803:
779:
791:
767:
751:
5838:
3314:
341:
3306:
3813:
5825:
20:
60:
3878:
218:. These five circles are separated from each other by six curved triangular regions, each bounded by the arcs from three pairwise-tangent circles. The construction continues by adding six more circles, one in each of these six curved triangles, tangent to its three sides. These in turn create 18 more curved triangles, and the construction continues by again filling these with tangent circles, ad infinitum.
2065:
608:
967:. The first few of these integral Apollonian gaskets are listed in the following table. The table lists the curvatures of the largest circles in the gasket. Only the first three curvatures (of the five displayed in the table) are needed to completely describe each gasket – all other curvatures can be derived from these three.
160:(black in the figure), that are each tangent to the other two, but that do not have a single point of triple tangency. These circles may be of different sizes to each other, and it is allowed for two to be inside the third, or for all three to be outside each other. As Apollonius discovered, there exist two more circles
1844:
3172:
The curvatures appearing in a primitive integral
Apollonian circle packing must belong to a set of six or eight possible residues classes modulo 24, and numerical evidence supported that any sufficiently large integer from these residue classes would also be present as a curvature within the packing.
734:
For
Euclidean symmetry transformations rather than Möbius transformations, in general, the Apollonian gasket will inherit the symmetries of its generating set of three circles. However, some triples of circles can generate Apollonian gaskets with higher symmetry than the initial triple; this happens
719:
centered at the point of tangency (a special case of a Möbius transformation) will transform these two circles into two parallel lines, and transform the rest of the gasket into the special form of a gasket between two parallel lines. Compositions of these inversions can be used to transform any two
386:
3239:
symmetry; such a symmetry consists of two reflections (perpendicular to each other) along diameters of the bounding circle, with a two-fold rotational symmetry of 180°. The gasket described by curvatures (−1, 2, 2, 3) is the only
Apollonian gasket (up to a scaling factor) to possess
714:
of the plane preserve the shapes and tangencies of circles, and therefore preserve the structure of an
Apollonian gasket. Any two triples of mutually tangent circles in an Apollonian gasket may be mapped into each other by a Möbius transformation, and any two Apollonian gaskets may be mapped into
3349:
integral
Apollonian gaskets. The sequence has some interesting properties, and the table lists a factorization of the curvatures, along with the multiplier needed to go from the previous set to the current one. The absolute values of the curvatures of the "a" disks obey the
739:), the Apollonian gasket generated by three congruent circles in an equilateral triangle (with the symmetry of the triangle), and the Apollonian gasket generated by two circles of radius 1 surrounded by a circle of radius 2 (with two lines of reflective symmetry).
610:
This equation may have a solution with a negative radius; this means that one of the circles (the one with negative radius) surrounds the other three. One or two of the initial circles of this construction, or the circles resulting from this construction, can
3275:
symmetry, which corresponds to three reflections along diameters of the bounding circle (spaced 120° apart), along with three-fold rotational symmetry of 120°. In this case the ratio of the curvature of the bounding circle to the three inner circles is
3852:
For example, the gaskets defined by (−2, 3, 6, 7), (−3, 4, 12, 13), (−8, 9, 72, 73), and (−9, 10, 90, 91) all follow this pattern. Because every interior circle that is defined by
6410:
2060:{\displaystyle {\begin{bmatrix}a\\b\\c\\d\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\-1&1&0&0\\-1&0&1&0\\-1&1&1&-2\end{bmatrix}}{\begin{bmatrix}x\\d_{1}\\d_{2}\\m\end{bmatrix}}}
603:{\displaystyle \left({\frac {1}{r_{1}}}+{\frac {1}{r_{2}}}+{\frac {1}{r_{3}}}+{\frac {1}{r_{4}}}\right)^{2}=2\left({\frac {1}{r_{1}^{2}}}+{\frac {1}{r_{2}^{2}}}+{\frac {1}{r_{3}^{2}}}+{\frac {1}{r_{4}^{2}}}\right).}
957:
802:
778:
2609:
2382:
790:
2663:
766:
750:
735:
when the same gasket has a different and more-symmetric set of generating circles. Particularly symmetric cases include the
Apollonian gasket between two parallel lines (with infinite
1728:
2172:
2530:
2277:
4283:
1839:
1781:
817:(the inverse of their radius) then all circles in the gasket will have integer curvature. Since the equation relating curvatures in an Apollonian gasket, integral or not, is
4050:
Ronald L. Graham, Jeffrey C. Lagarias, Colin M. Mallows, Alan R. Wilks, and
Catherine H. Yan; "Apollonian Circle Packings: Number Theory" J. Number Theory, 100 (2003), 1-45
2477:
2441:
252:
3328:
symmetry. The same figure is displayed at right, with labels indicating the curvatures of the interior circles, illustrating that the gasket actually possesses only the
2321:
2216:
2109:
1684:
311:
615:
to a straight line, which can be thought of as a circle with infinite radius. When there are two lines, they must be parallel, and are considered to be tangent at a
2737:
2710:
381:
212:
185:
158:
131:
104:
2683:
2405:
677:
657:
637:
331:
272:
2739:
one can find all the primitive root quadruples. The following Python code demonstrates this algorithm, producing the primitive root quadruples listed above.
724:, so in hyperbolic geometry all Apollonian gaskets are congruent. In a sense, there is therefore only one Apollonian gasket, up to (hyperbolic) isometry.
4594:
820:
5698:
4329:
4428:
5298:
4128:
Summer Haag; Clyde
Kertzer; James Rickards; Katherine E. Stange. "The Local-Global Conjecture for Apollonian circle packings is false".
4374:
3193:
If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group
5776:
4424:
4280:
3387:
4192:
5079:
4615:
3182:
6428:
4587:
4150:
5623:
43:
generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each
5353:
344:
In the limiting case (0,0,1,1), the two largest circles are replaced by parallel straight lines. This produces a family of
5318:
5686:
5089:
4338:. Newspaper story about an artwork in the form of a partial Apollonian gasket, with an outer circumference of nine miles.
5874:
5753:
4580:
4307:
2535:
2326:
333:
stages. In the limit, this set of circles is an
Apollonian gasket. In it, each pair of tangent circles has an infinite
5907:
4270:
4216:
4186:
3221:
symmetry, which corresponds to a reflection along a diameter of the bounding circle, with no rotational symmetry.
2614:
6434:
5722:
5655:
5288:
5168:
4561:
4493:
4367:
3988:
715:
each other by a Möbius transformation. In particular, for any two tangent circles in any
Apollonian gasket, an
1689:
6290:
6247:
2114:
6488:
6163:
5791:
5548:
5436:
4611:
4488:
4083:
Fuchs, Elena; Sanden, Katherine (2011-11-28). "Some Experiments with Integral Apollonian Circle Packings".
2482:
2221:
6368:
6450:
5934:
5500:
5431:
4263:
3283: − 3. As this ratio is not rational, no integral Apollonian circle packings possess this
5127:
4943:
4478:
4432:
4420:
1786:
4918:
4296:
4049:
3214:
Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have
6100:
5662:
5633:
4993:
4848:
4468:
4360:
3894:
3881:
1733:
694:
of about 1.3057. Because it has a well-defined fractional dimension, even though it is not precisely
3268:
If the three circles with smallest positive curvature have the same curvature, the gasket will have
5956:
5796:
5530:
5073:
4483:
4342:
4207:
2387:
This relationship can be used to find all the primitive root quadruples with a given negative bend
808:
Integral Apollonian circle packing defined by circle curvatures of (−10, 18, 23, 27)
784:
Integral Apollonian circle packing defined by circle curvatures of (−12, 25, 25, 28)
612:
2446:
2410:
796:
Integral Apollonian circle packing defined by circle curvatures of (−6, 10, 15, 19)
711:
224:
6493:
5758:
5734:
5608:
5543:
5484:
5421:
5411:
5147:
5066:
4928:
4838:
4718:
3862:
639:-axis and one unit above it, and a circle of unit diameter tangent to both lines centered on the
5028:
6442:
6393:
6001:
5867:
5781:
5729:
5628:
5456:
5406:
5391:
5386:
5157:
4958:
4893:
4883:
4833:
4442:
4062:
2282:
2177:
2070:
1645:
772:
Integral Apollonian circle packing defined by circle curvatures of (−3, 5, 8, 8)
277:
3900:
6227:
5919:
5829:
5681:
5525:
5466:
5343:
5266:
5214:
5016:
4923:
4773:
4473:
4036:
Counting circles and Ergodic theory of Kleinian groups by Hee Oh Brown. University Dec 2009
3932:
3921:
The Butterfly in the Iglesias Waseas World: The story of the most fascinating quantum fractal
3865:. This is demonstrated in the figure at right, which contains these sequential gaskets with
716:
353:
6388:
6383:
6173:
6105:
5806:
5746:
5710:
5553:
5376:
5323:
5293:
5283:
5192:
5055:
4948:
4863:
4818:
4798:
4643:
4628:
4015:
3971:
2715:
2688:
359:
190:
163:
136:
109:
82:
3824: > 0, there exists an Apollonian gasket defined by the following curvatures:
8:
6146:
6123:
6006:
5991:
5924:
5786:
5705:
5693:
5674:
5638:
5558:
5476:
5461:
5451:
5401:
5396:
5338:
5207:
5099:
4963:
4953:
4853:
4823:
4763:
4738:
4663:
4653:
4638:
4535:
4397:
4258:
3351:
721:
691:
48:
6058:
6373:
6353:
6317:
6312:
6075:
5842:
5801:
5741:
5669:
5535:
5510:
5328:
5303:
5271:
5109:
4868:
4813:
4778:
4673:
4129:
4092:
4019:
3984:
3945:
3888:
3173:
This conjecture, known as the local-global conjecture, was proved to be false in 2023.
