9336:
9329:
9325:
9241:
9235:
9231:
9224:
9220:
9212:
9208:
9204:
9198:
9194:
9179:
9175:
9171:
9169:Second, let
9168:
9162:
9156:
9149:
9111:
9105:
9094:
9090:
9086:
9083:
9077:
9074:
9053:
8884:
8865:
8680:
8677:
8115:
7937:
7931:
7925:
7918:
7774:
7771:
7675:
7672:
7589:
7532:
7509:
7497:
7486:
7390:
7384:
7380:
7309:
7306:
7299:
7263:
7260:
7208:
7205:
7096:
7093:
7063:
7055:
6993:
6990:
6913:
6907:
6901:
6887:
6881:
6866:
6783:
6773:In terms of
6772:
6599:
6596:
6386:
6380:
6374:
6371:
6320:
6313:
6310:
6293:
6007:
5813:
5275:
5260:
5253:
5244:
5235:
5232:
5223:
5216:
5212:
5203:
5194:
5086:
5077:
5068:
5055:
5046:
4966:
4959:
4950:
4941:
4938:
4931:
4854:
4574:
4568:
4564:
4561:
4462:
4453:
4444:
4441:
4342:
4332:
4325:
4314:
4307:
4297:
4288:
4285:
4234:
4213:
4204:
4187:
4178:
4175:
4168:
4008:
3882:
3863:
3752:
3749:
3477:
3474:
3438:
3435:
3027:
2940:
2709:
2542:
2331:
2183:
2172:
2166:
2162:
2158:
2149:
2145:
2141:
2138:
2135:
2084:
2078:
2075:
2034:
2031:
1952:
1949:
1864:
1862:, therefore
1858:
1855:
1849:
1843:
1828:
1813:
1798:
1794:
1790:
1786:
1782:
1779:
1773:
1767:
1761:
1752:
1742:
1735:
1729:
1723:
1717:
1711:
1705:
1699:
1693:
1687:
1677:
1669:
1659:
1652:
1644:
1637:
1631:
1625:
1616:
1610:
1604:
1587:
1584:
1536:
1522:
1507:
1503:
1499:
1495:
1491:
1487:
1483:
1476:
1452:
1431:
1409:
1393:
1377:
1346:
1281:
1105:
1024:
1006:, and where
884:
877:
845:
837:
829:anchor rodes
826:
686:
679:
672:
615:Gateway Arch
572:
554:
539:
532:
528:
524:
516:Gateway Arch
513:
503:
490:Nicolas Fuss
482:surface area
472:
461:
452:derived the
439:
434:
429:
419:
387:Robert Hooke
384:
373:
367:
364:equilibrium.
358:
353:Thomas Paine
340:
338:
330:Antoni Gaudí
312:
297:
281:Robert Hooke
270:
240:
236:Rope statics
235:
227:
223:
221:
210:
203:
86:
76:
67:spider's web
12359:G. H. Hardy
12162:Renaissance
12132:Polykleitos
12070:Tony Robbin
11981:John Ernest
11976:Jan Dibbets
11926:Crucifixion
11749:Renaissance
11603:Mathematics
11576:Celtic knot
11539:Fractal art
11441:Proportion
11415:Perspective
10639:Wagon, Stan
10627:Yates p. 80
9966:Hooke's law
9084:First, let
8866:Again, the
8266:from which
7772:from which
7520:Hooke's Law
4861:numerically
1821:-axis (see
1516:represents
1467:compression
851:chain boats
661:, like the
395:Taq-i Kisra
361:equilibrium
73:catenaries.
12568:Categories
12333:Owen Jones
12180:De pictura
12085:Oliver Sin
12035:Andy Lomas
11884:Relativity
11615:String art
11529:Fiber arts
11410:Paraboloid
11279:PlanetMath
11259:"Catenary"
11191:"Catenary"
11053:Following
10959:Following
10933:Following
10920:Following
10782:Following
10567:2019-09-21
10558:"Catenary"
10540:2023-12-08
10516:2010-11-17
9989:2010-11-17
9788:2010-11-17
9570:2019-05-03
9549:2020-06-04
9497:Structures
9452:Troposkein
9341:to obtain
9326:Divide by
6010:Lagrangian
3134:, giving:
2467:to obtain
2334:arc length
1518:arc length
1194:gives the
1101:arc length
773:cosh
726:cosh
132:-ən-err-ee
55:insulators
12144:Vitruvius
12118:Theorists
11898:Waterfall
11803:19th–20th
11731:Taj Mahal
11711:Parthenon
11688:Buildings
11642:Continuum
11610:Sculpture
11586:Interlace
11380:Algorithm
11196:MathWorld
10698:122116662
10684:: 63–64.
10584:MathWorld
10491:March 27,
10241:0032-4558
10235:(6): 89.
