Euclidean geometry

6235:, for which the geometry of the space part of space-time is not Euclidean geometry. For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. A relatively weak gravitational field, such as the Earth's or the Sun's, is represented by a metric that is approximately, but not exactly, Euclidean. Until the 20th century, there was no technology capable of detecting these deviations in rays of light from Euclidean geometry, but Einstein predicted that such deviations would exist. They were later verified by observations such as the slight bending of starlight by the Sun during a solar eclipse in 1919, and such considerations are now an integral part of the software that runs the 2508: 1746: 2712: 79: 2891: 1762: 1792: 1724: 2092: 2471: 2764: 2776: 9960: 9384: 2339: 2845: 2563: 2164: 2788: 2291: 2693: 2678: 3207: 6415: 1434: 2607: 6194: 3147: 1838: 10247: 9972: 2042:" refers to the idea that an entire figure is the same size and shape as another figure. Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. (Flipping it over is allowed.) Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as 9371: 2124: 4325: 4318: 38: 4311: 2204: 4290: 1595: 9996: 2015: 9984: 4304: 4297: 2866:), a colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians. Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original. He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the 7015:

the proposition being like that of a trestle bridge, with a ramp at each end which is more practicable the flatter the figure is drawn, the bridge is such that, while a horse could not surmount the ramp, an ass could; in other words, the term is meant to refer to the sure-footedness of the ass rather than to any want of intelligence on his part." (in "Excursis II", volume 1 of Heath's translation of

2434: 1712:(of finite length). Euclid, rather than discussing a ray as an object that extends to infinity in one direction, would normally use locutions such as "if the line is extended to a sufficient length", although he occasionally referred to "infinite lines". A "line" for Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary. 2734:. Certain practical results from Euclidean geometry (such as the right-angle property of the 3-4-5 triangle) were used long before they were proved formally. The fundamental types of measurements in Euclidean geometry are distances and angles, both of which can be measured directly by a surveyor. Historically, distances were often measured by chains, such as 3194:, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an 2241: 1994:

angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is

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Congruence of triangles is determined by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles unless

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Points are customarily named using capital letters of the alphabet. Other figures, such as lines, triangles, or circles, are named by listing a sufficient number of points to pick them out unambiguously from the relevant figure, e.g., triangle ABC would typically be a triangle with vertices at points

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with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are

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mentions another interpretation. This rests on the resemblance of the figure's lower straight lines to a steeply inclined bridge that could be crossed by an ass but not by a horse: "But there is another view (as I have learnt lately) which is more complimentary to the ass. It is that, the figure of

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were self-evident statements about physical reality. Euclid's proofs depend upon assumptions perhaps not obvious in Euclid's fundamental axioms, in particular that certain movements of figures do not change their geometrical properties such as the lengths of sides and interior angles, the so-called

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To the ancients, the parallel postulate seemed less obvious than the others. They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one

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Euclid's axioms: In his dissertation to Trinity College, Cambridge, Bertrand Russell summarized the changing role of Euclid's geometry in the minds of philosophers up to that time. It was a conflict between certain knowledge, independent of experiment, and empiricism, requiring experimental input.

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lines" (book I, proposition 12). However, he typically did not make such distinctions unless they were necessary. The postulates do not explicitly refer to infinite lines, although for example some commentators interpret postulate 3, existence of a circle with any radius, as implying that space is

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Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two

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in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space. Although he described a

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that, given a line segment, one may construct an equilateral triangle that includes the segment as one of its sides: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the

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can construct consistent systems of geometry (obeying the other axioms) in which the parallel postulate is true, and others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the

3184:. In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two, while doubling a cube requires the solution of a third-order equation. 6344:(410–485 CE), treated many questions about infinity as issues demanding proof and, e.g., Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it. 1876:(book I, proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). 1618:. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a 1891:

states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I, Prop. 32 after the manner of Euclid Book III, Prop. 31.

6116:'s algebra and revealing the geometric nature of these systems, especially in four dimensions. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modeled to new positions. The 1641:, intuitive, or practically useful. Euclid's constructive proofs often supplanted fallacious nonconstructive ones, e.g. some Pythagorean proofs that assumed all numbers are rational, usually requiring a statement such as "Find the greatest common measure of ..." 6286:

The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infinite (see below) and what its

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Within Euclid's assumptions, it is quite easy to give a formula for area of triangles and squares. However, in a more general context like set theory, it is not as easy to prove that the area of a square is the sum of areas of its pieces, for example. See

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is. Modern, more rigorous reformulations of the system typically aim for a cleaner separation of these issues. Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1–4 are consistent with either infinite or finite space (as in

6256:, which include translations, reflections and rotations of figures. Taken as a physical description of space, postulate 2 (extending a line) asserts that space does not have holes or boundaries; postulate 4 (equality of right angles) says that space is 1681:. Supplementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the straight angle (180 degree angle). The number of rays in between the two original rays is infinite. 3171:

with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until

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Ancient geometers may have considered the parallel postulate – that two parallel lines do not ever intersect – less certain than the others because it makes a statement about infinitely remote regions of space, and so cannot be physically verified.

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Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763, at least 28 different proofs had been published, but all were found incorrect.

1966:. Euclid proved these results in various special cases such as the area of a circle and the volume of a parallelepipedal solid. Euclid determined some, but not all, of the relevant constants of proportionality. For instance, it was his successor 1506:: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. 1745: 1437:

The parallel postulate (Postulate 5): If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far

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follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

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The sum of the angles of a triangle is equal to a straight angle (180 degrees). This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one

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is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost.

5827: 1454:, these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality. 5901: 5495: 6569:, and not about some one or more particular things, then our deductions constitute mathematics. Thus, mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. 1674:. Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays is infinite. 1828:

of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross.

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of the objects defined within the formal system, rather than instances of those objects. For example, a Euclidean straight line has no width, but any real drawn line will have. Though nearly all modern mathematicians consider

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axioms from which the most important geometric theorems could be deduced. The outstanding objectives were to make Euclidean geometry rigorous (avoiding hidden assumptions) and to make clear the ramifications of the parallel

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Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example, "In any triangle, two angles taken together in any manner are less than two right angles." (Book I proposition 17) and the

6157:. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself. The very first geometric proof in the 5744: 5419: 6085: 6035: 5231: 5558: 5356: 1626:, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory. Strictly speaking, the lines on paper are 5681: 5282: 3057: 4933: 6201:. The rays of starlight were bent by the Sun's gravity on their way to Earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry. 6150:, in which the parallel postulate is not valid. Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates. 5168: 5121: 5607: 6685:

for its logical basis, in contrast to Hilbert's axioms, which involve point sets. Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certain

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shown in the figure above, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third

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A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with short horizontal lines) were photographed during a solar

2423:, was integral to drafting practices. However, with the advent of modern CAD systems, such in-depth knowledge of these theorems is less necessary in contemporary design and manufacturing processes. 5074: 4762: 2775: 2038:

Euclid refers to a pair of lines, or a pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes are equal respectively, and similarly for angles. The stronger term "

4879: 6165:. His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the 5939: 4795: 3150:

Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized

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Giuseppe Veronese, On Non-Archimedean Geometry, 1908. English translation in Real Numbers, Generalizations of the Reals, and Theories of Continua, ed. Philip Ehrlich, Kluwer, 1994.

1342:, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects. This is in contrast to 6651:: Birkhoff proposed four postulates for Euclidean geometry that can be confirmed experimentally with scale and protractor. This system relies heavily on the properties of the 3112:

In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g.,

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in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another

1395:"In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." (Book I, proposition 47) 3715: 3705: 3695: 3676: 3666: 3656: 3637: 3627: 3617: 3598: 3588: 3578: 3559: 3549: 3539: 3520: 3510: 3500: 1308:

For more than two thousand years, the adjective "Euclidean" was unnecessary because Euclid's axioms seemed so intuitively obvious (with the possible exception of the

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was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e.g., the proof of the infinitude of primes.

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systems, Euclidean geometry is fundamental for creating accurate 3D models of mechanical parts. These models are crucial for visualizing and testing designs before

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Schläfli performed this work in relative obscurity and it was published in full only posthumously in 1901. It had little influence until it was rediscovered and

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Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of

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is the simplest and most symmetric flat embedding of the Cartesian product of two circles (in the same sense that the surface of a cylinder is "flat").

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A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A sphere and cylinder were placed on the tomb of Archimedes at his request.

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Because of Euclidean geometry's fundamental status in mathematics, it is impractical to give more than a representative sampling of applications here.

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As suggested by the etymology of the word, one of the earliest reasons for interest in and also one of the most common current uses of geometry is

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Although Euclid explicitly only asserts the existence of the constructed objects, in his reasoning he also implicitly assumes them to be unique.

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In the 19th century, it was also realized that Euclid's ten axioms and common notions do not suffice to prove all of the theorems stated in the

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Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern

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We regard as elementary that part of Euclidean geometry which can be formulated and established without the help of any set-theoretical devices

7922:. Heath's authoritative translation of Euclid's Elements, plus his extensive historical research and detailed commentary throughout the text. 5181: 3139:, introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry, 5508: 1312:) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Today, however, many other 7309: 5313: 2047: 2022:

to them. The last figure is neither. Congruences alter some properties, such as location and orientation, but leave others unchanged, like

5638: 5239: 2946: 8844: 6472:, in which every proposition is either true or false, i.e., for any proposition P, the proposition "P or not P" is automatically true. 3259: 1207: 6231:

However, the three-dimensional "space part" of the Minkowski space remains the space of Euclidean geometry. This is not the case with

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The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists.

1252:) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a 8625: 8161: 6622:

was not necessarily valid and its applicability was an empirical matter, deciding whether the applicable geometry was Euclidean or

17: 5129: 5082: 8133: 5566: 9169: 8108: 2179:- In optical engineering, Euclidean geometry is critical in the design of lenses, where precise geometric shapes determine the 7106: 8899: 7800: 7768: 7442: 7391: 7343: 6909: 6875: 6741: 307: 6690: 3143:, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced. 2731: 10232: 9783: 8864: 7271:

Birkhoff, G. D., 1932, "A Set of Postulates for Plane Geometry (Based on Scale and Protractors)", Annals of Mathematics 33.

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published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include

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of the undefined symbols; but..this interpretation can be ignored by the reader, who is free to replace it in his mind by

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Studies in Logic and the Foundations of Mathematics – The Axiomatic Method with Special Reference to Geometry and Physics

7236:; "It is actually Cayley whom we must thank for the correct development of quaternions as a representation of rotations." 2787: 1755:

states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees.

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Placing Euclidean geometry on a solid axiomatic basis was a preoccupation of mathematicians for centuries. The role of

4849: 10000: 7961: 7919: 7911: 7903: 7854: 7820: 7739: 7710: 7683: 7650: 7600: 7554: 7415: 7371: 7171: 2163: 273: 6398:, predated Euclid. Euclid avoided such discussions, giving, for example, the expression for the partial sums of the 2781:

The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry.

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Essai d'une théorie algébrique des nombre entiers, avec une Introduction logique à une théorie déductive quelconque

3286: 1421:. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The 5909: 3222:

systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.

9878: 2619: 2391:, and other similar Euclidean forms. Today, CAD/CAM is essential in the design of a wide range of products, from 1900:

In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions,

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models of Euclidean geometry, in which the distance between two points may be infinite or infinitesimal, in the

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between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).

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utilizes Euclidean geometry for the efficient placement and routing of components, ensuring functionality while

9976: 9268: 9201: 8834: 8714: 6689:: there is an algorithm that, for every proposition, can be shown either true or false. (This does not violate 6224:. This shows that non-Euclidean geometries, which had been introduced a few years earlier for showing that the 2754: 2315: 1200: 1154: 760: 219: 4770: 2507: 1587:, through a point not on a given straight line, at most one line can be drawn that never meets the given line. 1402:, with numbers treated geometrically as lengths of line segments or areas of surface regions. Notions such as 9406: 8095: 8077: 7432: 7302: 7224:, the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions. 6544:

that result when...the system of undefined symbols is successively replaced by each of the interpretations...