2668:
2390:
727:
The Apollonian gasket is the limit set of a group of Möbius transformations known as a
662:
642:
622:
316:
257:
4302:
4244:
4225:, The American Mathematical Monthly, Vol. 109, No. 4 (Apr., 2002), pp. 338–361, (
3313:
3235:
If two different curvatures are repeated within the first five, the gasket will have D
720:
points of tangency into each other. Möbius transformations are also isometries of the
6483:
6416:
6378:
6302:
6210:
6115:
6021:
5986:
5929:
5912:
5902:
5897:
5860:
5837:
5515:
5426:
5202:
5175:
4938:
4758:
4748:
4683:
4603:
4540:
4241:
4212:
4182:
4127:
4110:
3200:; the gasket described by curvatures (−10, 18, 23, 27) is an example.
616:
4023:
214:(red) that are tangent to all three of the original circles – these are called
6333:
6200:
6183:
6011:
5717:
5588:
5505:
5261:
5249:
5197:
4933:
4383:
4291:
4102:
4003:
3959:
340:
6348:
6285:
5946:
5358:
5348:
5242:
4968:
4287:
4274:
4202:
4154:
4106:
4011:
3967:
695:
6043:
6363:
6295:
6266:
6222:
6205:
6188:
6141:
6085:
6070:
6038:
5976:
5618:
5613:
5441:
5333:
5313:
5141:
4698:
4668:
4519:
4498:
4460:
4447:
4412:
3928:
3305:
736:
728:
6307:
4035:
3963:
813:
If any four mutually tangent circles in an Apollonian gasket all have integer
6477:
6217:
6193:
6063:
6033:
6016:
5981:
5966:
5578:
5446:
5416:
5237:
5045:
4988:
4556:
4503:
4198:
4114:
964:
960:
684:
4572:
3812:
959:
it follows that one may move from one quadruple of curvatures to another by
6462:
6457:
6358:
6338:
6095:
6028:
5520:
5308:
4983:
4743:
4733:
3989:"Hausdorff dimension and conformal dynamics, III: Computation of dimension"
334:
4007:
6423:
6343:
6053:
6048:
5134:
5022:
4728:
4713:
4402:
3950:
3321:
The figure at left is an integral Apollonian gasket that appears to have
680:
345:
28:
4322:
4226:
4195:". Computers and Graphics, Vol 30, Issue 1, January 2006, pages 134–136.
3185:
that can occur with a gasket depending on the curvature of the circles.
6276:
6261:
6256:
6237:
5971:
5120:
5039:
4978:
4973:
4913:
4898:
4843:
4828:
4783:
4723:
4708:
4688:
4658:
4623:
4314:
1686:
are a root quadruple (the smallest in some integral circle packing) if
356:, which states that, for any four mutually tangent circles, the radii
6232:
6178:
6090:
5941:
4873:
4858:
4808:
4703:
4693:
4678:
4648:
4281:
A Matlab script to plot 2D Apollonian gasket with n identical circles
4249:
3877:
814:
757:
79:
The construction of the Apollonian gasket starts with three circles
69:), there are in general two other circles mutually tangent to them (
59:
19:
6133:
5010:
4788:
4633:
4352:
4134:
4097:
3176:
1637:
952:{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=2ab+2ac+2ad+2bc+2bd+2cd,\,}
6080:
5883:
5583:
4908:
4903:
4803:
4793:
4768:
3924:
699:
44:
40:
3948:(1973), "The residual set dimension of the Apollonian packing",
3857: + 1 can become the bounding circle (defined by −
63:
Mutually tangent circles. Given three mutually tangent circles (
6151:
4878:
4753:
3903:, a self-similar fractal with a similar combinatorial structure
3891:, a graph derived from finite subsets of the Apollonian gasket
5254:
5232:
4888:
3897:, a three-dimensional generalization of the Apollonian gasket
221:
Continued stage by stage in this way, the construction adds
5852:
4323:
Computing Science: A Tisket, a Tasket, an Apollonian Gasket
3382:
3335:
symmetry common to many other integral Apollonian gaskets.
4239:
3390:), from which it follows that the multiplier converges to
4151:"Two Students Unravel a Widely Believed Math Conjecture"
47:
to another three. It is named after Greek mathematician
4221:
Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks:
4271:
An interactive Apollonian gasket running on pure HTML5
2008:
1896:
1853:
742:
4318:
Demonstration of an Apollonian gasket running on Java
2718:
2691:
2671:
2617:
2538:
2485:
2449:
2413:
2393:
2329:
2285:
2224:
2180:
2117:
2073:
1847:
1789:
1736:
1692:
1648:
823:
756:
Integral Apollonian circle packing defined by circle
665:
645:
625:
389:
362:
319:
280:
260:
227:
193:
166:
139:
112:
85:
2731:
2704:
2677:
2657:
2603:
2524:
2471:
2435:
2399:
2376:
2315:
2271:
2210:
2166:
2103:
2059:
1833:
1775:
1722:
1678:
951:
671:
651:
631:
602:
375:
325:
305:
266:
246:
206:
179:
152:
125:
98:
4330:"Sand drawing the world's largest single artwork"
2604:{\displaystyle 3m^{2}\leq d_{1}d_{2}-m^{2}=x^{2}}
2377:{\displaystyle x<0\leq 2m\leq d_{1}\leq d_{2}}
6475:
2225:
2111:satisfies the Descartes equation precisely when
1737:
659:-axis, then the circles that are tangent to the
337:of circles tangent to both circles in the pair.
3177:Symmetry of integral Apollonian circle packings
2665:. By iterating over all the possible values of
1638:Enumerating integral Apollonian circle packings
4045:
4043:
5868:
4602:
4588:
4368:
3290:symmetry, although many packings come close.
2611:. Therefore, any root quadruple will satisfy
352:The size of each new circle is determined by
4325:. American Scientist, January/February 2010.
619:. When the gasket includes two lines on the
4346:, Tartapelago by Giorgio Pietrocola, 2014.
4082:
4040:
2658:{\displaystyle 0\leq m\leq |x|/{\sqrt {3}}}
5875:
5861:
4595:
4581:
4375:
4361:
3861:) in another gasket, these gaskets can be
4208:Indra's Pearls: The Vision of Felix Klein
4133:
4096:
948:
3983:
3876:
3811:
3807:
3338:The following table lists more of these
3312:
3304:
1723:{\displaystyle a<0\leq b\leq c\leq d}
339:
58:
18:
6429:List of fractals by Hausdorff dimension
4193:An Introduction to the Apollony Fractal
6476:
5624:Latin translations of the 12th century
3402:Integral Apollonian gaskets with near-
2167:{\displaystyle x^{2}+m^{2}=d_{1}d_{2}}
5856:
5354:Straightedge and compass construction
4576:
4356:
4240:
3293:
2525:{\displaystyle 4m^{2}\leq d_{1}d_{2}}
2272:{\displaystyle \gcd(x,d_{1},d_{2})=1}
5319:Incircle and excircles of a triangle
4382:
4211:, Cambridge University Press, 2002,
4148:
4060:
3944:
970:
4227:arXiv:math.MG/0101066 v1 9 Jan 2001
4223:Beyond the Descartes Circle Theorem
2323:is a root quadruple precisely when
743:Integral Apollonian circle packings
16:Fractal composed of tangent circles
13:
4308:The Wolfram Demonstrations Project
3258:There are no integer gaskets with
1783:. Defining a new set of variables
383:of the circles obeys the equation
23:An example of an Apollonian gasket
14:
6505:
6411:How Long Is the Coast of Britain?
4233:
3850: + 1) + 1).
1834:{\displaystyle (x,d_{1},d_{2},m)}
5836:
5823:
4297:Online experiments with JSXGraph
4149:Levy, Max G. (August 10, 2023).
3520:
3517:
3514:
3511:
3317:(−15, 32, 32, 33)
3309:(−15, 32, 32, 33)
801:
789:
777:
765:
749:
4063:"Revisiting Apollonian Gaskets"
3996:American Journal of Mathematics
1776:{\displaystyle \gcd(a,b,c,d)=1}
963:, just as when finding a new
760:of (−1, 2, 2, 3)
54:
6435:The Fractal Geometry of Nature
5656:A History of Greek Mathematics
5169:The Quadrature of the Parabola
4504:Sphere-packing (Hamming) bound
4179:The Fractal Geometry of Nature
4142:
4121:
4076:
4054:
4029:
3977:
3938:
3913:
3397: + 2 ≈ 3.732050807.