10209:0002-9920
9924:144311552
9760:1094-4087
9711:126267453
9703:2330-4022
9659:1520-8532
9612:199492908
9515:148137330
9480:MathWorld
9303:≈
9297:Δ
9272:−
9263:Δ
8912:
8844:β
8755:α
8708:
8221:φ
8218:
8161:φ
8158:
8028:φ
8025:
7811:φ
7808:
7738:φ
7735:
7692:φ
7689:
7604:stiffness
7495:in 1836.
7454:φ
7445:π
7435:
7424:
7407:φ
7340:
7329:
7286:φ
7283:
7277:ρ
7240:φ
7237:
7231:∫
7225:φ
7222:
7174:∫
7154:φ
7151:
7117:φ
7114:
7042:β
6950:φ
6947:
6826:φ
6823:
6810:ρ
6554:∫
6530:φ
6527:
6452:∫
6445:φ
6442:
6403:φ
6400:
6329:∫
6277:−
6242:λ
6239:−
6170:∂
6158:∂
6144:−
6090:λ
6042:λ
6039:−
5942:∫
5869:∫
5781:
5487:φ
5484:
5468:φ
5465:
5332:φ
5329:
5304:φ
5301:
5280:- and/or
5161:
5110:
5010:−
4874:
4814:
4778:−
4709:
4672:−
4649:−
4629:
4592:−
4519:
4510:−
4482:
4399:
4390:−
4362:
4254:
4222:Translate
4113:
4101:φ
4098:
4044:
4032:φ
4029:
3979:−
3968:φ
3965:
3959:−
3956:φ
3953:
3908:φ
3905:
3896:φ
3893:
3846:β
3840:φ
3837:
3805:α
3796:φ
3793:
3784:φ
3781:
3772:
3732:φ
3729:
3723:φ
3720:
3708:φ
3705:
3689:⋅
3686:φ
3683:
3671:φ
3628:φ
3594:φ
3591:
3579:φ
3576:
3560:⋅
3557:φ
3554:
3542:φ
3499:φ
3461:φ
3458:
3342:δ
3339:−
3293:
3285:−
3272:
3260:δ
3257:−
3220:
3212:−
3185:−
3167:δ
3164:−
3155:
3147:−
3091:↔
3065:↔
3039:↔
3014:◻
3000:
2968:δ
2923:−
2907:δ
2887:δ
2865:−
2856:
2801:−
2792:
2676:−
2667:
2571:
2563:−
1989:φ
1986:
1923:φ
1920:
1881:φ
1878:
1410:A moving
1365:directrix
1316:φ
1313:
1294:ρ
1277:curvature
1263:κ
1205:κ
1182:φ
1153:φ
1150:
1120:φ
1079:φ
1043:φ
1040:
966:−
904:
783:−
579:Barcelona
574:Casa Milà
496:in 1796.
440:In 1691,
412:in Rome (
399:Ctesiphon
334:Casa Milà
295:in 1691.
232:funicular
228:chainette
185:) is the
12552:Category
12279:Romantic
11957:Max Bill
11891:Reptiles
11706:Pantheon
11663:Octacube
11629:Artworks
11571:Knotting
11559:Muqarnas
11457:Symmetry
11385:Catenary
11373:Concepts
11290:Catenary
11275:Catenary
11173:(1833).
11157:(1858).
11115:(1891).
11070:Art. 455
11066:Follows
11057:Art. 500
11044:Art. 494
11032:Art. 489
11017:26601283
10963:Art. 453
10937:Art. 452
10924:Art. 450
10883:and the
10870:art. 447
10598:Art. 444
10471:Lockwood
10085:25750153
9861:Lockwood
9768:30114126
9667:29240100
9540:Archived
9423:See also
7598:, where
7087:, where
6261:′
6177:′
6151:′
6061:′
5969:′
5902:′
5264:, where
4863:. Since
4194:and let
2082:. Then
1817:and the
1803:, where
1444:Analysis
1420:parabola
1373:tractrix
1369:involute
1361:envelope
1353:roulette
1349:parabola
1286:is then
863:Equation
819:A heavy
694:parabola
581:, Spain.
486:catenoid
454:equation
277:parabola
255:catenoid
217:catenoid
206:parabola
87:catenary
83:geometry
12511:Related
12125:Ancient
11859:Man Ray
11805:Century
11741:Artists
11598:Origami
11512:Pyramid
11390:Fractal
11008:4646797
10987:Bibcode
10895:YouTube
10808:Whewell
10771:Whewell
10663:2691240
10586:, eq. 7
10076:4360092
10055:Bibcode
9738:Bibcode
9639:Bibcode
9409:= (0, −
7602:is the
7512:elastic
7084:
7072:
6879:) from
6859:. Most
6102:is the
5929:(where
2612:(where
2336:to get
1682:
1666:
1526:is the
1463:tension
1453:In the
1440:-axis.
1406:Science
1402:-axis.
1363:of the
1347:When a
1339:-axis.