6143: 2152: 2622:- Euclidean geometry is integral in using Jacobian matrices for transformations and control systems in both 10276: 9988: 9337: 9094: 8982: 8157: 2818: 2372: 7763:(Proceedings of International Symposium at Berkeley 1957–8; Reprint ed.). Brouwer Press. p. 16. 10271: 9903: 9459: 9046: 8977: 8090: 8072: 6726: 6426: 2918: 2677: 8085: 8067: 6173:

in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of

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If equals are subtracted from equals, then the differences are equal (subtraction property of equality).

9474: 8673: 8489: 7595:. Vol. 3 (Reprint of Simon and Schuster 1956 ed.). Courier Dover Publications. p. 1577. 2834: 2076: 1175: 785: 8464: 3219: 2495:

helps in designing antennas, where the spatial arrangement and dimensions directly affect antenna and

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relies on Euclidean geometry. The design geometry in CAD/CAM typically consists of shapes bounded by

1193: 6296:), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a 2630:

fields, providing insights into system behavior and properties. The Jacobian serves as a linearized

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That is, mathematics is context-independent knowledge within a hierarchical framework. As said by

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copied onto the end of another line segment to extend its length, and similarly for subtraction.

1546: 1328: 588: 268: 125: 8574: 7700: 7306: 4456: 4426: 1622:. In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as 10128: 10027: 9916: 9813: 9793: 9788: 9717: 9442: 9327: 9275: 9174: 9002: 8952: 8937: 8932: 8703: 8504: 8439: 8429: 8379: 7934: 7846: 7840: 7588: 7204: 7200: 6778: 6751: 6623: 6465: 6402:

in IX.35 without commenting on the possibility of letting the number of terms become infinite.

6221: 6147: 6133: 5822:{\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118} 3231: 3106: 2890: 2627: 2623: 2420: 1645: 1634: 1465:(axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath): 1451: 1316: 664: 375: 253: 138: 7815:

Franzén, Torkel (2005). Gödel's Theorem: An Incomplete Guide to its Use and Abuse. AK Peters.

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are known, the first ones having been discovered in the early 19th century. An implication of

10199: 9943: 9873: 9750: 9674: 9613: 9598: 9593: 9570: 9452: 9375: 9227: 9071: 9012: 8889: 8812: 8760: 8562: 8469: 8319: 7756: 7634: 7381: 7327: 6261: 5896:{\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146} 5490:{\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366} 3351: 2567: 2544: 2532: 2451: 2356: 2043: 2039: 2019: 436: 397: 356: 351: 204: 7727: 7617: 6865: 10059: 9923: 9803: 9798: 9722: 9623: 9352: 9292: 9256: 9099: 8922: 8869: 8839: 8829: 8738: 8601: 8494: 8409: 8364: 8344: 8189: 8174: 6798: 6721: 6686: 6648: 6360: 6352: 6334: 6280: 6178: 3452: 3343: 3195: 2867: 2599: 2500: 2470: 2156: 2080: 1705: 1678: 1671: 1628: 1370: 1331:

is a good approximation for it only over short distances (relative to the strength of the

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Euclid sometimes distinguished explicitly between "finite lines" (e.g., Postulate 2) and "

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interactions in three-dimensional space. The relationship of which is characterized by an

2091: 1450:("true statements") are derived from a small number of simple axioms. Until the advent of 8: 10168: 10163: 10148: 10143: 10054: 9938: 9848: 9770: 9669: 9603: 9560: 9550: 9530: 9332: 9251: 9239: 9220: 9184: 9104: 9022: 9007: 8997: 8947: 8942: 8884: 8753: 8645: 8509: 8499: 8399: 8369: 8309: 8284: 8209: 8199: 8184: 7212: 6808: 6731: 6629: 6446: 6269: 6174: 3438: 3331: 3322: 3181: 3168: 3140: 2933: 2873: 2830: 2727: 2635: 2416: 2327: 2140: 2051: 1873: 1769: 1638: 1576: 1572: 1532:

If equals are added to equals, then the wholes are equal (Addition property of equality).

1522: 1392: 1109: 1053: 966: 820: 800: 725: 615: 486: 476: 339: 214: 209: 192: 167: 155: 107: 102: 83: 6496:...when we begin to formulate the theory, we can imagine that the undefined symbols are 5290: 2018:

An example of congruence. The two figures on the left are congruent, while the third is

10194: 10153: 9964: 9883: 9823: 9755: 9745: 9659: 9535: 9492: 9487: 9388: 9347: 9287: 9215: 9081: 9056: 8874: 8817: 8655: 8414: 8359: 8324: 8219: 8048: 7995: 7939: 6942: 6793: 6761: 6736: 6698: 6666: 6619: 6388: 6329:, but nobody had been able to put them on a firm logical basis, with paradoxes such as 6265: 6232: 6225: 6209: 6182: 5970: 5947: 5615: 4826: 4803: 4702: 4679: 4656: 4633: 4610: 4587: 3300: 2512: 2229: 2148: 2116: 2072: 1983: 1884: 1798: 1614: 1558: 1526: 1339: 1324: 1309: 1282: 1258: 1068: 795: 635: 263: 187: 177: 148: 60: 7208: 6166: 4545: 4538: 4462: 4438: 4432: 3462: 3266: 10250: 10138: 10096: 10086: 10020: 9959: 9679: 9664: 9608: 9555: 9383: 9061: 8972: 8822: 8748: 8721: 8484: 8304: 8294: 8229: 8149: 7957: 7950: 7926: 7915: 7907: 7899: 7890: 7850: 7816: 7796: 7764: 7735: 7706: 7679: 7646: 7596: 7550: 7438: 7411: 7387: 7367: 7339: 7167: 6905: 6871: 6803: 6788: 6783: 6716: 6711: 6678: 6454: 6395: 6368: 6356: 6330: 6293: 6113: 6109: 3177: 2911: 2735: 2108: 1888: 1411: 1355: 1343: 1224: 1139: 927: 905: 830: 689: 415: 344: 236: 182: 143: 8052: 4559: 2907: 2338: 1538:

Things that coincide with one another are equal to one another (reflexive property).

1347: 1129: 1058: 855: 765: 10101: 9893: 9868: 9740: 9588: 9525: 9263: 9134: 9051: 8807: 8795: 8743: 8479: 8102: 8038: 8030: 7864: 7836: 7110: 6970: 6934: 6756: 6694: 6558: 6485: 6481: 6450: 6399: 6372: 6138:

The century's most influential development in geometry occurred when, around 1830,

3359: 3132: 2651: 2551: 2463: 2376: 2184: 1584: 1479: 1443: 1290: 1278: 1274: 1119: 860: 570: 448: 383: 241: 226: 91: 2562: 10220: 10158: 10091: 9833: 9760: 9689: 9482: 8904: 8894: 8788: 8514: 8112: 7313: 6583: 6469: 6276: 6217: 6205: 3487: 3318: 3270: 3255: 3191: 2844: 2746: 2647: 2591: 2488: 2144: 1999: 1701: 1407: 1320: 1313: 542: 405: 248: 231: 172: 78: 7359: 6604: 6333:

occurring that had not been resolved to universal satisfaction. Euclid used the

6139: 1824:. Its name may be attributed to its frequent role as the first real test in the 1114: 1083: 1017: 865: 810: 745: 10118: 9911: 9838: 9545: 9164: 9159: 8987: 8879: 8859: 8687: 8244: 8214: 7972: 6489: 6326: 6322: 6117: 6098: 5739:{\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693} 5414:{\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48} 4573: 4566: 4532: 3297:. He developed their theory and discovered all the regular polytopes, i.e. the 3251: 3239: 3187: 3173: 2750: 2702: 2698: 2587: 2583: 2540: 2536: 2524: 2458:. Efficient layout of electronic components on PCBs is critical for minimizing 2331: 2283: 2180: 2136: 1991: 1690: 1422: 1418: 1286: 1170: 1078: 1022: 987: 895: 805: 775: 735: 640: 8034: 4552: 1240:. Euclid's approach consists in assuming a small set of intuitively appealing 1144: 755: 10265: 9699: 9631: 9583: 9124: 8992: 8962: 8783: 8591: 8534: 7982: 7930: 6670: 4517: 4500: 3338: 3243: 3215: 3210:

Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions

2639: 2631: 2528: 2496: 2492: 2360: 2323: 1813: 1731: 1575:

to the parallel postulate (in the context of the other axioms). For example,

1475: 1403: 1399: 1302: 1149: 1134: 1063: 880: 840: 790: 565: 528: 495: 333: 329: 8118: 6080:{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193} 6030:{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863} 4450: 4444: 2290: 1970:

who proved that a sphere has 2/3 the volume of the circumscribing cylinder.

10184: 9641: 9636: 9540: 9066: 8854: 8529: 8289: 8279: 6746: 6652: 6468:, and therefore the traditional presentation of Euclidean geometry assumes 6364: 6170: 4492: 3206: 3101: 2799: 2595: 2548: 2455: 2279: 2260: 2256: 2245: 1857: 1709: 1521:

Things that are equal to the same thing are also equal to one another (the

1486: 1414:

are introduced. It is proved that there are infinitely many prime numbers.

1088: 1037: 850: 705: 620: 410: 9398: 7410:(Reprint of 1939 Macmillan Company ed.). Courier Dover. p. 167. 4524: 4510: 4355: 1847:

equal sides and an adjacent angle are not necessarily equal or congruent.

1739:

states that in an isosceles triangle, α = β and γ = δ.

9843: 9507: 9430: 8680: 8568: 8274: 8259: 7885: 7255: 7011: 6766: 3386: 3367: 2684: 1987: 1861: 1500: 1351: 1245: 1124: 997: 815: 750: 678: 650: 625: 6414: 6193: 6162: 2240: 2111:, relies heavily on Euclidean geometry to ensure proper tooth shape and 1433: 9828: 9707: 9502: 8666: 8585: 8524: 8519: 8459: 8444: 8389: 8374: 8329: 8269: 8254: 8234: 8204: 8169: 8043: 7216: 7196: 6946: 6682: 6348: 6213: 5226:{\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825} 3400: 3294: 3235: 2895: 2852: 2739: 2659: 2655: 2606: 2404: 2311: 2233: 2225: 2112: 2084: 1967: 1781:) of a right triangle equals the area of the square on the hypotenuse ( 1623: 982: 961: 951: 941: 900: 845: 740: 730: 630: 481: 56: 5553:{\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329} 3146: 2732:

cognitive and computational approaches to visual perception of objects

2232:, aiding in the design of systems that can withstand or utilize these 1837: 1802:

states that if AC is a diameter, then the angle at B is a right angle.

10189: 8419: 8404: 8354: 8249: 8239: 8224: 8194: 6862:

The assumptions of Euclid are discussed from a modern perspective in

6697:

for the theorem to apply.) This is equivalent to the decidability of

6257: 6228:

cannot be proved, are also useful for describing the physical world.

5351:{\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569} 2723: 2415:: Historically, advanced Euclidean geometry, including theorems like 2319: 2275: 2224:- Euclidean geometry is essential in analyzing and understanding the 2221: 2007: 1568:: his first 28 propositions are those that can be proved without it. 1462: 992: 710: 673: 537: 509: 6938: 6693:, because Euclidean geometry cannot describe a sufficient amount of 5676:{\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314} 5277:{\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713} 4324: 4317: 3052:{\displaystyle |PQ|={\sqrt {(p_{x}-q_{x})^{2}+(p_{y}-q_{y})^{2}}}\,} 2123: 10111: 10043: 9732: 9651: 9578: 8556: 8334: 8179: 6925:

Florence P. Lewis (Jan 1920), "History of the Parallel Postulate",

6314: 6288: 6121: 3446: 3432: 3282: 3247: 2396: 2380: 2264: 2023: 1637:

just as sound as constructive ones, they are often considered less

1231: 1073: 1032: 1002: 890: 885: 835: 560: 519: 467: 361: 324: 70: 37: 7632: 6599:

Geometry is the science of correct reasoning on incorrect figures.