3247:
3224:
3203:
3188:
2639:
2631:
2310:
2286:
2260:
2228:
2205:
2181:
2098:
2074:
1828:
1790:
1764:
1740:
1673:
1649:
1:
4315:Interactive Apollonian Gasket
4171:
705:
5882:
5437:Intersecting secants theorem
4107:10.1080/10586458.2011.565255
3181:There are multiple types of
2472:{\displaystyle 2m\leq d_{2}}
2436:{\displaystyle 2m\leq d_{1}}
2218:is primitive precisely when
1617:
1604:
1591:
1578:
1565:
1552:
1539:
1526:
1513:
1500:
1487:
1474:
1461:
1448:
1435:
1422:
1409:
1396:
1383:
1370:
1357:
1344:
1331:
1318:
1305:
1292:
1279:
1266:
1253:
1240:
1227:
1214:
1201:
1188:
1175:
1162:
1149:
1136:
1123:
1110:
1097:
1084:
1071:
1058:
1045:
1032:
1019:
1006:
993:
698:, it can be thought of as a
690:The Apollonian gasket has a
247:{\displaystyle 2\cdot 3^{n}}
7:
6451:Chaos: Making a New Science
5432:Intersecting chords theorem
5299:Doctrine of proportionality
3872:
3869:running from 2 through 20.
977:Integral Apollonian gaskets
10:
6510:
5128:On the Sphere and Cylinder
5081:On the Sizes and Distances
4343:Dynamic apollonian gaskets
4277: (archived 2011-05-02)
1730:. They are primitive when
6402:
6326:
6275:
6246:
6162:
6132:
6114:
5955:
5890:
5830:Ancient Greece portal
5819:
5769:
5647:
5634:Philosophy of mathematics
5604:
5597:
5571:
5549:Ptolemy's table of chords
5493:
5475:
5374:
5367:
5223:
5185:
5002:
4610:
4604:Ancient Greek mathematics
4549:
4528:
4512:
4459:
4411:
4390:
3964:10.1112/S0025579300004745
3895:Apollonian sphere packing
3882:Apollonian sphere packing
3816:Nested Apollonian gaskets
3509:
3498:
3424:
3422:
3419:
3417:
3414:
2316:{\displaystyle (a,b,c,d)}
2211:{\displaystyle (a,b,c,d)}
2104:{\displaystyle (a,b,c,d)}
1679:{\displaystyle (a,b,c,d)}
306:{\displaystyle 3^{n+1}+2}
5501:Aristarchus's inequality
5074:On Conoids and Spheroids
4429:isosceles right triangle
4085:Experimental Mathematics
3907:
2741:
5609:Ancient Greek astronomy
5422:Inscribed angle theorem
5412:Greek geometric algebra
5067:Measurement of a Circle
1841:by the matrix equation
6443:The Beauty of Fractals
5843:Mathematics portal
5629:Non-Euclidean geometry
5584:Mouseion of Alexandria
5457:Tangent-secant theorem
5407:Geometric mean theorem
5392:Exterior angle theorem
5387:Angle bisector theorem
5091:On Sizes and Distances
4443:Circle packing theorem
4177:Benoit B. Mandelbrot:
3884:
3842: + 1),
3817:
3318:
3310:
2733:
2706:
2679:
2659:
2605:
2526:
2473:
2437:
2401:
2378:
2317:
2273:
2212:
2168:
2105:
2061:
1835:
1777:
1724:
1680:
1563:−15, 17, 128, 128, 132
1550:−15, 16, 240, 241, 241
1485:−14, 15, 210, 211, 211
1433:−13, 14, 182, 183, 183
1355:−12, 13, 156, 157, 157
1303:−11, 12, 132, 133, 133
1264:−10, 11, 110, 111, 111
953:
712:Möbius transformations
673:
653:
633:
604:
377:
349:
327:
307:
268:
248:
208:
181:
154:
127:
100:
76:
24:
5531:Pappus's area theorem
5467:Theorem of the gnomon
5344:Quadratrix of Hippias
5267:Circles of Apollonius
5215:Problem of Apollonius
5193:Constructible numbers
5017:Archimedes Palimpsest
4306:by Michael Screiber,
4181:, W H Freeman, 1982,
4008:10.1353/ajm.1998.0031
3880:
3834: + 1,
3815:
3808:Sequential curvatures
3316:
3308:
2734:
2732:{\displaystyle d_{2}}
2707:
2705:{\displaystyle d_{1}}
2680:
2660:
2606:
2527:
2474:
2438:
2402:
2379:
2318:
2274:
2213:
2169:
2106:
2067:gives a system where
2062:
1836:
1778:
1725:
1681:
983:Beginning curvatures
954:
717:inversion in a circle
674:
654:
634:
605:
378:
376:{\displaystyle r_{i}}
343:
328:
308:
269:
254:new circles at stage
249:
209:
207:{\displaystyle C_{5}}
182:
180:{\displaystyle C_{4}}
155:
153:{\displaystyle C_{3}}
128:
126:{\displaystyle C_{2}}
101:
99:{\displaystyle C_{1}}
62:
22:
6389:Lewis Fry Richardson
6384:Hamid Naderi Yeganeh
6174:Burning Ship fractal
6106:Weierstrass function
5747:prehistoric counting
5544:Ptolemy's inequality
5485:Apollonius's theorem
5324:Method of exhaustion
5294:Diophantine equation
5284:Circumscribed circle
5101:On the Moving Sphere
4425:equilateral triangle
2716:
2689:
2669:
2615:
2536:
2483:
2447:
2411:
2391:
2327:
2283:
2222:
2178:
2115:
2071:
1845:
1787:
1734:
1690:
1646:
1446:−13, 15, 98, 98, 102
821:
663:
643:
623:
387:
360:
317:
278:
274:, giving a total of
258:
225:
191:
164:
137:
110:
83:
6489:Hyperbolic geometry
6147:Space-filling curve
6124:Multifractal system
6007:Space-filling curve
5992:Sierpinski triangle
5833: •
5639:Neusis construction
5559:Spiral of Theodorus
5452:Pythagorean theorem
5397:Euclidean algorithm
5339:Lune of Hippocrates
5208:Squaring the circle
4964:Theon of Alexandria
4639:Aristaeus the Elder
4562:Slothouber–Graatsma
4259:Alexander Bogomolny
4245:"Apollonian Gasket"
3985:McMullen, Curtis T.
3901:Sierpiński triangle
3411:
3352:recurrence relation
3146:get_primitive_bends
2753:get_primitive_bends
1615:−15, 32, 32, 33, 65
1602:−15, 28, 33, 40, 52
1589:−15, 24, 41, 44, 56
1576:−15, 24, 40, 49, 49
1537:−14, 27, 31, 34, 54
1524:−14, 22, 39, 43, 51
1511:−14, 19, 54, 55, 63
1498:−14, 18, 63, 67, 67
1472:−13, 23, 30, 38, 42
1459:−13, 18, 47, 50, 54
1420:−12, 25, 25, 28, 48
1407:−12, 21, 29, 32, 44
1394:−12, 21, 28, 37, 37
1381:−12, 17, 41, 44, 48
1368:−12, 16, 49, 49, 57
1342:−11, 21, 24, 28, 40
1329:−11, 16, 36, 37, 45
1316:−11, 13, 72, 72, 76
1290:−10, 18, 23, 27, 35
1277:−10, 14, 35, 39, 39
979:
692:Hausdorff dimension
589:
564:
539:
514:
49:Apollonius of Perga
6374:Aleksandr Lyapunov
6354:Desmond Paul Henry
6318:Self-avoiding walk
6313:Percolation theory
5957:Iterated function
5898:Fractal dimensions
5526:Menelaus's theorem
5516:Irrational numbers
5329:Parallel postulate
5304:Euclidean geometry
5272:Apollonian circles
4814:Isidore of Miletus
4286:2008-10-07 at the
4242:Weisstein, Eric W.
3889:Apollonian network
3885:
3818:
3400:
3319:
3311:
2729:
2702:
2675:
2655:
2601:
2522:
2469:
2433:
2407:. It follows from
2397:
2374:
2313:
2269:
2208:
2164:
2101:
2057:
2051:
1997:
1882:
1831:
1773:
1720:
1676:
1251:−9, 18, 19, 22, 34
1238:−9, 14, 26, 27, 35
1225:−9, 11, 50, 50, 54
1212:−9, 10, 90, 91, 91
1199:−8, 13, 21, 24, 28
1186:−8, 12, 25, 25, 33
1160:−7, 12, 17, 20, 24
1121:−6, 11, 14, 15, 23
1108:−6, 10, 15, 19, 19
975:
949:
669:
649:
629:
600:
575:
550:
525:
500:
373:
354:Descartes' theorem
350:
323:
303:
264:
244:
216:Apollonian circles
204:
177:
150:
123:
96:
77:
25:
6471:
6470:
6417:Coastline paradox
6394:Wacław Sierpiński
6379:Benoit Mandelbrot
6303:Fractal landscape
6211:Misiurewicz point
6116:Strange attractor
5997:Apollonian gasket
5987:Sierpinski carpet
5850:
5849:
5815:
5814:
5567:
5566:
5554:Ptolemy's theorem
5427:Intercept theorem
5277:Apollonian gasket
5203:Doubling the cube
5176:The Sand Reckoner
4570:
4569:
4529:Other 3-D packing
4513:Other 2-D packing
4438:Apollonian gasket
4303:Apollonian Gasket
4264:Apollonian Gasket
4191:Paul D. Bourke: "
4061:Bradford, Alden.