1275:is the
1091:is the
1012:similar
1002:is the
426:anagram
416:, 1748)
370:Galileo
321:History
289:Huygens
285:Leibniz
257:, is a
245:of the
224:alysoid
182:-nər-ee
79:physics
71:elastic
25:Cinsaut
12326:Modern
11962:Martin
11825:Newton
11620:Tiling
11564:Zellij
11534:4D art
11222:
11015:
11005:
10833:Maurer
10721:
10696:
10661:
10473:p. 122
10424:
10399:
10372:
10345:
10314:
10289:
10268:
10239:
10207:
10172:
10145:
10083:
10073:
10026:
9945:
9922:
9916:532102
9914:
9863:p. 124
9816:
9766:
9758:
9709:
9701:
9665:
9657:
9631:JOSA A
9610:
9600:
9566:. 2015
9513:
9503:
9150:where
9058:where
8909:arsinh
8705:arsinh
8116:Then
7919:where
7590:where
7516:spring
7510:In an
7300:where
6895:where
6199:where
6082:where
5158:arcosh
4934:> 0
3324:which
2899:(with
2833:which
1833:where
1750:where
1585:where
1512:where
1412:charge
1333:normal
1255:where
1071:where
998:where
840:anchor
821:anchor
448:, and
341:catēna
300:arches
213:arches
199:weight
12137:Canon
11593:Music
11549:Girih
11485:Forms
11450:Human
11107:–289.
11068:Routh
11055:Routh
11042:Routh
11030:Routh
10961:Routh
10948:Routh
10935:Routh
10922:Routh
10909:Routh
10868:Routh
10837:Routh
10820:Routh
10810:p. 67
10796:Routh
10784:Routh
10773:p. 65
10738:Routh
10694:S2CID
10659:JSTOR
10596:Routh
10455:: 51.
10125:Routh
9920:S2CID
9912:JSTOR
9883:–360.
9707:S2CID
9608:S2CID
9543:(PDF)
9532:(PDF)
9464:Notes
9056:limit
5662:But,
4562:When
2758:, so
2639:is a
2463:then
1827:(0, −
1643:(0, −
1591:is a
1494:) = (
1459:curve
1357:focus
569:Gaudí
531:cosh(
509:kilns
494:force
484:(the
474:Euler
345:chain
243:graph
195:cable
191:chain
187:curve
39:chain
11964:and
11554:Jali
11220:ISBN
11013:PMID
10866:See
10719:ISBN
10493:2009
10422:ISBN
10397:ISBN
10370:ISBN
10343:ISBN
10312:ISBN
10287:ISBN
10262:and
10237:ISSN
10205:ISSN
10170:ISBN
10143:ISBN
10081:PMID
10024:ISBN
9943:ISBN
9814:ISBN
9764:PMID
9756:ISSN
9699:ISSN
9663:PMID
9655:ISSN
9598:ISBN
9511:OCLC
9501:ISBN
9491:e.g.
9192:and
9160:and
8878:and
8870:and
7935:and
7721:and
6905:and
6428:and
6378:and
6314:Let
5778:cosh
5242:and
5201:and
5138:and
5107:sinh
5062:and
5053:and
4948:and
4914:>
4871:sinh
4811:sinh
4700:sinh
4626:cosh
4516:sinh
4479:sinh
4396:cosh
4359:cosh
4322:and
4295:and
4251:cosh
4110:sinh
4078:and
4041:cosh
3876:and
3868:and
3817:and
3606:and
3281:sinh
3269:cosh
3208:sinh
3143:cosh
3080:and
2997:cosh
2853:cosh
2789:sinh
2710:But
2664:sinh
2559:sinh
2265:and
2161:sin
2144:cos
1906:and
1847:and
1797:sin
1789:cos
1748:, 0)
1520:and
1432:The
1282:The
1099:the
1095:and
1025:The
1000:cosh
901:cosh
849:and
742:and
613:The
593:The
514:The
291:and
85:, a
81:and
11292:at
11277:at
11105:287
11003:PMC
10995:doi
10758:387
10686:doi
10651:doi
10453:207
10233:120
10107:doi
10071:PMC
10063:doi
10051:373
9962:cf.
9904:doi
9881:359
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1308:2
1300:a
1297:=
1243:,
1235:2
1231:a
1227:+
1222:2
1218:s
1213:a
1208:=
1162:,
1157:a
1145:2
1134:=
1128:s
1125:d
1117:d
1097:s
1059:,
1054:a
1051:s
1046:=
1016:a
1008:a
986:,
982:)
974:a
971:x
962:e
958:+
952:a
949:x
944:e
939:(
933:2
930:a
925:=
921:)
916:a
913:x
908:(
898:a
895:=
892:y
873:a
795:1
792:+
789:]
786:1
780:)
777:1
769:(
766:[
761:2
757:x
753:=
750:y
730:x
722:=
719:y
617:(
601:.
535:)
529:A
525:y
478:x
172:/
169:i
163:n
157:t
154:ˈ
151:ə
148:k
145:/
123:/
120:i
111:t
108:æ
105:k
102:ˈ
99:/
89:(
27:.
20:.
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