10106: 9517: 9129: 8454: 8449: 8349: 8339: 8314: 7434:

The Road to Reality: A Complete Guide to the Laws of the Universe

6341: 6198: 3422: 3394: 3290: 2929:) coordinates, a line is represented by its equation, and so on. 2806: 2271: 2267: 2196: 1773:

states that the sum of the areas of the two squares on the legs (

1447: 1332: 1298: 1249: 1007: 720: 514: 458: 258: 42: 8115:(a treatment using analytic geometry; PDF format, GFDL licensed) 7569: 4310: 8424: 8299: 7379: 7166:

Theorem 120, Elements of Abstract Algebra, Allan Clark, Dover,

6933:(1), The American Mathematical Monthly, Vol. 27, No. 1: 16–23, 6248: 4928:{\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270} 4289: 3408: 3380: 2003: 1694: 1594: 1493: 1227: 956: 946: 825: 770: 645: 608: 596: 551: 504: 422: 87: 52: 7227: 2014: 2011:

implied, for example in the proof of book IX, proposition 20.

1978:

Euclidean geometry has two fundamental types of measurements:

10215: 8800: 8778: 8434: 6297: 6242: 2388: 2307: 2306:- Euclidean geometry helps in calculating and predicting the 2303: 2188: 2027: 1979: 1700:

Modern school textbooks often define separate figures called

1253: 1241: 1012: 936: 870: 715: 319: 314: 8016:"On Cayley's Factorization of 4D Rotations and Applications" 7845:(4th ed. ed.). New York: Dover Publications. pp. 7407:

Elementary Mathematics from an Advanced Standpoint: Geometry

6677:

Euclidean geometry as the geometry that can be expressed in

6536:

The system of undefined symbols can then be regarded as the

6375:

provided a rigorous logical foundation for Veronese's work.

5163:{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926} 5116:{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926} 4303: 4296: 2717:

A parabolic mirror brings parallel rays of light to a focus.

2433: 2203: 1689:

In modern terminology, angles would normally be measured in

51:

featuring a Greek mathematician – perhaps representing

10012: 7360:

Luciano da Fontoura Costa; Roberto Marcondes Cesar (2001).

7195:, p. 18–21; In four-dimensional Euclidean geometry, a 6260:

and figures may be moved to any location while maintaining

5602:{\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333} 3414: 2917:

In this approach, a point on a plane is represented by its

2887:) is mainly known for his investigation of conic sections. 2400: 2384: 2192: 2176: 2104: 603: 453: 7186: 7010:

Ignoring the alleged difficulty of Book I, Proposition 5,

2079:

in mechanical components, which is essential for ensuring

2392: 1665: 7759:. In Leon Henkin; Patrick Suppes; Alfred Tarski (eds.). 3190:

discussed a generalization of Euclidean geometry called

2478: 2147:, where the geometric configuration greatly influences 1973: 1715: 1684: 7970: 7363:

Shape analysis and classification: theory and practice

7001:

Euclid, book I, proposition 5, tr. Heath, p. 251.

6618:

This issue became clear as it was discovered that the

6394:

Supposed paradoxes involving infinite series, such as

6048: 5998: 5914: 5864: 5845: 5764: 5701: 5650: 5578: 5520: 5439: 5376: 5326: 5252: 5193: 5135: 5088: 5048: 5015: 4983: 4951: 4892: 4854: 4736: 7987:

A Decision Method for Elementary Algebra and Geometry

6046: 5996: 5973: 5950: 5912: 5843: 5755: 5692: 5641: 5618: 5569: 5511: 5430: 5367: 5316: 5293: 5242: 5184: 5132: 5085: 5045: 5013: 4981: 4949: 4890: 4852: 4829: 4806: 4773: 4733: 4705: 4682: 4659: 4636: 4613: 4590: 3303: 2949: 2738:, and angles using graduated circles and, later, the 2535:, Euclidean geometry aids in visualizing and solving 1939: 1906: 30:"Plane geometry" redirects here. For other uses, see 6701:, of which elementary Euclidean geometry is a model. 6484:, or undefined concepts, was clearly put forward by 5069:{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707} 4757:{\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581} 2749:, such as the problem of finding the most efficient 1571:

Many alternative axioms can be formulated which are

1327:

is that physical space itself is not Euclidean, and

7925: 7260:

Foundations and Fundamental Concepts of Mathematics

1986:. The angle scale is absolute, and Euclid uses the 8013: 7949: 7896:(2nd ed. ed.). New York: Dover Publications. 7889: 7325: 7233: 6079: 6029: 5979: 5956: 5933: 5895: 5821: 5738: 5675: 5624: 5601: 5552: 5489: 5413: 5350: 5299: 5276: 5225: 5162: 5115: 5068: 5028: 4996: 4964: 4927: 4873: 4835: 4812: 4789: 4756: 4711: 4688: 4665: 4642: 4619: 4596: 3309: 3051: 2753:in n dimensions. This problem has applications in 2745:An application of Euclidean solid geometry is the 1958: 1925: 1517:also include the following five "common notions": 7725: 6924: 6632:: Hilbert's axioms had the goal of identifying a 6325:had previously been discussed extensively by the 2107:- The design of gears, a crucial element in many 10263: 7615: 7586: 7505:Robinson, Abraham (1966). Non-standard analysis. 7456: 7454: 7352: 7292:Misner, Thorne, and Wheeler (1973), p. 191. 6525:questions thus become completely independent of 4874:{\displaystyle {\tfrac {1}{\phi }}\approx 0.618} 1549:use more extensive and complete sets of axioms. 7702:Elementary geometry from an advanced standpoint 7199:is simply a (w, x, y, z) Cartesian coordinate. 6863: 6853:Misner, Thorne, and Wheeler (1973), p. 47. 3317:-dimensional analogues of regular polygons and 2075:- Euclidean geometry is pivotal in determining 1656: 1223:is a mathematical system attributed to ancient 7698: 7665: 7536: 7279: 7277: 6500:and that the unproved propositions are simply 6188: 2539:problems. This is essential for understanding 1990:as his basic unit, so that, for example, a 45- 10028: 9414: 8148: 8134: 8014:Perez-Gracia, Alba; Thomas, Federico (2017). 7842:A Short Account of the History of Mathematics 7786: 7754: 7633:George David Birkhoff; Ralph Beatley (1999). 7451: 7430: 7402:underlie the metric notions of geometry. See 7222:ordered four-element multiple of real numbers 7095:Eves, vol. 1, p. 5; Mlodinow, p. 7. 6989:Solved and Unsolved Problems in Number Theory 6986: 2666: 2034:and studying them is the essence of geometry. 1895: 1670:Angles whose sum is a right angle are called 1201: 7635:"Chapter 2: The five fundamental principles" 7573:Revue de métaphysique et de morale, Volume 8 6475: 6359:, and others produced controversial work on 6169:property of the real numbers. Starting with 5934:{\displaystyle {\tfrac {2}{3}}\approx 0.667} 2839: 1457:Near the beginning of the first book of the 1354:to express geometric properties by means of 1262:from axioms and previously proved theorems. 9428: 7641:(3rd ed.). AMS Bookstore. pp. 38 7403: 7274: 6582:Such foundational approaches range between 2811:classical construction problems of geometry 2586:- The application of Euclidean geometry in 2259:- The application of Euclidean geometry in 2030:. The latter sort of properties are called 2006:are derived from distances. For example, a 1651: 10035: 10021: 9421: 9407: 8141: 8127: 7668:"Chapter 3: Elementary Euclidean Geometry" 7380:Helmut Pottmann; Johannes Wallner (2010). 7215:, publishing his discovery of the regular 6825: 6823: 6243:As a description of the structure of space 3323:regular convex polytopes in dimension four 3260:rotations in 4-dimensional Euclidean space 2902: 2499:performance in transmitting and receiving 2050:in a pair of similar shapes are equal and 1832: 1208: 1194: 77: 8042: 7993: 7719: 7616:Bertrand Russell (1897). "Introduction". 7570:Société française de philosophie (1900). 7192: 6962: 6593: 6492:delegation at the 1900 Paris conference: 6300:for two-dimensional Euclidean geometry). 6127: 3062:defining the distance between two points 3048: 2769:Geometry is used in art and architecture. 1933:, and the volume of a solid to the cube, 1677:Angles whose sum is a straight angle are 1346:, introduced almost 2,000 years later by 7947: 7626: 7531: 7529: 6511:that we have initially chosen is simply 6303: 6192: 4790:{\displaystyle {\sqrt {2}}\approx 1.414} 3205: 3145: 2889: 2843: 2605: 2561: 2506: 2469: 2432: 2337: 2289: 2239: 2202: 2122: 2090: 2013: 1836: 1593: 1432: 1230:, which he described in his textbook on 36: 27:Mathematical model of the physical space 7892:The Thirteen Books of Euclid's Elements 7863: 7791:. In Dov M. Gabbay; John Woods (eds.). 7780: 7748: 7619:An essay on the foundations of geometry 7609: 7424: 7319: 6820: 6112:, unifying Hamilton's quaternions with 2793:Geometry can be used to design origami. 2566:Potential Flow Around a Source without 2278:where geometric shape directly impacts 2143:, Euclidean geometry is used to design 14: 10264: 9170:Latin translations of the 12th century 7981: 7659: 7580: 7535:A detailed discussion can be found in 6899: 6867:Introduction to Non-Euclidean Geometry 3325:, and three in all higher dimensions. 2062: 1867: 1666:Complementary and supplementary angles 1496:with any centre and distance (radius). 1297:states results of what are now called 308:Straightedge and compass constructions 10016: 9402: 8900:Straightedge and compass construction 8122: 7884: 7692: 7674:. John Wiley & Sons. pp. 84 7563: 7526: 7107:"Origami and Geometric Constructions" 6772: 6742:List of interactive geometry software 6673:(1902–1983) and his students defined 6464:Euclid frequently used the method of 6378: 2835:Non-Euclidean geometry § History 2747:determination of packing arrangements 2479:Electromagnetic and Fluid Flow Fields 1850: 1842:the angle specified is a right angle. 1552: 1305:, explained in geometrical language. 1244:(postulates) and deducing many other 10233:List of differential geometry topics 9983: 8865:Incircle and excircles of a triangle 7875: 7835: 7589:"Mathematics and the metaphysicians" 7104: 6856: 6841: 6829: 6519:.. that satisfies the conditions... 6504:imposed upon the undefined symbols. 6409: 6340:Later ancient commentators, such as 3225: 2594:, particularly in understanding and 2590:helps in the analysis and design of 2167:U-Tube Shell and Tube Heat Exchanger 1974:System of measurement and arithmetic 1716:Some important or well known results 1685:Modern versions of Euclid's notation 1620:compass and an unmarked straightedge 1338:Euclidean geometry is an example of 9995: 6308: 2932:In Euclid's original approach, the 2515:, Extremely high gain ~70 dBi. 1607: 1541:The whole is greater than the part. 24: 7666:James T. Smith (10 January 2000). 7514:Nagel and Newman, 1958, p. 9. 7017:The Thirteen Books of the Elements 6576:Mathematics and the metaphysicians 6459: 6283:significantly modifies this view. 3273:, extending Euclidean geometry to 2726:. In addition it has been used in 2452:Printed Circuit Board (PCB) Design 2373:CAM (computer-aided manufacturing) 2342:Animation of Orbit by Eccentricity 2162: 1879: 25: 10293: 8060: 7971:Nagel, E.; Newman, J. R. (1958). 7878:A Survey of Geometry (Volume One) 7203:did not see them as such when he 6927:The American Mathematical Monthly 6347:At the turn of the 20th century, 3269:developed the general concept of 2441: 2057: 2054:are in proportion to each other. 274:Noncommutative algebraic geometry 63:to draw a geometric construction. 10246: 10245: 9994: 9982: 9971: 9970: 9958: 9382: 9369: 7994:Stillwell, John (January 2001). 7705:(3rd ed.). Addison–Wesley. 7213:four-dimensional Euclidean space 7068:Euclid, book XI, proposition 33. 7059:Euclid, book XII, proposition 2. 6991:. American Mathematical Society. 6413: 6405: 4323: 4316: 4309: 4302: 4295: 4288: 3718: 3713: 3708: 3703: 3698: 3693: 3688: 3679: 3674: 3669: 3664: 3659: 3654: 3649: 3640: 3635: 3630: 3625: 3620: 3615: 3610: 3601: 3596: 3591: 3586: 3581: 3576: 3571: 3562: 3557: 3552: 3547: 3542: 3537: 3532: 3523: 3518: 3513: 3508: 3503: 3498: 3493: 2824: 2798:Geometry is used extensively in 2786: 2774: 2762: 2710: 2691: 2676: 1807: 1790: 1760: 1744: 1722: 1489:continuously in a straight line. 1469:Let the following be postulated: 9879:Computational complexity theory 7809: 7517: 7508: 7499: 7490: 7481: 7472: 7463: 7295: 7286: 7265: 7248: 7239: 7211:would be the first to consider 7177: 7160: 7151: 7142: 7133: 7124: 7098: 7089: 7080: 7071: 7062: 7053: 7044: 7031: 7028:Euclid, book I, proposition 32. 7022: 7004: 6995: 6980: 5029:{\displaystyle {\tfrac {1}{2}}} 4997:{\displaystyle {\tfrac {1}{2}}} 4965:{\displaystyle {\tfrac {1}{4}}} 3201: 3158: 2894:René Descartes. Portrait after 2805:Geometry can be used to design 2646:. The Jacobian is also used in 2413:Evolution of Drafting Practices 1361: 32:Plane geometry (disambiguation) 9202:A History of Greek Mathematics 8715:The Quadrature of the Parabola 7234:Perez-Gracia & Thomas 2017 6953: 6918: 6893: 6884: 6847: 6835: 6108:introduced what is now termed 3037: 3010: 2998: 2971: 2962: 2951: 2755:error detection and correction 2346: 2153:shell-and-tube heat exchangers 1959:{\displaystyle V\propto L^{3}} 1926:{\displaystyle A\propto L^{2}} 667:- / other-dimensional 13: 1: 7829: 7757:"What is elementary geometry" 7622:. Cambridge University Press. 7591:. In James Roy Newman (ed.). 7437:. Vintage Books. p. 29. 7303:University of New South Wales 6268:) that space is flat (has no 6146:separately published work on 6144:Nikolai Ivanovich Lobachevsky 3135:, motivated by the theory of 2877: 2856: 10042: 8983:Intersecting secants theorem 8023:Adv. Appl. Clifford Algebras 7989:. Univ. of California Press. 7977:. New York University Press. 7793:Logic from Russell to Church 7732:Geometry: ancient and modern 7332:The Non-Euclidean Revolution 7157:Hofstadter 1979, p. 91. 6904:, Prentice-Hall, p. 8, 6498:completely devoid of meaning 6337:rather than infinitesimals. 6212:involves a four-dimensional 3287:higher-dimensional analogues 1657:Naming of points and figures 1398:Books V and VII–X deal with 7: 8978:Intersecting chords theorem 8845:Doctrine of proportionality 8091:Encyclopedia of Mathematics 8073:Encyclopedia of Mathematics 7996:"The Story of the 120-Cell" 7734:. Oxford University Press. 