3805:
3804:
3183:dihedral symmetry
2678:{\displaystyle m}
2653:
2532:, and hence that
2400:{\displaystyle x}
1634:
1633:
1627:
1626:
1173:−8, 9, 72, 73, 73
1147:−7, 9, 32, 32, 36
1134:−7, 8, 56, 57, 57
1095:−6, 7, 42, 43, 43
1082:−5, 7, 18, 18, 22
1069:−5, 6, 30, 31, 31
1043:−4, 5, 20, 21, 21
1017:−3, 4, 12, 13, 13
737:dihedral symmetry
672:{\displaystyle x}
652:{\displaystyle y}
632:{\displaystyle x}
617:point at infinity
590:
565:
540:
515:
471:
451:
431:
411:
326:{\displaystyle n}
267:{\displaystyle n}
33:Apollonian gasket
6501:
6334:Michael Barnsley
6201:Lyapunov fractal
6059:Sierpiński curve
6012:Blancmange curve
5877:
5870:
5863:
5854:
5853:
5841:
5840:
5828:
5827:
5826:
5602:
5601:
5589:Platonic Academy
5536:Problem II.8 of
5506:Crossbar theorem
5462:Thales's theorem
5402:Euclid's theorem
5372:
5371:
5289:Commensurability
5250:Axiomatic system
5198:Angle trisection
5163:
5153:
5115:
5105:
5095:
5085:
5061:
5051:
5034:
4597:
4590:
4583:
4574:
4573:
4451:
4391:Abstract packing
4384:Packing problems
4377:
4370:
4363:
4354:
4353:
4349:
4337:
4321:Dana Mackenzie.
4292:circle inversion
4255:
4254:
4205:, David Wright:
4166:
4165:
4163:
4161:
4146:
4140:
4139:
4137:
4125:
4119:
4118:
4100:
4080:
4074:
4073:
4071:
4069:
4058:
4052:
4047:
4038:
4033:
4027:
4026:
3993:
3981:
3975:
3974:
3942:
3936:
3917:
3820:For any integer
3412:
3399:
3396:
3395:
3385:
3379:
3282:
3281:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2738:
2736:
2735:
2730:
2728:
2727:
2711:
2709:
2708:
2703:
2701:
2700:
2684:
2682:
2681:
2676:
2664:
2662:
2661:
2656:
2654:
2649:
2647:
2642:
2634:
2610:
2608:
2607:
2602:
2600:
2599:
2587:
2586:
2574:
2573:
2564:
2563:
2551:
2550:
2531:
2529:
2528:
2523:
2521:
2520:
2511:
2510:
2498:
2497:
2478:
2476:
2475:
2470:
2468:
2467:
2442:
2440:
2439:
2434:
2432:
2431:
2406:
2404:
2403:
2398:
2383:
2381:
2380:
2375:
2373:
2372:
2360:
2359:
2322:
2320:
2319:
2314:
2278:
2276:
2275:
2270:
2259:
2258:
2246:
2245:
2217:
2215:
2214:
2209:
2173:
2171:
2170:
2165:
2163:
2162:
2153:
2152:
2140:
2139:
2127:
2126:
2110:
2108:
2107:
2102:
2066:
2064:
2063:
2058:
2056:
2055:
2041:
2040:
2027:
2026:
2002:
2001:
1887:
1886:
1840:
1838:
1837:
1832:
1821:
1820:
1808:
1807:
1782:
1780:
1779:
1774:
1729:
1727:
1726:
1721:
1685:
1683:
1682:
1677:
980:
974:
971:
958:
956:
955:
950:
872:
871:
859:
858:
846:
845:
833:
832:
805:
793:
781:
769:
753:
722:hyperbolic plane
678:
676:
675:
670:
658:
656:
655:
650:
638:
636:
635:
630:
609:
607:
606:
601:
596:
592:
591:
588:
583:
571:
566:
563:
558:
546:
541:
538:
533:
521:
516:
513:
508:
496:
483:
482:
477:
473:
472:
470:
469:
457:
452:
450:
449:
437:
432:
430:
429:
417:
412:
410:
409:
397:
382:
380:
379:
374:
372:
371:
332:
330:
329:
324:
312:
310:
309:
304:
296:
295:
273:
271:
270:
265:
253:
251:
250:
245:
243:
242:
213:
211:
210:
205:
203:
202:
186:
184:
183:
178:
176:
175:
159:
157:
156:
151:
149:
148:
132:
130:
129:
124:
122:
121:
105:
103:
102:
97:
95:
94:
74:
68:
6509:
6508:
6504:
6503:
6502:
6500:
6499:
6498:
6474:
6473:
6472:
6467:
6398:
6349:Felix Hausdorff
6322:
6286:Brownian motion
6271:
6242:
6165:
6158:
6128:
6110:
6101:Pythagoras tree
5958:
5951:
5947:Self-similarity
5891:Characteristics
5886:
5881:
5851:
5846:
5835:
5824:
5822:
5811:
5777:Arabian/Islamic
5765:
5754:numeral systems
5643:
5593:
5563:
5511:Heron's formula
5489:
5471:
5363:
5359:Triangle center
5349:Regular polygon
5226:and definitions
5225:
5219:
5181:
5161:
5151:
5113:
5103:
5093:
5083:
5059:
5049:
5032:
4998:
4969:Theon of Smyrna
4614:
4606:
4601:
4571:
4566:
4545:
4524:
4508:
4455:
4449:
4448:Tammes problem
4407:
4386:
4381:
4347:
4328:
4288:Wayback Machine
4275:Wayback Machine
4236:
4203:Caroline Series
4174:
4169:
4159:
4157:
4155:Quanta Magazine
4147:
4143:
4126:
4122:
4081:
4077:
4067:
4065:
4059:
4055:
4048:
4041:
4034:
4030:
3991:
3982:
3978:
3943:
3939:
3919:Satija, I. I.,
3918:
3914:
3910:
3875:
3851:
3825:
3810:
3408:
3393:
3391:
3381:
3354:
3348:
3334:
3327:
3303:
3300:
3289:
3279:
3277:
3274:
3264:
3256:
3253:
3243:
3238:
3233:
3230:
3220:
3212:
3209:
3199:
3191:
3179:
3170:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2723:
2719:
2717:
2714:
2713:
2696:
2692:
2690:
2687:
2686:
2670:
2667:
2666:
2648:
2643:
2638:
2630:
2616:
2613:
2612:
2595:
2591:
2582:
2578:
2569:
2565:
2559:
2555:
2546:
2542:
2537:
2534:
2533:
2516:
2512:
2506:
2502:
2493:
2489:
2484:
2481:
2480:
2463:
2459:
2448:
2445:
2444:
2427:
2423:
2412:
2409:
2408:
2392:
2389:
2388:
2368:
2364:
2355:
2351:
2328:
2325:
2324:
2284:
2281:
2280:
2254:
2250:
2241:
2237:
2223:
2220:
2219:
2179:
2176:
2175:
2174:. Furthermore,
2158:
2154:
2148:
2144:
2135:
2131:
2122:
2118:
2116:
2113:
2112:
2072:
2069:
2068:
2050:
2049:
2043:
2042:
2036:
2032:
2029:
2028:
2022:
2018:
2015:
2014:
2004:
2003:
1996:
1995:
1987:
1982:
1977:
1968:
1967:
1962:
1957:
1952:
1943:
1942:
1937:
1932:
1927:
1918:
1917:
1912:
1907:
1902:
1892:
1891:
1881:
1880:
1874:
1873:
1867:
1866:
1860:
1859:
1849:
1848:
1846:
1843:
1842:
1816:
1812:
1803:
1799:
1788:
1785:
1784:
1735:
1732:
1731:
1691:
1688:
1687:
1647:
1644:
1643:
1642:The curvatures
1640:
1635:
1623:
1610:
1597:
1584:
1571:
1558:
1545:
1532:
1519:
1506:
1493:
1480:
1467:
1454:
1441:
1428:
1415:
1402:
1389:
1376:
1363:
1350:
1337:
1324:
1311:
1298:
1285:
1272:
1259:
1246:
1233:
1220:
1207:
1194:
1181:
1168:
1155:
1142:
1129:
1116:
1103:
1090:
1077:
1064:
1056:−4, 8, 9, 9, 17
1051:
1038:
1030:−3, 5, 8, 8, 12
1025:
1012:
999:
867:
863:
854:
850:
841:
837:
828:
824:
822:
819:
818:
809:
806:
797:
794:
785:
782:
773:
770:
761:
754:
745:
708:
683:, important in
664:
661:
660:
644:
641:
640:
624:
621:
620:
584:
579:
570:
559:
554:
545:
534:
529:
520:
509:
504:
495:
494:
490:
478:
465:
461:
456:
445:
441:
436:
425:
421:
416:
405:
401:
396:
395:
391:
390:
388:
385:
384:
367:
363:
361:
358:
357:
318:
315:
314:
285:
281:
279:
276:
275:
259:
256:
255:
238:
234:
226:
223:
222:
198:
194:
192:
189:
188:
171:
167:
165:
162:
161:
144:
140:
138:
135:
134:
117:
113:
111:
108:
107:
90:
86:
84:
81:
80:
70:
64:
57:
17:
12:
11:
5:
6507:
6497:
6496:
6494:Circle packing
6491:
6486:
6469:
6468:
6466:
6465:
6460:
6455:
6447:
6439:
6431:
6426:
6421:
6420:
6419:
6406:
6404:
6400:
6399:
6397:
6396:
6391:
6386:
6381:
6376:
6371:
6366:
6364:Helge von Koch
6361:
6356:
6351:
6346:
6341:
6336:
6330:
6328:
6324:
6323:
6321:
6320:
6315:
6310:
6305:
6300:
6299:
6298:
6296:Brownian motor
6293:
6282:
6280:
6273:
6272:
6270:
6269:
6267:Pickover stalk
6264:
6259:
6253:
6251:
6244:
6243:
6241:
6240:
6235:
6230:
6225:
6223:Newton fractal
6220:
6215:
6214:
6213:
6206:Mandelbrot set
6203:
6198:
6197:
6196:
6191:
6189:Newton fractal
6186:
6176:
6170:
6168:
6160:
6159:
6157:
6156:
6155:
6154:
6144:
6142:Fractal canopy
6138:
6136:
6130:
6129:
6127:
6126:
6120:
6118:
6112:
6111:
6109:
6108:
6103:
6098:
6093:
6088:
6086:Vicsek fractal
6083:
6078:
6073:
6068:
6067:
6066:
6061:
6056:
6051:
6046:
6041:
6036:
6031:
6026:
6025:
6024:
6014:
6004:
6002:Fibonacci word
5999:
5994:
5989:
5984:
5979:
5977:Koch snowflake
5974:
5969:
5963:
5961:
5953:
5952:
5950:
5949:
5944:
5939:
5938:
5937:
5932:
5927:
5922:
5917:
5916:
5915:
5905:
5894:
5892:
5888:
5887:
5880:
5879:
5872:
5865:
5857:
5848:
5847:
5820:
5817:
5816:
5813:
5812:
5810:
5809:
5804:
5799:
5794:
5789:
5784:
5779:
5773:
5771:
5770:Other cultures
5767:
5766:
5764:
5763:
5762:
5761:
5751:
5750:
5749:
5739:
5738:
5737:
5727:
5726:
5725:
5715:
5714:
5713:
5703:
5702:
5701:
5691:
5690:
5689:
5679:
5678:
5677:
5667:
5666:
5665:
5651:
5649:
5645:
5644:
5642:
5641:
5636:
5631:
5626:
5621:
5619:Greek numerals
5616:
5614:Attic numerals
5611:
5605:
5599:
5595:
5594:
5592:
5591:
5586:
5581:
5575:
5573:
5569:
5568:
5565:
5564:
5562:
5561:
5556:
5551:
5546:
5541:
5533:
5528:
5523:
5518:
5513:
5508:
5503:
5497:
5495:
5491:
5490:
5488:
5487:
5481:
5479:
5473:
5472:
5470:
5469:
5464:
5459:
5454:
5449:
5444:
5442:Law of cosines
5439:
5434:
5429:
5424:
5419:
5414:
5409:
5404:
5399:
5394:
5389:
5383:
5381:
5369:
5365:
5364:
5362:
5361:
5356:
5351:
5346:
5341:
5336:
5334:Platonic solid
5331:
5326:
5321:
5316:
5314:Greek numerals
5311:
5306:
5301:
5296:
5291:
5286:
5281:
5280:
5279:
5274:
5264:
5259:
5258:
5257:
5247:
5246:
5245:
5240:
5229:
5227:
5221:
5220:
5218:
5217:
5212:
5211:
5210:
5205:
5200:
5189:
5187:
5183:
5182:
5180:
5179:
5172:
5165:
5155:
5145:
5142:Planisphaerium
5138:
5131:
5124:
5117:
5107:
5097:
5087:
5077:
5070:
5063:
5053:
5043:
5036:
5026:
5019:
5014:
5006:
5004:
5000:
4999:
4997:
4996:
4991:
4986:
4981:
4976:
4971:
4966:
4961:
4956:
4951:
4946:
4941:
4936:
4931:
4926:
4921:
4916:
4911:
4906:
4901:
4896:
4891:
4886:
4881:
4876:
4871:
4866:
4861:
4856:
4851:
4846:
4841:
4836:
4831:
4826:
4821:
4816:
4811:
4806:
4801:
4796:
4791:
4786:
4781:
4776:
4771:
4766:
4761:
4756:
4751:
4746:
4741:
4736:
4731:
4726:
4721:
4716:
4711:
4706:
4701:
4696:
4691:
4686:
4681:
4676:
4671:
4666:
4661:
4656:
4651:
4646:
4641:
4636:
4631:
4626:
4620:
4618:
4612:Mathematicians
4608:
4607:
4600:
4599:
4592:
4585:
4577:
4568:
4567:
4565:
4564:
4559:
4553:
4551:
4547:
4546:
4544:
4543:
4538:
4532:
4530:
4526:
4525:
4523:
4522:
4520:Square packing
4516:
4514:
4510:
4509:
4507:
4506:
4501:
4499:Kissing number
4496:
4491:
4486:
4481:
4476:
4471:
4465:
4463:
4461:Sphere packing
4457:
4456:
4454:
4453:
4445:
4440:
4435:
4417:
4415:
4413:Circle packing
4409:
4408:
4406:
4405:
4400:
4394:
4392:
4388:
4387:
4380:
4379:
4372:
4365:
4357:
4351:
4350:
4339:
4326:
4319:
4311:
4299:
4294:
4278:
4268:
4267:, cut-the-knot
4256:
4235:
4234:External links
4232:
4231:
4230:
4219:
4196:
4189:
4173:
4170:
4168:
4167:
4141:
4120:
4091:(4): 380–399.
4075:
4053:
4039:
4028:
4002:(4): 691–721,
3976:
3958:(2): 170–174,
3946:Boyd, David W.