7728:"§1.4 Hilbert and Birkhoff" 7523:Cajori (1918), p. 197. 7383:Computational Line Geometry 7326:Richard J. Trudeau (2008). 6727:Cartesian coordinate system 6705: 6565:If our hypothesis is about 6189:20th century and relativity 3214:In the early 19th century, 3105:, and other metrics define 2573: 2357:CAD (computer-aided design) 2314:, essential for successful 2211: 1277:(high school) as the first 10: 10298: 9929:Films about mathematicians 8674:On the Sphere and Cylinder 8627:On the Sizes and Distances 7726:John R. Silvester (2001). 7366:. CRC Press. p. 314. 7334:. Birkhäuser. pp. 39 7205:discovered the quaternions 6900:Venema, Gerard A. (2006), 6663:become primitive concepts. 6444: 6387:The modern formulation of 6371:sense. Fifty years later, 6131: 3332:Regular convex 4-polytopes 3131:Also in the 17th century, 2828: 2667:Other general applications 2426: 1896:Scaling of area and volume 1753:triangle angle sum theorem 1556: 1383:There are 13 books in the 1368: 1281:and the first examples of 29: 10241: 10208: 10177: 10127: 10075: 10050: 9952: 9902: 9859: 9769: 9731: 9698: 9650: 9622: 9569: 9516: 9498:Philosophy of mathematics 9473: 9438: 9376:Ancient Greece portal 9365: 9315: 9193: 9180:Philosophy of mathematics 9150: 9143: 9117: 9095:Ptolemy's table of chords 9039: 9021: 8920: 8913: 8769: 8731: 8548: 8156: 8150:Ancient Greek mathematics 8035:10.1007/s00006-016-0683-9 7795:. Elsevier. p. 574. 7587:Bertrand Russell (2000). 6870:. Mill Press. p. 9. 6476:Modern standards of rigor 6247:Euclid believed that his 3366: 3350: 3330: 3321:. He found there are six 2840:Archimedes and Apollonius 2556:conservative vector field 2187:analyzes the focusing of 1503:are equal to one another. 1442:Euclidean geometry is an 1428: 9934:Recreational mathematics 9047:Aristarchus's inequality 8620:On Conoids and Spheroids 7898:In 3 vols.: vol. 1 7869:Introduction to Geometry 7593:The world of mathematics 7576:. Hachette. p. 592. 7539:"Chapter 2: Foundations" 7386:. Springer. p. 60. 7183:Eves (1963), p. 64. 6864:Harold E. Wolfe (2007). 6814: 6323:infinitesimal quantities 6106:William Kingdon Clifford 6095:fully documented in 1948 3152:compass and straightedge 3107:non-Euclidean geometries 2815:compass and straightedge 2644:non-linear least squares 2369:Design and Manufacturing 1652:Notation and terminology 1317:non-Euclidean geometries 1256:in which each result is 163:Non-Archimedean geometry 18:Euclidean plane geometry 10228:List of geometry topics 9819:Mathematical statistics 9809:Mathematical psychology 9779:Engineering mathematics 9713:Algebraic number theory 9155:Ancient Greek astronomy 8968:Inscribed angle theorem 8958:Greek geometric algebra 8613:Measurement of a Circle 7935:Wheeler, John Archibald 7699:Edwin E. Moise (1990). 7537:James T. Smith (2000). 6902:Foundations of Geometry 6890:tr. Heath, pp. 195–202. 6681:and does not depend on 6264:; and postulate 5 (the 3098:) is then known as the 2903:17th century: Descartes 2580:Control System Analysis 2429:History of CAD software 1833:Congruence of triangles 1737:bridge of asses theorem 269:Noncommutative geometry 9965:Mathematics portal 9814:Mathematical sociology 9794:Mathematical economics 9789:Mathematical chemistry 9718:Analytic number theory 9599:Differential equations 9389:Mathematics portal 9175:Non-Euclidean geometry 9130:Mouseion of Alexandria 9003:Tangent-secant theorem 8953:Geometric mean theorem 8938:Exterior angle theorem 8933:Angle bisector theorem 8637:On Sizes and Distances 7787:Keith Simmons (2009). 7755:Alfred Tarski (2007). 7431:Roger Penrose (2007). 6987:Daniel Shanks (2002). 6779:Angle bisector theorem 6752:Non-Euclidean geometry 6614: 6594:Axiomatic formulations 6580: 6555: 6517:another interpretation 6466:proof by contradiction 6202: 6148:non-Euclidean geometry 6134:Non-Euclidean geometry 6128:Non-Euclidean geometry 6120:on the surface of the 6081: 6031: 5981: 5958: 5935: 5897: 5823: 5740: 5677: 5626: 5603: 5554: 5491: 5415: 5352: 5301: 5278: 5227: 5164: 5117: 5070: 5030: 4998: 4966: 4929: 4875: 4837: 4814: 4791: 4758: 4713: 4690: 4667: 4644: 4621: 4598: 3311: 3232:William Rowan Hamilton 3211: 3155: 3053: 2910:(1596–1650) developed 2899: 2849: 2701:applies to a stack of 2628:electrical engineering 2611: 2570: 2533:electromagnetic fields 2525:Complex Potential Flow 2516: 2475: 2438: 2343: 2295: 2248: 2208: 2168: 2128: 2096: 2035: 1960: 1927: 1843: 1646:proof by contradiction 1635:nonconstructive proofs 1612:Euclidean Geometry is 1604: 1598:A proof from Euclid's 1485:To produce (extend) a 1452:non-Euclidean geometry 1439: 1417:Books XI–XIII concern 237:Discrete/Combinatorial 64: 10200:Differential geometry 9944:Mathematics education 9874:Theory of computation 9594:Hypercomplex analysis 9077:Pappus's area theorem 9013:Theorem of the gnomon 8890:Quadratrix of Hippias 8813:Circles of Apollonius 8761:Problem of Apollonius 8739:Constructible numbers 8563:Archimedes Palimpsest 7876:Eves, Howard (1963). 7545:. Wiley. pp. 19 7307:GPS Satellite Signals 6975:Banach–Tarski paradox 6597: 6563: 6494: 6304:Treatment of infinity 6196: 6082: 6032: 5982: 5959: 5936: 5898: 5824: 5741: 5678: 5627: 5604: 5555: 5492: 5416: 5353: 5302: 5279: 5228: 5165: 5118: 5071: 5031: 4999: 4967: 4930: 4876: 4838: 4815: 4792: 4759: 4714: 4691: 4668: 4645: 4622: 4599: 3312: 3209: 3149: 3054: 2893: 2847: 2813:are impossible using 2609: 2598:system stability and 2565: 2545:electromagnetic field 2510: 2501:electromagnetic waves 2473: 2436: 2341: 2322:operations. Also see 2293: 2243: 2206: 2166: 2157:plate heat exchangers 2133:Heat Exchanger Design 2126: 2094: 2017: 1961: 1928: 1840: 1708:(semi-infinite), and 1597: 1436: 220:Discrete differential 40: 9924:Informal mathematics 9804:Mathematical physics 9799:Mathematical finance 9784:Mathematical biology 9723:Diophantine geometry 9293:prehistoric counting 9090:Ptolemy's inequality 9031:Apollonius's theorem 8870:Method of exhaustion 8840:Diophantine equation 8830:Circumscribed circle 8647:On the Moving Sphere 8086:"Plane trigonometry" 8068:"Euclidean geometry" 7469:e.g., Tarski (1951). 7404:Felix Klein (2004). 7037:Heath, p. 135. 6542:specialized theories 6353:Paul du Bois-Reymond 6335:method of exhaustion 6281:theory of relativity 6275:As discussed above, 6044: 5994: 5971: 5948: 5910: 5841: 5753: 5690: 5639: 5616: 5567: 5509: 5428: 5365: 5314: 5291: 5240: 5182: 5130: 5083: 5043: 5011: 4979: 4947: 4888: 4850: 4827: 4804: 4771: 4731: 4703: 4680: 4657: 4634: 4611: 4588: 3301: 3196:equivalence relation 2947: 2868:Archimedean property 2819:solved using origami 2610:Basic feedback loop. 2493:geometry of antennas 2294:Airfoil Nomenclature 2257:Aircraft Wing Design 2081:structural integrity 2048:Corresponding angles 1937: 1904: 1573:logically equivalent 1487:finite straight line 1461:, Euclid gives five 1285:. It goes on to the 48:The School of Athens 10277:Elementary geometry 9939:Mathematics and art 9849:Operations research 9604:Functional analysis 9379: • 9185:Neusis construction 9105:Spiral of Theodorus 8998:Pythagorean theorem 8943:Euclidean algorithm 8885:Lune of Hippocrates 8754:Squaring the circle 8510:Theon of Alexandria 8185:Aristaeus the Elder 7543:Methods of geometry 7487:Heath, p. 268. 7460:Heath, p. 200. 7050:Heath, p. 318. 7039:Extract of page 135 7012:Sir Thomas L. Heath 6809:Pythagorean theorem 6270:intrinsic curvature 3182:squaring the circle 3169:trisecting an angle 3141:projective geometry 2934:Pythagorean theorem 2874:Apollonius of Perga 2870:of finite numbers. 2831:History of geometry 2728:classical mechanics 2474:PCB of a DVD Player 2464:circuit performance 2460:signal interference 2421:Brianchon's theorem 2328:celestial mechanics 2141:thermal engineering 2077:stress distribution 2063:Design and Analysis 2052:corresponding sides 1874:Pythagorean theorem 1868:Pythagorean theorem 1770:Pythagorean theorem 1523:transitive property 1393:Pythagorean theorem 1333:gravitational field 1283:mathematical proofs 1225:Greek mathematician 487:Pythagorean theorem 10272:Euclidean geometry 10195:Algebraic geometry 9884:Numerical analysis 9493:Mathematical logic 9488:Information theory 9072:Menelaus's theorem 9062:Irrational numbers 8875:Parallel postulate 8850:Euclidean geometry 8818:Apollonian circles 8360:Isidore of Miletus 8111:2011-10-26 at the 8003:Notices of the AMS 7956:. The Free Press. 7927:Misner, Charles W. 7880:. Allyn and Bacon. 7871:. New York: Wiley. 7358:See, for example: 7312:2010-06-12 at the 7245:Ball, p. 485. 6773:Classical theorems 6762:Parallel postulate 6737:Incidence geometry 6699:real closed fields 6620:parallel postulate 6574:Bertrand Russell, 6540:obtained from the 6513:one interpretation 6425:. You can help by 6389:proof by induction 6379:Infinite processes 6266:parallel postulate 6233:general relativity 6226:parallel postulate 6210:special relativity 6203: 6077: 6069: 6027: 6019: 5977: 5954: 5931: 5923: 5893: 5875: 5856: 5819: 5807: 5736: 5724: 5673: 5661: 5622: 5599: 5587: 5550: 5538: 5487: 5475: 5411: 5399: 5348: 5335: 5300:{\displaystyle 24} 5297: 5274: 5261: 5223: 5211: 5160: 5151: 5113: 5104: 5066: 5057: 5026: 5024: 4994: 4992: 4962: 4960: 4925: 4917: 4871: 4863: 4833: 4810: 4787: 4754: 4745: 4709: 4686: 4663: 4640: 4617: 4594: 4525:irregular hexagons 3307: 3212: 3156: 3049: 2900: 2850: 2751:packing of spheres 2683:A surveyor uses a 2612: 2571: 2527:- In the study of 2517: 2476: 2439: 2344: 2296: 2249: 2230:mechanical systems 2218:Vibration Analysis 2209: 2169: 2149:thermal efficiency 2129: 2117:power transmission 2109:mechanical systems 2097: 2036: 1956: 1923: 1851:Triangle angle sum 1844: 1644:Euclid often used 1605: 1559:Parallel postulate 1553:Parallel postulate 1527:Euclidean relation 1440: 1412:irrational numbers 1356:algebraic formulas 1340:synthetic geometry 1325:general relativity 1310:parallel postulate 1273:, still taught in 1221:Euclidean geometry 65: 10259: 10258: 10010: 10009: 9609:Harmonic analysis 9396: 9395: 9361: 9360: 9113: 9112: 9100:Ptolemy's theorem 8973:Intercept theorem 8823:Apollonian gasket 8749:Doubling the cube 8722:The Sand Reckoner 7948:Mlodinow (2001). 