3937:
3929:IOP Publishing
3911:
3909:
3906:
3905:
3904:
3898:
3892:
3874:
3871:
3809:
3806:
3803:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3768:
3767:
3764:
3761:
3758:
3755:
3752:
3749:
3746:
3743:
3740:
3737:
3733:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3698:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3663:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3628:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3593:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3558:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3523:
3522:
3519:
3516:
3513:
3510:
3508:
3505:
3502:
3499:
3497:
3494:
3491:
3488:
3484:
3483:
3478:
3473:
3468:
3463:
3458:
3453:
3448:
3443:
3438:
3433:
3427:
3426:
3423:
3421:
3418:
3416:
3406:
3346:
3332:
3325:
3302:
3298:
3292:
3287:
3272:
3262:
3255:
3251:
3246:
3241:
3236:
3232:
3228:
3223:
3218:
3211:
3207:
3202:
3197:
3190:
3187:
3178:
3175:
2742:
2726:
2722:
2699:
2695:
2674:
2652:
2646:
2641:
2637:
2633:
2629:
2626:
2623:
2620:
2598:
2594:
2590:
2585:
2581:
2577:
2572:
2568:
2562:
2558:
2554:
2549:
2545:
2541:
2519:
2515:
2509:
2505:
2501:
2496:
2492:
2488:
2466:
2462:
2458:
2455:
2452:
2430:
2426:
2422:
2419:
2416:
2396:
2371:
2367:
2363:
2358:
2354:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2268:
2265:
2262:
2257:
2253:
2249:
2244:
2240:
2236:
2233:
2230:
2227:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2161:
2157:
2151:
2147:
2143:
2138:
2134:
2130:
2125:
2121:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2054:
2048:
2045:
2044:
2039:
2035:
2031:
2030:
2025:
2021:
2017:
2016:
2013:
2010:
2009:
2007:
2000:
1994:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1970:
1969:
1966:
1963:
1961:
1958:
1956:
1953:
1951:
1948:
1945:
1944:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1920:
1919:
1916:
1913:
1911:
1908:
1906:
1903:
1901:
1898:
1897:
1895:
1890:
1885:
1879:
1876:
1875:
1872:
1869:
1868:
1865:
1862:
1861:
1858:
1855:
1854:
1852:
1830:
1827:
1824:
1819:
1815:
1811:
1806:
1802:
1798:
1795:
1792:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1639:
1636:
1632:
1631:
1625:
1624:
1621:
1616:
1612:
1611:
1608:
1603:
1599:
1598:
1595:
1590:
1586:
1585:
1582:
1577:
1573:
1572:
1569:
1564:
1560:
1559:
1556:
1551:
1547:
1546:
1543:
1538:
1534:
1533:
1530:
1525:
1521:
1520:
1517:
1512:
1508:
1507:
1504:
1499:
1495:
1494:
1491:
1486:
1482:
1481:
1478:
1473:
1469:
1468:
1465:
1460:
1456:
1455:
1452:
1447:
1443:
1442:
1439:
1434:
1430:
1429:
1426:
1421:
1417:
1416:
1413:
1408:
1404:
1403:
1400:
1395:
1391:
1390:
1387:
1382:
1378:
1377:
1374:
1369:
1365:
1364:
1361:
1356:
1352:
1351:
1348:
1343:
1339:
1338:
1335:
1330:
1326:
1325:
1322:
1317:
1313:
1312:
1309:
1304:
1300:
1299:
1296:
1291:
1287:
1286:
1283:
1278:
1274:
1273:
1270:
1265:
1261:
1260:
1257:
1252:
1248:
1247:
1244:
1239:
1235:
1234:
1231:
1226:
1222:
1221:
1218:
1213:
1209:
1208:
1205:
1200:
1196:
1195:
1192:
1187:
1183:
1182:
1179:
1174:
1170:
1169:
1166:
1161:
1157:
1156:
1153:
1148:
1144:
1143:
1140:
1135:
1131:
1130:
1127:
1122:
1118:
1117:
1114:
1109:
1105:
1104:
1101:
1096:
1092:
1091:
1088:
1083:
1079:
1078:
1075:
1070:
1066:
1065:
1062:
1057:
1053:
1052:
1049:
1044:
1040:
1039:
1036:
1031:
1027:
1026:
1023:
1018:
1014:
1013:
1010:
1005:
1004:−2, 3, 6, 7, 7
1001:
1000:
997:
992:
991:−1, 2, 2, 3, 3
988:
987:
984:
969:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
870:
866:
862:
857:
853:
849:
844:
840:
836:
831:
827:
811:
810:
807:
800:
798:
795:
788:
786:
783:
776:
774:
771:
764:
762:
755:
748:
744:
741:
729:Kleinian group
707:
704:
679:-axis are the
668:
648:
628:
599:
595:
587:
582:
578:
574:
569:
562:
557:
553:
549:
544:
537:
532:
528:
524:
519:
512:
507:
503:
499:
493:
489:
486:
481:
476:
468:
464:
460:
455:
448:
444:
440:
435:
428:
424:
420:
415:
408:
404:
400:
394:
370:
366:
322:
313:circles after
302:
299:
294:
291:
288:
284:
263:
241:
237:
233:
230:
201:
197:
174:
170:
147:
143:
120:
116:
93:
89:
56:
53:
37:Apollonian net
15:
9:
6:
4:
3:
2:
6506:
6495:
6492:
6490:
6487:
6485:
6482:
6481:
6479:
6464:
6461:
6459:
6456:
6453:
6452:
6448:
6445:
6444:
6440:
6437:
6436:
6432:
6430:
6427:
6425:
6422:
6418:
6415:
6414:
6412:
6408:
6407:
6405:
6401:
6395:
6392:
6390:
6387:
6385:
6382:
6380:
6377:
6375:
6372:
6370:
6367:
6365:
6362:
6360:
6357:
6355:
6352:
6350:
6347:
6345:
6342:
6340:
6337:
6335:
6332:
6331:
6329:
6325:
6319:
6316:
6314:
6311:
6309:
6306:
6304:
6301:
6297:
6294:
6292:
6291:Brownian tree
6289:
6288:
6287:
6284:
6283:
6281:
6278:
6274:
6268:
6265:
6263:
6260:
6258:
6255:
6254:
6252:
6249:
6245:
6239:
6236:
6234:
6231:
6229:
6226:
6224:
6221:
6219:
6218:Multibrot set
6216:
6212:
6209:
6208:
6207:
6204:
6202:
6199:
6195:
6194:Douady rabbit
6192:
6190:
6187:
6185:
6182:
6181:
6180:
6177:
6175:
6172:
6171:
6169:
6167:
6161:
6153:
6150:
6149:
6148:
6145:
6143:
6140:
6139:
6137:
6135:
6131:
6125:
6122:
6121:
6119:
6117:
6113:
6107:
6104:
6102:
6099:
6097:
6094:
6092:
6089:
6087:
6084:
6082:
6079:
6077:
6074:
6072:
6069:
6065:
6064:Z-order curve
6062:
6060:
6057:
6055:
6052:
6050:
6047:
6045:
6042:
6040:
6037:
6035:
6034:Hilbert curve
6032:
6030:
6027:
6023:
6020:
6019:
6018:
6017:De Rham curve
6015:
6013:
6010:
6009:
6008:
6005:
6003:
6000:
5998:
5995:
5993:
5990:
5988:
5985:
5983:
5982:Menger sponge
5980:
5978:
5975:
5973:
5970:
5968:
5967:Barnsley fern
5965:
5964:
5962:
5960:
5954:
5948:
5945:
5943:
5940:
5936:
5933:
5931:
5928:
5926:
5923:
5921:
5918:
5914:
5911:
5910:
5909:
5906:
5904:
5901:
5900:
5899:
5896:
5895:
5893:
5889:
5885:
5878:
5873:
5871:
5866:
5864:
5859:
5858:
5855:
5845:
5844:
5839:
5832:
5831:
5818:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5775:
5774:
5772:
5768:
5760:
5757:
5756:
5755:
5752:
5748:
5745:
5744:
5743:
5740:
5736:
5733:
5732:
5731:
5728:
5724:
5721:
5720:
5719:
5716:
5712:
5709:
5708:
5707:
5704:
5700:
5697:
5696:
5695:
5692:
5688:
5685:
5684:
5683:
5680:
5676:
5673:
5672:
5671:
5668:
5664:
5660:
5659:
5658:
5657:
5653:
5652:
5650:
5646:
5640:
5637:
5635:
5632:
5630:
5627:
5625:
5622:
5620:
5617:
5615:
5612:
5610:
5607:
5606:
5603:
5600:
5596:
5590:
5587:
5585:
5582:
5580:
5577:
5576:
5574:
5570:
5560:
5557:
5555:
5552:
5550:
5547:
5545:
5542:
5540:
5539:
5534:
5532:
5529:
5527:
5524:
5522:
5519:
5517:
5514:
5512:
5509:
5507:
5504:
5502:
5499:
5498:
5496:
5492:
5486:
5483:
5482:
5480:
5478:
5474:
5468:
5465:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5447:Pons asinorum
5445:
5443:
5440:
5438:
5435:
5433:
5430:
5428:
5425:
5423:
5420:
5418:
5417:Hinge theorem
5415:
5413:
5410:
5408:
5405:
5403:
5400:
5398:
5395:
5393:
5390:
5388:
5385:
5384:
5382:
5380:
5379:
5373:
5370:
5366:
5360:
5357:
5355:
5352:
5350:
5347:
5345:
5342:
5340:
5337:
5335:
5332:
5330:
5327:
5325:
5322:
5320:
5317:
5315:
5312:
5310:
5307:
5305:
5302:
5300:
5297:
5295:
5292:
5290:
5287:
5285:
5282:
5278:
5275:
5273:
5270:
5269:
5268:
5265:
5263:
5260:
5256:
5253:
5252:
5251:
5248:
5244:
5241:
5239:
5236:
5235:
5234:
5231:
5230:
5228:
5222:
5216:
5213:
5209:
5206:
5204:
5201:
5199:
5196:
5195:
5194:
5191:
5190:
5188:
5184:
5178:
5177:
5173:
5171:
5170:
5166:
5164:
5160:
5156:
5154:
5150:
5146:
5144:
5143:
5139:
5137:
5136:
5132:
5130:
5129:
5125:
5123:
5122:
5118:
5116:
5112:
5108:
5106:
5102:
5098:
5096:
5092:
5088:
5086:
5084:(Aristarchus)
5082:
5078:
5076:
5075:
5071:
5069:
5068:
5064:
5062:
5058:
5054:
5052:
5048:
5044:
5042:
5041:
5037:
5035:
5031:
5027:
5025:
5024:
5020:
5018:
5015:
5013:
5012:
5008:
5007:
5005:
5001:
4995:
4992:
4990:
4989:Zeno of Sidon
4987:
4985:
4982:
4980:
4977:
4975:
4972:
4970:
4967:
4965:
4962:
4960:
4957:
4955:
4952:
4950:
4947:
4945:
4942:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4872:
4870:
4867:
4865:
4862:
4860:
4857:
4855:
4852:
4850:
4847:
4845:
4842:
4840:
4837:
4835:
4832:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4697:
4695:
4692:
4690:
4687:
4685:
4682:
4680:
4677:
4675:
4672:
4670:
4667:
4665:
4662:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4635:
4632:
4630:
4627:
4625:
4622:
4621:
4619:
4617:
4613:
4609:
4605:
4598:
4593:
4591:
4586:
4584:
4579:
4578:
4575:
4563:
4560:
4558:
4555:
4554:
4552:
4548:
4542:
4539:
4537:
4534:
4533:
4531:
4527:
4521:
4518:
4517:
4515:
4511:
4505:
4502:
4500:
4497:
4495:
4494:Close-packing
4492:
4490:
4489:In a cylinder
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4466:
4464:
4462:
4458:
4452:
4446:
4444:
4441:
4439:
4436:
4434:
4430:
4426:
4422:
4419:
4418:
4416:
4414:
4410:
4404:
4401:
4399:
4396:
4395:
4393:
4389:
4385:
4378:
4373:
4371:
4366:
4364:
4359:
4358:
4355:
4345:
4344:
4340:
4336:, 16 Dec 2009
4335:
4334:The Telegraph
4331:
4327:
4324:
4320:
4317:
4316:
4312:
4309:
4305:
4304:
4300:
4298:
4295:
4293:
4289:
4285:
4282:
4279:
4276:
4272:
4269:
4266:
4265:
4260:
4257:
4252:
4251:
4246:
4243:
4238:
4237:
4228:
4224:
4220:
4218:
4217:0-521-35253-3
4214:
4210:
4209:
4204:
4200:
4199:David Mumford
4197:
4194:
4190:
4188:
4187:0-7167-1186-9
4184:
4180:
4176:
4175:
4156:
4152:
4145:
4136:
4131:
4124:
4116:
4112:
4108:
4104:
4099:
4094:
4090:
4086:
4079:
4064:
4057:
4051:
4046:
4044:
4037:
4032:
4025:
4021:
4017:
4013:
4009:
4005:
4001:
3997:
3990:
3986:
3980:
3973:
3969:
3965:
3961:
3957:
3953:
3952:
3947:
3941:
3934:
3930:
3926:
3922:
3916:
3912:
3902:
3899:
3896:
3893:
3890:
3887:
3886:
3883:
3879:
3870:
3868:
3864:
3860:
3856:
3849:
3845:
3841:
3837:
3833:
3829:
3823:
3814:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3769:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3734:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3699:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3664:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3629:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3594:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3559:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3524:
3506:
3503:
3500:
3495:
3492:
3489:
3486:
3485:
3482:
3479:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3459:
3457:
3454:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3428:
3413:
3410:
3405:
3398:
3389:
3384:
3377:
3373:
3369:
3365:
3361:
3357:
3353:
3345:
3341:
3336:
3331:
3324:
3315:
3307:
3297:
3291:
3286:
3271:
3266:
3261:
3250:
3245:
3227:
3222:
3217:
3206:
3201:
3196:
3186:
3184:
3174:
2740:
2724:
2720:
2697:
2693:
2672:
2650:
2644:
2635:
2627:
2624:
2621:
2618:
2596:
2592:
2588:
2583:
2579:
2575:
2570:
2566:
2560:
2556:
2552:
2547:
2543:
2539:
2517:
2513:
2507:
2503:
2499:
2494:
2490:
2486:
2464:
2460:
2456:
2453:
2450:
2428:
2424:
2420:
2417:
2414:
2394:
2385:
2369:
2365:
2361:
2356:
2352:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2266:
2263:
2255:
2251:
2247:
2242:
2238:
2234:
2231:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2159:
2155:
2149:
2145:
2141:
2136:
2132:
2128:
2123:
2119:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2052:
2046:
2037:
2033:
2023:
2019:
2011:
2005:
1998:
1992:
1989:
1984:
1979:
1974:
1971:
1964:
1959:
1954:
1949:
1946:
1939:
1934:
1929:
1924:
1921:
1914:
1909:
1904:
1899:
1893:
1888:
1883:
1877:
1870:
1863:
1856:
1850:
1825:
1822:
1817:
1813:
1809:
1804:
1800:
1796:
1793:
1770:
1767:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1630:
1620:
1614:
1613:
1607:
1601:
1600:
1594:
1588:
1587:
1581:
1575:
1574:
1568:
1562:
1561:
1555:
1549:
1548:
1542:
1536:
1535:
1529:
1523:
1522:
1516:
1510:
1509:
1503:
1497:
1496:
1490:
1484:
1483:
1477:
1471:
1470:
1464:
1458:
1457:
1451:
1445:
1444:
1438:
1432:
1431:
1425:
1419:
1418:
1412:
1406:
1405:
1399:
1393:
1392:
1386:
1380:
1379:
1373:
1367:
1366:
1360:
1354:
1353:
1347:
1341:
1340:
1334:
1328:
1327:
1321:
1315:
1314:
1308:
1302:
1301:
1295:
1289:
1288:
1282:
1276:
1275:
1269:
1263:
1262:
1256:
1250:
1249:
1243:
1237:
1236:
1230:
1224:
1223:
1217:
1211:
1210:
1204:
1198:
1197:
1191:
1185:
1184:
1178:
1172:
1171:
1165:
1159:
1158:
1152:
1146:
1145:
1139:
1133:
1132:
1126:
1120:
1119:
1113:
1107:
1106:
1100:
1094:
1093:
1087:
1081:
1080:
1074:
1068:
1067:
1061:
1055:
1054:
1048:
1042:
1041:
1035:
1029:
1028:
1022:
1016:
1015:
1009:
1003:
1002:
996:
990:
989:
985:
982:
981:
978:
973:
972:
968:
966:
965:Markov number
962:
961:Vieta jumping
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
868:
864:
860:
855:
851:
847:
842:
838:
834:
829:
825:
816:
804:
799:
792:
787:
780:
775:
768:
763:
759:
752:
747:
746:
740:
738:
732:
730:
725:
723:
718:
713:
703:
701:
697:
693:
688:
686:
685:number theory
682:
666:
646:
626:
618:
614:
597:
593:
585:
580:
576:
572:
567:
560:
555:
551:
547:
542:
535:
530:
526:
522:
517:
510:
505:
501:
497:
491:
487:
484:
479:
474:
466:
462:
458:
453:
446:
442:
438:
433:
426:
422:
418:
413:
406:
402:
398:
392:
368:
364:
355:
347:
342:
338:
336:
320:
300:
297:
292:
289:
286:
282:
261:
239:
235:
231:
228:
219:
217:
199:
195:
172:
168:
145:
141:
118:
114:
91:
87:
73:
67:
61:
52:
50:
46:
42:
38:
34:
30:
21:
6463:Chaos theory
6458:Kaleidoscope
6449:
6441:
6433:
6359:Gaston Julia
6339:Georg Cantor
6164:Escape-time
6096:Gosper curve
6044:Lévy C curve
6029:Dragon curve
5996:
5908:Box-counting
5834:
5821:
5663:Thomas Heath
5654:
5537:
5521:Law of sines
5377:
5309:Golden ratio
5276:
5174:
5167:
5158:
5152:(Theodosius)
5148:
5140:
5133:
5126:
5119:
5110:
5100:
5094:(Hipparchus)
5090:
5080:
5072:
5065:
5056:
5046:
5038:
5033:(Apollonius)
5029:
5021:
5009:
4984:Zeno of Elea
4744:Eratosthenes
4734:Dionysodorus
4437:
4431: /
4427: /
4423: /
4348:(in Italian)
4341:
4333:
4313:
4301:
4262:
4248:
4222:
4206:
4178:
4158:. Retrieved
4144:
4123:
4088:
4084:
4078:
4066:. Retrieved
4056:
4031:
3999:
3995:
3979:
3955:
3949:
3940:
3920:
3915:
3866:
3858:
3854:
3847:
3843:
3839:
3835:
3831:
3827:
3821:
3819:
3801:3.732026144
3766:3.731707317
3731:3.731707317
3696:3.727272727
3661:3.727272727
3626:3.666666667
3591:3.666666667
3556:3.000000000
3480:
3475:
3470:
3465:
3460:
3455:
3450:
3445:
3440:
3435:
3430:
3403:
3401:
3375:
3371:
3367:
3363:
3359:
3355:
3343:
3339:
3337:
3329:
3322:
3320:
3295:
3284:
3269:
3267:
3259:
3257:
3248:
3234:
3225:
3215:
3213:
3204:
3194:
3192:
3180:
3171:
2386:
1641:
1628:
1618:
1605:
1592:
1579:
1566:
1553:
1540:
1527:
1514:
1501:
1488:
1475:
1462:
1449:
1436:
1423:
1410:
1397:
1384:
1371:
1358:
1345:
1332:
1319:
1306:
1293:
1280:
1267:
1254:
1241:
1228:
1215:
1202:
1189:
1176:
1163:
1150:
1137:
1124:
1111:
1098:
1085:
1072:
1059:
1046:
1033:
1020:
1007:
994:
976:
812:
733:
726:
709:
696:self-similar
689:
681:Ford circles
351:
346:Ford circles
335:Pappus chain
220:
215:
78:
71:
65:
55:Construction
36:
32:
26:
6454:(1987 book)
6446:(1986 book)
6438:(1982 book)
6424:Fractal art
6344:Bill Gosper
6308:Lévy flight
6054:Peano curve
6049:Moore curve
5935:Topological
5920:Correlation
5730:mathematics
5538:Arithmetica
5135:Ostomachion
5104:(Autolycus)
5023:Arithmetica
4799:Hippocrates
4729:Dinostratus
4714:Dicaearchus
4644:Aristarchus
4536:Tetrahedron
4479:In a sphere
4450:(on sphere)
4421:In a circle
3951:Mathematika
3798:3.731983425
3795:3.732142857
3792:3.732050810
3763:3.732302296
3760:3.732142857
3757:3.732050842
3728:3.731112433
3725:3.733333333
3722:3.732051282
3693:3.735555556
3690:3.733333333
3687:3.732057416
3658:3.719008264
3655:3.750000000
3652:3.732142857
3623:3.781250000
3620:3.750000000
3617:3.733333333
3588:3.555555556
3585:4.000000000
3582:3.750000000
3553:4.500000000
3550:4.000000000
3547:4.000000000
3425:Multiplier
3189:No symmetry
29:mathematics
6478:Categories
6262:Orbit trap
6257:Buddhabrot
6250:techniques
6238:Mandelbulb
6039:Koch curve
5972:Cantor set
5782:Babylonian
5682:arithmetic
5648:History of
5477:Apollonius
5162:(Menelaus)
5121:On Spirals
5040:Catoptrics
4979:Xenocrates
4974:Thymaridas
4959:Theodosius
4944:Theaetetus
4924:Simplicius
4914:Pythagoras
4899:Posidonius
4884:Philonides
4844:Nicomachus
4839:Metrodorus
4829:Menaechmus
4784:Hipparchus
4774:Heliodorus
4724:Diophantus
4709:Democritus
4689:Chrysippus
4659:Archimedes
4654:Apollonius
4624:Anaxagoras
4616:(timeline)
4469:Apollonian
4172:References
4160:August 14,
4135:2307.02749
3415:Curvature
3380:(sequence
3265:symmetry.