7865:Coxeter, H. S. M. 7837:Ball, W. W. Rouse 7802:978-0-444-51620-6 7770:978-1-4067-5355-4 7478:Ball, p. 31. 7444:978-0-679-77631-4 7393:978-3-642-04017-7 7345:978-0-8176-4782-7 7328:"Euclid's axioms" 7220:quaternion as an 7130:Eves, p. 27. 7077:Ball, p. 66. 6959:Ball, p. 56. 6911:978-0-13-143700-5 6877:978-1-4067-1852-2 6804:Nine-point circle 6799:Menelaus' theorem 6784:Butterfly theorem 6722:Birkhoff's axioms 6717:Analytic geometry 6712:Absolute geometry 6679:first-order logic 6655:. The notions of 6649:Birkhoff's axioms 6482:primitive notions 6455:Real closed field 6443: 6442: 6357:Giuseppe Veronese 6294:elliptic geometry 6254:Euclidean motions 6114:Hermann Grassmann 6110:geometric algebra 6091: 6090: 6087: 6068: 6062: 6054: 6037: 6018: 6012: 6004: 5987: 5980:{\displaystyle 2} 5964: 5957:{\displaystyle 1} 5941: 5922: 5903: 5874: 5870: 5855: 5851: 5829: 5806: 5803: 5781: 5746: 5723: 5707: 5683: 5660: 5656: 5632: 5625:{\displaystyle 8} 5609: 5586: 5560: 5537: 5531: 5497: 5474: 5458: 5456: 5421: 5398: 5382: 5358: 5336: 5334: 5307: 5284: 5262: 5260: 5233: 5210: 5204: 5170: 5152: 5150: 5123: 5105: 5103: 5076: 5058: 5056: 5036: 5023: 5004: 4991: 4972: 4959: 4935: 4916: 4913: 4881: 4862: 4843: 4836:{\displaystyle 1} 4820: 4813:{\displaystyle 1} 4797: 4779: 4764: 4746: 4744: 4719: 4712:{\displaystyle 1} 4696: 4689:{\displaystyle 1} 4673: 4666:{\displaystyle 1} 4650: 4643:{\displaystyle 1} 4627: 4620:{\displaystyle 1} 4604: 4597:{\displaystyle 1} 4506:4 rectangles x 4 3728:Mirror dihedrals 3310:{\displaystyle n} 3275:higher dimensions 3254:which extend the 3226:Higher dimensions 3178:doubling the cube 3120:+ 1 (a line), or 3046: 2912:analytic geometry 2616:Calculation Tools 2159:for more details. 2095:Mechanical Stress 1889:Thales of Miletus 1425:are constructed. 1371:Euclid's Elements 1344:analytic geometry 1218: 1217: 1183: 1182: 906:List of geometers 589:Three-dimensional 578: 577: 16:(Redirected from 10289: 10282:Greek inventions 10249: 10248: 10037: 10030: 10023: 10014: 10013: 9998: 9997: 9986: 9985: 9974: 9973: 9963: 9962: 9894:Computer algebra 9869:Computer science 9589:Complex analysis 9423: 9416: 9409: 9400: 9399: 9387: 9386: 9374: 9373: 9372: 9148: 9147: 9135:Platonic Academy 9082:Problem II.8 of 9052:Crossbar theorem 9008:Thales's theorem 8948:Euclid's theorem 8918: 8917: 8835:Commensurability 8796:Axiomatic system 8744:Angle trisection 8709: 8699: 8661: 8651: 8641: 8631: 8607: 8597: 8580: 8143: 8136: 8129: 8120: 8119: 8105:Geometry Unbound 8099: 8081: 8056: 8046: 8020: 8010: 8000: 7990: 7978: 7967: 7955: 7944: 7943:. W. H. Freeman. 7897: 7895: 7886:Heath, Thomas L. 7881: 7872: 7860: 7824: 7813: 7807: 7806: 7789:"Tarski's logic" 7784: 7778: 7777: 7752: 7746: 7745: 7723: 7717: 7716: 7696: 7690: 7689: 7663: 7657: 7656: 7630: 7624: 7623: 7613: 7607: 7606: 7584: 7578: 7577: 7567: 7561: 7560: 7533: 7524: 7521: 7515: 7512: 7506: 7503: 7497: 7494: 7488: 7485: 7479: 7476: 7470: 7467: 7461: 7458: 7449: 7448: 7428: 7422: 7421: 7400:group of motions 7397: 7377: 7356: 7350: 7349: 7323: 7317: 7299: 7293: 7290: 7284: 7281: 7272: 7269: 7263: 7252: 7246: 7243: 7237: 7231: 7225: 7190: 7184: 7181: 7175: 7164: 7158: 7155: 7149: 7146: 7140: 7139:Ball, pp. 268ff. 7137: 7131: 7128: 7122: 7121: 7119: 7118: 7109:. Archived from 7102: 7096: 7093: 7087: 7086:Ball, p. 5. 7084: 7078: 7075: 7069: 7066: 7060: 7057: 7051: 7048: 7042: 7035: 7029: 7026: 7020: 7008: 7002: 6999: 6993: 6992: 6984: 6978: 6971:Lebesgue measure 6966: 6960: 6957: 6951: 6950: 6922: 6916: 6914: 6897: 6891: 6888: 6882: 6881: 6860: 6854: 6851: 6845: 6839: 6833: 6827: 6757:Ordered geometry 6732:Hilbert's axioms 6630:Hilbert's axioms 6612: 6578: 6559:Bertrand Russell 6553: 6486:Alessandro Padoa 6451:Axiomatic system 6447:Hilbert's axioms 6438: 6435: 6417: 6410: 6400:geometric series 6373:Abraham Robinson 6309:Infinite objects 6086: 6084: 6083: 6078: 6070: 6064: 6063: 6060: 6055: 6052: 6049: 6040: 6036: 6034: 6033: 6028: 6020: 6014: 6013: 6010: 6005: 6002: 5999: 5990: 5986: 5984: 5983: 5978: 5967: 5963: 5961: 5960: 5955: 5944: 5940: 5938: 5937: 5932: 5924: 5915: 5906: 5902: 5900: 5899: 5894: 5886: 5885: 5880: 5876: 5866: 5865: 5857: 5847: 5846: 5837: 5828: 5826: 5825: 5820: 5812: 5808: 5805: 5804: 5799: 5797: 5796: 5783: 5782: 5777: 5765: 5749: 5745: 5743: 5742: 5737: 5729: 5725: 5722: 5721: 5720: 5703: 5702: 5686: 5682: 5680: 5679: 5674: 5666: 5662: 5652: 5651: 5635: 5631: 5629: 5628: 5623: 5612: 5608: 5606: 5605: 5600: 5592: 5588: 5579: 5563: 5559: 5557: 5556: 5551: 5543: 5539: 5533: 5532: 5527: 5521: 5505: 5496: 5494: 5493: 5488: 5480: 5476: 5473: 5472: 5471: 5457: 5452: 5441: 5440: 5424: 5420: 5418: 5417: 5412: 5404: 5400: 5397: 5396: 5395: 5378: 5377: 5361: 5357: 5355: 5354: 5349: 5341: 5337: 5327: 5325: 5310: 5306: 5304: 5303: 5298: 5287: 5283: 5281: 5280: 5275: 5267: 5263: 5253: 5251: 5236: 5232: 5230: 5229: 5224: 5216: 5212: 5206: 5205: 5200: 5194: 5178: 5169: 5167: 5166: 5161: 5153: 5146: 5145: 5136: 5134: 5126: 5122: 5120: 5119: 5114: 5106: 5099: 5098: 5089: 5087: 5079: 5075: 5073: 5072: 5067: 5059: 5049: 5047: 5039: 5035: 5033: 5032: 5027: 5025: 5016: 5007: 5003: 5001: 5000: 4995: 4993: 4984: 4975: 4971: 4969: 4968: 4963: 4961: 4952: 4943: 4934: 4932: 4931: 4926: 4918: 4915: 4914: 4909: 4907: 4906: 4893: 4884: 4880: 4878: 4877: 4872: 4864: 4855: 4846: 4842: 4840: 4839: 4834: 4823: 4819: 4817: 4816: 4811: 4800: 4796: 4794: 4793: 4788: 4780: 4775: 4767: 4763: 4761: 4760: 4755: 4747: 4737: 4735: 4727: 4718: 4716: 4715: 4710: 4699: 4695: 4693: 4692: 4687: 4676: 4672: 4670: 4669: 4664: 4653: 4649: 4647: 4646: 4641: 4630: 4626: 4624: 4623: 4618: 4607: 4603: 4601: 4600: 4595: 4584: 4475:675 in 120-cell 4472:120 in 120-cell 4421:120 dodecahedra 4375:1200 triangular 4350:600 tetrahedral 4347:120 icosahedral 4327: 4320: 4313: 4306: 4299: 4292: 4280: 4278: 4277: 4274: 4271: 4265: 4263: 4262: 4259: 4256: 4250: 4248: 4247: 4244: 4241: 4235: 4233: 4232: 4229: 4226: 4220: 4218: 4217: 4214: 4211: 4205: 4203: 4202: 4199: 4196: 4188: 4186: 4185: 4182: 4179: 4173: 4171: 4170: 4167: 4164: 4158: 4156: 4155: 4152: 4149: 4143: 4141: 4140: 4137: 4134: 4128: 4126: 4125: 4122: 4119: 4113: 4111: 4110: 4107: 4104: 4096: 4094: 4093: 4090: 4087: 4081: 4079: 4078: 4075: 4072: 4066: 4064: 4063: 4060: 4057: 4051: 4049: 4048: 4045: 4042: 4036: 4034: 4033: 4030: 4027: 4021: 4019: 4018: 4015: 4012: 4004: 4002: 4001: 3998: 3995: 3989: 3987: 3986: 3983: 3980: 3974: 3972: 3971: 3968: 3965: 3959: 3957: 3956: 3953: 3950: 3944: 3942: 3941: 3938: 3935: 3929: 3927: 3926: 3923: 3920: 3912: 3910: 3909: 3906: 3903: 3897: 3895: 3894: 3891: 3888: 3882: 3880: 3879: 3876: 3873: 3867: 3865: 3864: 3861: 3858: 3852: 3850: 3849: 3846: 3843: 3837: 3835: 3834: 3831: 3828: 3820: 3818: 3817: 3814: 3811: 3805: 3803: 3802: 3799: 3796: 3790: 3788: 3787: 3784: 3781: 3775: 3773: 3772: 3769: 3766: 3760: 3758: 3757: 3754: 3751: 3745: 3743: 3742: 3739: 3736: 3723: 3722: 3721: 3717: 3716: 3712: 3711: 3707: 3706: 3702: 3701: 3697: 3696: 3692: 3691: 3684: 3683: 3682: 3678: 3677: 3673: 3672: 3668: 3667: 3663: 3662: 3658: 3657: 3653: 3652: 3645: 3644: 3643: 3639: 3638: 3634: 3633: 3629: 3628: 3624: 3623: 3619: 3618: 3614: 3613: 3606: 3605: 3604: 3600: 3599: 3595: 3594: 3590: 3589: 3585: 3584: 3580: 3579: 3575: 3574: 3567: 3566: 3565: 3561: 3560: 3556: 3555: 3551: 3550: 3546: 3545: 3541: 3540: 3536: 3535: 3528: 3527: 3526: 3522: 3521: 3517: 3516: 3512: 3511: 3507: 3506: 3502: 3501: 3497: 3496: 3328: 3327: 3316: 3314: 3313: 3308: 3285:, which are the 3133:Girard Desargues 3128:= 7 (a circle). 3058: 3056: 3055: 3050: 3047: 3045: 3044: 3035: 3034: 3022: 3021: 3006: 3005: 2996: 2995: 2983: 2982: 2970: 2965: 2954: 2886: 2882: 2879: 2865: 2861: 2858: 2790: 2778: 2766: 2714: 2695: 2680: 2552:solenoidal field 2456:optimizing space 2417:Pascal's theorem 2304:Satellite Orbits 2300:Satellite Orbits 2286:characteristics. 2185:Geometric optics 1998:Measurements of 1965: 1963: 1962: 1957: 1955: 1954: 1932: 1930: 1929: 1924: 1922: 1921: 1794: 1764: 1748: 1726: 1608:Methods of proof 1577:Playfair's axiom 1444:axiomatic system 1291:three dimensions 1279:axiomatic system 1275:secondary school 1210: 1203: 1196: 924: 923: 443: 442: 376:Zero-dimensional 81: 67: 66: 59: – using a 21: 10297: 10296: 10292: 10291: 10290: 10288: 10287: 10286: 10262: 10261: 10260: 10255: 10237: 10204: 10173: 10130: 10123: 10078: 10071: 10046: 10041: 10011: 10006: 9957: 9948: 9898: 9855: 9834:Systems science 9765: 9761:Homotopy theory 9727: 9694: 9646: 9618: 9565: 9512: 9483:Category theory 9469: 9434: 9427: 9397: 9392: 9381: 9370: 9368: 9357: 9323:Arabian/Islamic 9311: 9300:numeral systems 9189: 9139: 9109: 9057:Heron's formula 9035: 9017: 8909: 8905:Triangle center 8895:Regular polygon 8772:and definitions 8771: 8765: 8727: 8707: 8697: 8659: 8649: 8639: 8629: 8605: 8595: 8578: 8544: 8515:Theon of Smyrna 8160: 8152: 8147: 8113:Wayback Machine 8103:Kiran Kedlaya, 8084: 8066: 8063: 8018: 7998: 7964: 7952:Euclid's Window 7857: 7832: 7827: 7814: 7810: 7803: 7785: 7781: 7771: 7753: 7749: 7742: 7724: 7720: 7713: 7697: 7693: 7686: 7664: 7660: 7653: 7631: 7627: 7614: 7610: 7603: 7585: 7581: 7568: 7564: 7557: 7534: 7527: 7522: 7518: 7513: 7509: 7504: 7500: 7495: 7491: 7486: 7482: 7477: 7473: 7468: 7464: 7459: 7452: 7445: 7429: 7425: 7418: 7394: 7374: 7357: 7353: 7346: 7324: 7320: 7314:Wayback Machine 7300: 7296: 7291: 7287: 7282: 7275: 7270: 7266: 7258:, 1997 (1958). 7253: 7249: 7244: 7240: 7232: 7228: 7191: 7187: 7182: 7178: 7165: 7161: 7156: 7152: 7147: 7143: 7138: 7134: 7129: 7125: 7116: 7114: 7103: 7099: 7094: 7090: 7085: 7081: 7076: 7072: 7067: 7063: 7058: 7054: 7049: 7045: 7036: 7032: 7027: 7023: 7009: 7005: 7000: 6996: 6985: 6981: 6967: 6963: 6958: 6954: 6939:10.2307/2973238 6923: 6919: 6912: 6898: 6894: 6889: 6885: 6878: 6861: 6857: 6852: 6848: 6840: 6836: 6828: 6821: 6817: 6794:Heron's formula 6775: 6708: 6691:Gödel's theorem 6667:Tarski's axioms 6613: 6609:How to Solve It 6603: 6596: 6584:foundationalism 6579: 6573: 6554: 6548: 6509:system of ideas 6478: 6470:classical logic 6462: 6460:Classical logic 6457: 6439: 6433: 6430: 6423:needs expansion 6408: 6381: 6361:non-Archimedean 6311: 6306: 6277:Albert Einstein 6245: 6218:Minkowski space 6191: 6179:George Birkhoff 6136: 6130: 6059: 6051: 6050: 6047: 6045: 6042: 6041: 6009: 6001: 6000: 5997: 5995: 5992: 5991: 5972: 5969: 5968: 5949: 5946: 5945: 5913: 5911: 5908: 5907: 5881: 5863: 5859: 5858: 5844: 5842: 5839: 5838: 5798: 5792: 5788: 5784: 5776: 5766: 5763: 5759: 5754: 5751: 5750: 5716: 5712: 5708: 5700: 5696: 5691: 5688: 5687: 5649: 5645: 5640: 5637: 5636: 5617: 5614: 5613: 5577: 5573: 5568: 5565: 5564: 5526: 5522: 5519: 5515: 5510: 5507: 5506: 5467: 5463: 5459: 5451: 5438: 5434: 5429: 5426: 5425: 5391: 5387: 5383: 5375: 5371: 5366: 5363: 5362: 5324: 5320: 5315: 5312: 5311: 5292: 5289: 5288: 5250: 5246: 5241: 5238: 5237: 5199: 5195: 5192: 5188: 5183: 5180: 5179: 5141: 5137: 5133: 5131: 5128: 5127: 5094: 5090: 5086: 5084: 5081: 5080: 5046: 5044: 5041: 5040: 5014: 5012: 5009: 5008: 4982: 4980: 4977: 4976: 4950: 4948: 4945: 4944: 4908: 4902: 4898: 4897: 4891: 4889: 4886: 4885: 4853: 4851: 4848: 4847: 4828: 4825: 4824: 4805: 4802: 4801: 4774: 4772: 4769: 4768: 4734: 4732: 4729: 4728: 4704: 4701: 4700: 4681: 4678: 4677: 4658: 4655: 4654: 4635: 4632: 4631: 4612: 4609: 4608: 4589: 4586: 4585: 4533:Petrie polygons 4463:10-dodecahedron 4418:600 tetrahedra 4395:1200 triangles 4372:720 pentagonal 4341:16 tetrahedral 4275: 4272: 4269: 4268: 4266: 4260: 4257: 4254: 4253: 4251: 4245: 4242: 4239: 4238: 4236: 4230: 4227: 4224: 4223: 4221: 4215: 4212: 4209: 4208: 4206: 4200: 4197: 4194: 4193: 4191: 4183: 4180: 4177: 4176: 4174: 4168: 4165: 4162: 4161: 4159: 4153: 4150: 4147: 4146: 4144: 4138: 4135: 4132: 4131: 4129: 4123: 4120: 4117: 4116: 4114: 4108: 4105: 4102: 4101: 4099: 4091: 4088: 4085: 4084: 4082: 4076: 4073: 4070: 4069: 4067: 4061: 4058: 4055: 4054: 4052: 4046: 4043: 4040: 4039: 4037: 4031: 4028: 4025: 4024: 4022: 4016: 4013: 4010: 4009: 4007: 3999: 3996: 3993: 3992: 3990: 3984: 3981: 3978: 3977: 3975: 3969: 3966: 3963: 3962: 3960: 3954: 3951: 3948: 3947: 3945: 3939: 3936: 3933: 3932: 3930: 3924: 3921: 3918: 3917: 3915: 3907: 3904: 3901: 3900: 3898: 3892: 3889: 3886: 3885: 3883: 3877: 3874: 3871: 3870: 3868: 3862: 3859: 3856: 3855: 3853: 3847: 3844: 3841: 3840: 3838: 3832: 3829: 3826: 3825: 3823: 3815: 3812: 3809: 3808: 3806: 3800: 3797: 3794: 3793: 3791: 3785: 3782: 3779: 3778: 3776: 3770: 3767: 3764: 3763: 3761: 3755: 3752: 3749: 3748: 3746: 3740: 3737: 3734: 3733: 3731: 3719: 3714: 3709: 3704: 3699: 3694: 3689: 3687: 3680: 3675: 3670: 3665: 3660: 3655: 3650: 3648: 3641: 3636: 3631: 3626: 3621: 3616: 3611: 3609: 3602: 3597: 3592: 3587: 3582: 3577: 3572: 3570: 3563: 3558: 3553: 3548: 3543: 3538: 3533: 3531: 3524: 3519: 3514: 3509: 3504: 3499: 3494: 3492: 3488:Coxeter mirrors 3463:Schläfli symbol 3455: 3449: 3441: 3435: 3427: 3425: 3417: 3411: 3403: 3397: 3389: 3383: 3371: 3363: 3355: 3347: 3319:Platonic solids 3302: 3299: 3298: 3281:, later called 3271:Euclidean space 3267:Ludwig Schläfli 3265:At mid-century 3256:complex numbers 3252:normed algebras 3228: 3204: 3192:affine geometry 3161: 3096: 3089: 3078: 3071: 3040: 3036: 3030: 3026: 3017: 3013: 3001: 2997: 2991: 2987: 2978: 2974: 2969: 2961: 2950: 2948: 2945: 2944: 2905: 2884: 2880: 2863: 2859: 2842: 2837: 2827: 2794: 2791: 2782: 2779: 2770: 2767: 2718: 2715: 2706: 2696: 2687: 2681: 2669: 2648:random matrices 2634:in statistical 2592:control systems 2584:Control Systems 2576: 2481: 2462:and optimizing 2444: 2431: 2349: 2214: 2207:Types of Lenses 2145:heat exchangers 2073:Stress Analysis 2069:Stress Analysis 2065: 2060: 1976: 1950: 1946: 1938: 1935: 1934: 1917: 1913: 1905: 1902: 1901: 1898: 1885:Thales' theorem 1882: 1880:Thales' theorem 1872:The celebrated 1870: 1853: 1835: 1818:bridge of asses 1810: 1803: 1799:Thales' theorem 1795: 1786: 1765: 1756: 1749: 1740: 1727: 1718: 1687: 1668: 1659: 1654: 1610: 1561: 1555: 1446:, in which all 1431: 1423:platonic solids 1373: 1367: 1329:Euclidean space 1321:Albert Einstein 1314:self-consistent 1214: 1185: 1184: 921: 920: 911: 910: 701: 700: 684: 683: 669: 668: 656: 655: 592: 591: 580: 579: 440: 439: 437:Two-dimensional 428: 427: 401: 400: 398:One-dimensional 389: 388: 379: 378: 367: 366: 300: 299: 298: 281: 280: 129: 128: 117: 94: 35: 28: 23: 22: 15: 12: 11: 5: 10295: 10285: 10284: 10279: 10274: 10257: 10256: 10254: 10253: 10242: 10239: 10238: 10236: 10235: 10230: 10225: 10224: 10223: 10212: 10210: 10206: 10205: 10203: 10202: 10197: 10192: 10187: 10181: 10179: 10175: 10174: 10172: 10171: 10166: 10161: 10156: 10151: 10146: 10141: 10135: 10133: 10129:Non-Euclidean 10125: 10124: 10122: 10121: 10119:Solid geometry 10116: 10115: 10114: 10109: 10102:Plane geometry 10099: 10094: 10089: 10083: 10081: 10073: 10072: 10070: 10069: 10064: 10063: 10062: 10051: 10048: 10047: 10040: 10039: 10032: 10025: 10017: 10008: 10007: 10005: 10004: 9992: 9980: 9968: 9953: 9950: 9949: 9947: 9946: 9941: 9936: 9931: 9926: 9921: 9920: 9919: 9912:Mathematicians 9908: 9906: 9904:Related topics 9900: 9899: 9897: 9896: 9891: 9886: 9881: 9876: 9871: 9865: 9863: 9857: 9856: 9854: 9853: 9852: 9851: 9846: 9841: 9839:Control theory 9831: 9826: 9821: 9816: 9811: 9806: 9801: 9796: 9791: 9786: 9781: 9775: 9773: 9767: 9766: 9764: 9763: 9758: 9753: 9748: 9743: 9737: 9735: 9729: 9728: 9726: 9725: 9720: 9715: 9710: 9704: 9702: 9696: 9695: 9693: 9692: 9687: 9682: 9677: 9672: 9667: 9662: 9656: 9654: 9648: 9647: 9645: 9644: 9639: 9634: 9628: 9626: 9620: 9619: 9617: 9616: 9614:Measure theory 9611: 9606: 9601: 9596: 9591: 9586: 9581: 9575: 9573: 9567: 9566: 9564: 9563: 9558: 9553: 9548: 9543: 9538: 9533: 9528: 9522: 9520: 9514: 9513: 9511: 9510: 9505: 9500: 9495: 9490: 9485: 9479: 9477: 9471: 9470: 9468: 9467: 9462: 9457: 9456: 9455: 9450: 9439: 9436: 9435: 9426: 9425: 9418: 9411: 9403: 9394: 9393: 9366: 9363: 9362: 9359: 9358: 9356: 9355: 9350: 9345: 9340: 9335: 9330: 9325: 9319: 9317: 9316:Other cultures 9313: 9312: 9310: 9309: 9308: 9307: 9297: 9296: 9295: 9285: 9284: 9283: 9273: 9272: 9271: 9261: 9260: 9259: 9249: 9248: 9247: 9237: 9236: 9235: 9225: 9224: 9223: 9213: 9212: 9211: 9197: 9195: 9191: 9190: 9188: 9187: 9182: 9177: 9172: 9167: 9165:Greek numerals 9162: 9160:Attic numerals 9157: 9151: 9145: 9141: 9140: 9138: 9137: 9132: 9127: 9121: 9119: 9115: 9114: 9111: 9110: 9108: 9107: 9102: 9097: 9092: 9087: 9079: 9074: 9069: 9064: 9059: 9054: 9049: 9043: 9041: 9037: 9036: 9034: 9033: 9027: 9025: 9019: 9018: 9016: 9015: 9010: 9005: 9000: 8995: 8990: 8988:Law of cosines 8985: 8980: 8975: 8970: 8965: 8960: 8955: 8950: 8945: 8940: 8935: 8929: 8927: 8915: 8911: 8910: 8908: 8907: 8902: 8897: 8892: 8887: 8882: 8880:Platonic solid 8877: 8872: 8867: 8862: 8860:Greek numerals 8857: 8852: 8847: 8842: 8837: 8832: 8827: 8826: 8825: 8820: 8810: 8805: 8804: 8803: 8793: 8792: 8791: 8786: 8775: 8773: 8767: 8766: 8764: 8763: 8758: 8757: 8756: 8751: 8746: 8735: 8733: 8729: 8728: 8726: 8725: 8718: 8711: 8701: 8691: 8688:Planisphaerium 8684: 8677: 8670: 8663: 8653: 8643: 8633: 8623: 8616: 8609: 8599: 8589: 8582: 8572: 8565: 8560: 8552: 8550: 8546: 8545: 8543: 8542: 8537: 8532: 8527: 8522: 8517: 8512: 8507: 8502: 8497: 8492: 8487: 8482: 8477: 8472: 8467: 8462: 8457: 8452: 8447: 8442: 8437: 8432: 8427: 8422: 8417: 8412: 8407: 8402: 8397: 8392: 8387: 8382: 8377: 8372: 8367: 8362: 8357: 8352: 8347: 8342: 8337: 8332: 8327: 8322: 8317: 8312: 8307: 8302: 8297: 8292: 8287: 8282: 8277: 8272: 8267: 8262: 8257: 8252: 8247: 8242: 8237: 8232: 8227: 8222: 8217: 8212: 8207: 8202: 8197: 8192: 8187: 8182: 8177: 8172: 8166: 8164: 8158:Mathematicians 8154: 8153: 8146: 8145: 8138: 8131: 8123: 8117: 8116: 8100: 8082: 8062: 8061:External links 8059: 8058: 8057: 8011: 7991: 7983:Tarski, Alfred 7979: 7968: 7962: 7945: 7931:Thorne, Kip S. 7923: 7914:, vol. 3 7906:, vol. 2 7882: 7873: 7861: 7855: 7831: 7828: 7826: 7825: 7808: 7801: 7779: 7769: 7747: 7740: 7718: 7711: 7691: 7684: 7658: 7651: 7639:Basic Geometry 7625: 7608: 7601: 7579: 7562: 7555: 7525: 7516: 7507: 7498: 7489: 7480: 7471: 7462: 7450: 7443: 7423: 7416: 7392: 7372: 7351: 7344: 7318: 7301:Rizos, Chris. 7294: 7285: 7283:Tarski (1951). 