3244:symmetry.
758:curvatures
706:Symmetries
613:degenerate
6369:Paul Lévy
6248:Rendering
6233:Mandelbox
6179:Julia set
6091:Hexaflake
6022:Minkowski
5942:Recursion
5925:Hausdorff
5243:Inscribed
5003:Treatises
4994:Zenodorus
4954:Theodorus
4929:Sosigenes
4874:Philolaus
4859:Oenopides
4854:Nicoteles
4849:Nicomedes
4809:Hypsicles
4704:Ctesibius
4694:Cleomedes
4679:Callippus
4664:Autolycus
4649:Aristotle
4629:Anthemius
4541:Ellipsoid
4484:In a cube
4250:MathWorld
4115:1058-6458
4098:1001.1406
3931:, 2016),
3002:remainder
2975:remainder
2628:≤
2622:≤
2576:−
2553:≤
2500:≤
2457:≤
2421:≤
2362:≤
2349:≤
2340:≤
1990:−
1972:−
1947:−
1922:−
1715:≤
1709:≤
1703:≤
986:Symmetry
815:curvature
232:⋅
6484:Fractals
6279:fractals
6166:fractals
6134:L-system
6076:T-square
5884:Fractals
5807:Japanese
5792:Egyptian
5735:timeline
5723:timeline
5711:timeline
5706:geometry
5699:timeline
5694:calculus
5687:timeline
5675:timeline
5378:Elements
5224:Concepts
5186:Problems
5159:Spherics
5149:Spherics
5114:(Euclid)
5060:(Euclid)
5057:Elements
5050:(Euclid)
5011:Almagest
4919:Serenus
4894:Porphyry
4834:Menelaus
4789:Hippasus
4764:Eutocius
4739:Domninus
4634:Archytas
4284:Archived
4068:7 August
4024:15928775
3987:(1998),
3873:See also
3420:Factors
3409:symmetry
3301:symmetry
3254:symmetry
3231:symmetry
3210:symmetry
6228:Tricorn
6081:n-flake
5930:Packing
5913:Higuchi
5903:Assouad
5787:Chinese
5742:numbers
5670:algebra
5598:Related
5572:Centers
5368:Results
5238:Central
4909:Ptolemy
4904:Proclus
4869:Perseus
4824:Marinus
4804:Hypatia
4794:Hippias
4769:Geminus
4759:Eudoxus
4749:Eudemus
4719:Diocles
4550:Puzzles
4273:at the
4016:1637951
3972:0493763
3925:Bristol
3830:,
3789:153×571
3786:209×418
3783:153×265
3754:153×153
3751:112×209
3392:√
3386:in the
3383:A001353
3370:− 1) −
3294:Almost-
3278:√
700:fractal
45:tangent
41:fractal
6327:People
6277:Random
6184:Filled
6152:H tree
6071:String
5959:system
5802:Indian
5579:Cyrene
5111:Optics
5030:Conics
4949:Theano
4939:Thales
4934:Sporus
4879:Philon
4864:Pappus
4754:Euclid
4684:Carpus
4674:Bryson
4557:Conway
4474:Finite
4433:square
4290:using
4215:
4185:
4113:
4022:
4014:
3970:
3863:nested
3771:−40545
3748:112×97
3736:−10864
3719:41×153
3716:56×112
3340:almost
2981:divmod
2810:return
2744:import
2712:, and
2279:, and
1629:
133:, and
6403:Other
5797:Incan
5718:logic
5494:Other
5262:Chord
5255:Axiom
5233:Angle
4889:Plato
4779:Heron
4699:Conon
4130:arXiv
4093:arXiv
4020:S2CID
3992:(PDF)
3908:Notes
3780:87363
3777:87362
3774:87362
3745:23409
3742:23409
3739:23408
3713:41×71
3701:−2911
3684:41×41
3681:30×56
3678:30×26
3649:11×41
3646:15×30
3643:11×19
3614:11×11
3362:) = 4
3164:bends
3158:print
3140:bends
3125:range
3053:yield
2936:floor
2900:range
2822:range
2786:yield
2479:that
66:black
39:is a
31:, an
5759:list
5047:Data
4819:Leon
4669:Bion
4213:ISBN
4183:ISBN
4162:2023
4111:ISSN
4070:2022
3933:p. 5
3710:6273
3707:6272
3704:6272
3675:1681
3672:1681
3669:1680
3666:−780
3631:−209
3611:8×15
3579:3×11
3388:OEIS
3378:− 2)
3014:math
2948:sqrt
2942:math
2930:math
2861:))):
2852:sqrt
2846:math
2834:ceil
2828:math
2747:math
2443:and
2334:<
1697:<
710:The
187:and
5661:by
5375:In
4403:Set
4398:Bin
4103:doi
4004:doi
4000:120
3960:doi
3640:451
3637:450
3634:450
3608:8×7
3605:121
3602:121
3599:120
3596:−56
3576:4×8
3573:3×5
3561:−15
3544:3×3
3541:2×4
3538:2×2
3507:1×3
3504:1×2
3501:1×1
3137:for
3116:for
3020:gcd
3011:and
2906:max
2891:for
2813:for
2765:int
2750:def
2226:gcd
1738:gcd
72:red
35:or
27:In
6480::
6413:"
4332:,
4261:,
4247:.
4201:,
4153:.
4109:.
4101:.
4089:20
4087:.
4042:^
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4012:MR
4010:,
3998:,
3994:,
3968:MR
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3956:20
3954:,
3927::
3826:(−
3570:33
3567:32
3564:32
3526:−4
3521:—
3487:−1
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3122:in
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3089:d1
3077:d2
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3867:n
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3832:n
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3532:9
3529:8
3518:—
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3407:3
3404:D
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3376:n
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3368:n
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3276:2
3273:3
3270:D
3263:3
3260:D
3252:3
3249:D
3242:2
3240:D
3237:2
3229:2
3226:D
3219:1
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3161:(
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3149:(
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3026:n
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2882:n
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2849:.
2843:/
2840:n
2837:(
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2807:1
2804:,
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2759:n
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2645:/
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2584:2
2580:m
2571:2
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2491:m
2487:4
2465:2
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2429:1
2425:d
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2370:2
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2308:d
2305:,
2302:c
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2256:2
2252:d
2248:,
2243:1
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2229:(
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2203:d
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2197:c
2194:,
2191:b
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2137:2
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2084:b
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2038:2
2034:d
2024:1
2020:d
2012:x
2006:[
1999:]
1993:2
1985:1
1980:1
1975:1
1965:0
1960:1
1955:0
1950:1
1940:0
1935:0
1930:1
1925:1
1915:0
1910:0
1905:0
1900:1
1894:[
1889:=
1884:]
1878:d
1871:c
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1851:[
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1826:m
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1609:1
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861:+
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