7273: 7264: 7247: 7238: 7226: 7193:Stillwell 2001 7185: 7176: 7159: 7150: 7141: 7132: 7123: 7097: 7088: 7079: 7070: 7061: 7052: 7043: 7030: 7021: 7003: 6994: 6979: 6961: 6952: 6917: 6910: 6892: 6883: 6876: 6855: 6846: 6834: 6818: 6816: 6813: 6812: 6811: 6806: 6801: 6796: 6791: 6789:Ceva's theorem 6786: 6781: 6774: 6771: 6770: 6769: 6764: 6759: 6754: 6749: 6744: 6739: 6734: 6729: 6724: 6719: 6714: 6707: 6704: 6703: 6702: 6664: 6646: 6627: 6601: 6595: 6592: 6571: 6546: 6477: 6474: 6461: 6458: 6441: 6440: 6420: 6418: 6407: 6404: 6396:Zeno's paradox 6380: 6377: 6331:Zeno's paradox 6327:Eleatic School 6321:The notion of 6310: 6307: 6305: 6302: 6244: 6241: 6190: 6187: 6132:Main article: 6129: 6126: 6118:Clifford torus 6099:H.S.M. Coxeter 6089: 6088: 6076: 6073: 6067: 6058: 6038: 6026: 6023: 6017: 6008: 5988: 5976: 5965: 5953: 5942: 5930: 5927: 5921: 5918: 5904: 5892: 5889: 5884: 5879: 5873: 5869: 5862: 5854: 5850: 5835: 5831: 5830: 5818: 5815: 5811: 5802: 5795: 5791: 5787: 5780: 5775: 5772: 5769: 5762: 5758: 5747: 5735: 5732: 5728: 5719: 5715: 5711: 5706: 5699: 5695: 5684: 5672: 5669: 5665: 5659: 5655: 5648: 5644: 5633: 5621: 5610: 5598: 5595: 5591: 5585: 5582: 5576: 5572: 5561: 5549: 5546: 5542: 5536: 5530: 5525: 5518: 5514: 5503: 5499: 5498: 5486: 5483: 5479: 5470: 5466: 5462: 5455: 5450: 5447: 5444: 5437: 5433: 5422: 5410: 5407: 5403: 5394: 5390: 5386: 5381: 5374: 5370: 5359: 5347: 5344: 5340: 5333: 5330: 5323: 5319: 5308: 5296: 5285: 5273: 5270: 5266: 5259: 5256: 5249: 5245: 5234: 5222: 5219: 5215: 5209: 5203: 5198: 5191: 5187: 5176: 5172: 5171: 5159: 5156: 5149: 5144: 5140: 5124: 5112: 5109: 5102: 5097: 5093: 5077: 5065: 5062: 5055: 5052: 5037: 5022: 5019: 5005: 4990: 4987: 4973: 4958: 4955: 4941: 4937: 4936: 4924: 4921: 4912: 4905: 4901: 4896: 4882: 4870: 4867: 4861: 4858: 4844: 4832: 4821: 4809: 4798: 4786: 4783: 4778: 4765: 4753: 4750: 4743: 4740: 4725: 4721: 4720: 4708: 4697: 4685: 4674: 4662: 4651: 4639: 4628: 4616: 4605: 4593: 4582: 4578: 4577: 4570: 4563: 4556: 4549: 4542: 4535: 4529: 4528: 4521: 4514: 4507: 4504: 4497: 4495: 4493:Great polygons 4489: 4488: 4485: 4482: 4479: 4476: 4473: 4470: 4466: 4465: 4459: 4457:30-tetrahedron 4453: 4447: 4441: 4435: 4429: 4423: 4422: 4419: 4416: 4413: 4410: 4409:16 tetrahedra 4407: 4404: 4400: 4399: 4398:720 pentagons 4396: 4393: 4390: 4387: 4384: 4381: 4377: 4376: 4373: 4370: 4369:96 triangular 4367: 4366:32 triangular 4364: 4361: 4360:10 triangular 4358: 4352: 4351: 4348: 4345: 4342: 4339: 4336: 4335:5 tetrahedral 4333: 4329: 4328: 4321: 4314: 4307: 4300: 4293: 4286: 4282: 4281: 4189: 4097: 4005: 3913: 3821: 3729: 3725: 3724: 3685: 3646: 3607: 3568: 3529: 3490: 3484: 3483: 3480: 3477: 3474: 3471: 3468: 3465: 3459: 3458: 3444: 3430: 3420: 3406: 3392: 3378: 3374: 3373: 3369: 3365: 3361: 3357: 3353: 3349: 3345: 3341: 3339:Symmetry group 3335: 3334: 3306: 3240:John T. Graves 3234:developed the 3227: 3224: 3203: 3200: 3174:Pierre Wantzel 3160: 3157: 3094: 3087: 3076: 3069: 3060: 3059: 3043: 3039: 3033: 3029: 3025: 3020: 3016: 3012: 3009: 3004: 3000: 2994: 2990: 2986: 2981: 2977: 2973: 2968: 2964: 2960: 2957: 2953: 2908:René Descartes 2904: 2901: 2885: 190 BCE 2881: 240 BCE 2864: 212 BCE 2860: 287 BCE 2841: 2838: 2826: 2823: 2796: 2795: 2792: 2785: 2783: 2780: 2773: 2771: 2768: 2761: 2736:Gunter's chain 2720: 2719: 2716: 2709: 2707: 2699:Sphere packing 2697: 2690: 2688: 2682: 2675: 2668: 2665: 2664: 2663: 2604: 2603: 2588:control theory 2575: 2572: 2560: 2559: 2541:fluid velocity 2537:potential flow 2505: 2504: 2489:Antenna Design 2485:Antenna Design 2480: 2477: 2468: 2467: 2443: 2442:Circuit Design 2440: 2425: 2424: 2409: 2408: 2365: 2364: 2348: 2345: 2336: 2335: 2332:elliptic orbit 2316:space missions 2288: 2287: 2263:is evident in 2238: 2237: 2213: 2210: 2201: 2200: 2161: 2160: 2137:Heat exchanger 2121: 2120: 2115:for efficient 2089: 2088: 2064: 2061: 2059: 2058:In engineering 2056: 1975: 1972: 1953: 1949: 1945: 1942: 1920: 1916: 1912: 1909: 1897: 1894: 1887:, named after 1881: 1878: 1869: 1866: 1852: 1849: 1834: 1831: 1820:) states that 1809: 1806: 1805: 1804: 1796: 1789: 1787: 1766: 1759: 1757: 1750: 1743: 1741: 1728: 1721: 1717: 1714: 1686: 1683: 1667: 1664: 1658: 1655: 1653: 1650: 1609: 1606: 1589: 1588: 1557:Main article: 1554: 1551: 1543: 1542: 1539: 1536: 1533: 1530: 1508: 1507: 1504: 1497: 1492:To describe a 1490: 1483: 1471: 1470: 1430: 1427: 1419:solid geometry 1369:Main article: 1366: 1360: 1348:René Descartes 1293:. Much of the 1287:solid geometry 1271:plane geometry 1254:logical system 1216: 1215: 1213: 1212: 1205: 1198: 1190: 1187: 1186: 1181: 1180: 1179: 1178: 1173: 1165: 1164: 1160: 1159: 1158: 1157: 1152: 1147: 1142: 1137: 1132: 1127: 1122: 1117: 1112: 1107: 1099: 1098: 1094: 1093: 1092: 1091: 1086: 1081: 1076: 1071: 1066: 1061: 1056: 1048: 1047: 1043: 1042: 1041: 1040: 1035: 1030: 1025: 1020: 1015: 1010: 1005: 1000: 995: 990: 985: 977: 976: 972: 971: 970: 969: 964: 959: 954: 949: 944: 939: 931: 930: 922: 918: 917: 916: 913: 912: 909: 908: 903: 898: 893: 888: 883: 878: 873: 868: 863: 858: 853: 848: 843: 838: 833: 828: 823: 818: 813: 808: 803: 798: 793: 788: 783: 778: 773: 768: 763: 758: 753: 748: 743: 738: 733: 728: 723: 718: 713: 708: 702: 698: 697: 696: 693: 692: 686: 685: 682: 681: 676: 670: 663: 662: 661: 658: 657: 654: 653: 648: 643: 641:Platonic Solid 638: 633: 628: 623: 618: 613: 612: 611: 600: 599: 593: 587: 586: 585: 582: 581: 576: 575: 574: 573: 568: 563: 555: 554: 548: 547: 546: 545: 540: 532: 531: 525: 524: 523: 522: 517: 512: 507: 499: 498: 492: 491: 490: 489: 484: 479: 471: 470: 464: 463: 462: 461: 456: 451: 441: 435: 434: 433: 430: 429: 426: 425: 420: 419: 418: 413: 402: 396: 395: 394: 391: 390: 387: 386: 380: 374: 373: 372: 369: 368: 365: 364: 359: 354: 348: 347: 342: 337: 327: 322: 317: 311: 310: 301: 297: 296: 293: 289: 288: 287: 286: 283: 282: 279: 278: 277: 276: 266: 261: 256: 251: 246: 245: 244: 234: 229: 224: 223: 222: 217: 212: 202: 201: 200: 195: 185: 180: 175: 170: 165: 160: 159: 158: 153: 152: 151: 136: 130: 124: 123: 122: 119: 118: 116: 115: 105: 99: 96: 95: 82: 74: 73: 26: 9: 6: 4: 3: 2: 10294: 10283: 10280: 10278: 10275: 10273: 10270: 10269: 10267: 10252: 10244: 10243: 10240: 10234: 10231: 10229: 10226: 10222: 10219: 10218: 10217: 10214: 10213: 10211: 10207: 10201: 10198: 10196: 10193: 10191: 10188: 10186: 10183: 10182: 10180: 10176: 10170: 10167: 10165: 10162: 10160: 10157: 10155: 10152: 10150: 10147: 10145: 10142: 10140: 10137: 10136: 10134: 10132: 10126: 10120: 10117: 10113: 10110: 10108: 10105: 10104: 10103: 10100: 10098: 10095: 10093: 10090: 10088: 10087:Combinatorial 10085: 10084: 10082: 10080: 10074: 10068: 10065: 10061: 10058: 10057: 10056: 10053: 10052: 10049: 10045: 10038: 10033: 10031: 10026: 10024: 10019: 10018: 10015: 10003: 10002: 9993: 9991: 9990: 9981: 9979: 9978: 9969: 9967: 9966: 9961: 9955: 9954: 9951: 9945: 9942: 9940: 9937: 9935: 9932: 9930: 9927: 9925: 9922: 9918: 9915: 9914: 9913: 9910: 9909: 9907: 9905: 9901: 9895: 9892: 9890: 9887: 9885: 9882: 9880: 9877: 9875: 9872: 9870: 9867: 9866: 9864: 9862: 9861:Computational 9858: 9850: 9847: 9845: 9842: 9840: 9837: 9836: 9835: 9832: 9830: 9827: 9825: 9822: 9820: 9817: 9815: 9812: 9810: 9807: 9805: 9802: 9800: 9797: 9795: 9792: 9790: 9787: 9785: 9782: 9780: 9777: 9776: 9774: 9772: 9768: 9762: 9759: 9757: 9754: 9752: 9749: 9747: 9744: 9742: 9739: 9738: 9736: 9734: 9730: 9724: 9721: 9719: 9716: 9714: 9711: 9709: 9706: 9705: 9703: 9701: 9700:Number theory 9697: 9691: 9688: 9686: 9683: 9681: 9678: 9676: 9673: 9671: 9668: 9666: 9663: 9661: 9658: 9657: 9655: 9653: 9649: 9643: 9640: 9638: 9635: 9633: 9632:Combinatorics 9630: 9629: 9627: 9625: 9621: 9615: 9612: 9610: 9607: 9605: 9602: 9600: 9597: 9595: 9592: 9590: 9587: 9585: 9584:Real analysis 9582: 9580: 9577: 9576: 9574: 9572: 9568: 9562: 9559: 9557: 9554: 9552: 9549: 9547: 9544: 9542: 9539: 9537: 9534: 9532: 9529: 9527: 9524: 9523: 9521: 9519: 9515: 9509: 9506: 9504: 9501: 9499: 9496: 9494: 9491: 9489: 9486: 9484: 9481: 9480: 9478: 9476: 9472: 9466: 9463: 9461: 9458: 9454: 9451: 9449: 9446: 9445: 9444: 9441: 9440: 9437: 9432: 9424: 9419: 9417: 9412: 9410: 9405: 9404: 9401: 9391: 9390: 9385: 9378: 9377: 9364: 9354: 9351: 9349: 9346: 9344: 9341: 9339: 9336: 9334: 9331: 9329: 9326: 9324: 9321: 9320: 9318: 9314: 9306: 9303: 9302: 9301: 9298: 9294: 9291: 9290: 9289: 9286: 9282: 9279: 9278: 9277: 9274: 9270: 9267: 9266: 9265: 9262: 9258: 9255: 9254: 9253: 9250: 9246: 9243: 9242: 9241: 9238: 9234: 9231: 9230: 9229: 9226: 9222: 9219: 9218: 9217: 9214: 9210: 9206: 9205: 9204: 9203: 9199: 9198: 9196: 9192: 9186: 9183: 9181: 9178: 9176: 9173: 9171: 9168: 9166: 9163: 9161: 9158: 9156: 9153: 9152: 9149: 9146: 9142: 9136: 9133: 9131: 9128: 9126: 9123: 9122: 9120: 9116: 9106: 9103: 9101: 9098: 9096: 9093: 9091: 9088: 9086: 9085: 9080: 9078: 9075: 9073: 9070: 9068: 9065: 9063: 9060: 9058: 9055: 9053: 9050: 9048: 9045: 9044: 9042: 9038: 9032: 9029: 9028: 9026: 9024: 9020: 9014: 9011: 9009: 9006: 9004: 9001: 8999: 8996: 8994: 8993:Pons asinorum 8991: 8989: 8986: 8984: 8981: 8979: 8976: 8974: 8971: 8969: 8966: 8964: 8963:Hinge theorem 8961: 8959: 8956: 8954: 8951: 8949: 8946: 8944: 8941: 8939: 8936: 8934: 8931: 8930: 8928: 8926: 8925: 8919: 8916: 8912: 8906: 8903: 8901: 8898: 8896: 8893: 8891: 8888: 8886: 8883: 8881: 8878: 8876: 8873: 8871: 8868: 8866: 8863: 8861: 8858: 8856: 8853: 8851: 8848: 8846: 8843: 8841: 8838: 8836: 8833: 8831: 8828: 8824: 8821: 8819: 8816: 8815: 8814: 8811: 8809: 8806: 8802: 8799: 8798: 8797: 8794: 8790: 8787: 8785: 8782: 8781: 8780: 8777: 8776: 8774: 8768: 8762: 8759: 8755: 8752: 8750: 8747: 8745: 8742: 8741: 8740: 8737: 8736: 8734: 8730: 8724: 8723: 8719: 8717: 8716: 8712: 8710: 8706: 8702: 8700: 8696: 8692: 8690: 8689: 8685: 8683: 8682: 8678: 8676: 8675: 8671: 8669: 8668: 8664: 8662: 8658: 8654: 8652: 8648: 8644: 8642: 8638: 8634: 8632: 8630:(Aristarchus) 8628: 8624: 8622: 8621: 8617: 8615: 8614: 8610: 8608: 8604: 8600: 8598: 8594: 8590: 8588: 8587: 8583: 8581: 8577: 8573: 8571: 8570: 8566: 8564: 8561: 8559: 8558: 8554: 8553: 8551: 8547: 8541: 8538: 8536: 8535:Zeno of Sidon 8533: 8531: 8528: 8526: 8523: 8521: 8518: 8516: 8513: 8511: 8508: 8506: 8503: 8501: 8498: 8496: 8493: 8491: 8488: 8486: 8483: 8481: 8478: 8476: 8473: 8471: 8468: 8466: 8463: 8461: 8458: 8456: 8453: 8451: 8448: 8446: 8443: 8441: 8438: 8436: 8433: 8431: 8428: 8426: 8423: 8421: 8418: 8416: 8413: 8411: 8408: 8406: 8403: 8401: 8398: 8396: 8393: 8391: 8388: 8386: 8383: 8381: 8378: 8376: 8373: 8371: 8368: 8366: 8363: 8361: 8358: 8356: 8353: 8351: 8348: 8346: 8343: 8341: 8338: 8336: 8333: 8331: 8328: 8326: 8323: 8321: 8318: 8316: 8313: 8311: 8308: 8306: 8303: 8301: 8298: 8296: 8293: 8291: 8288: 8286: 8283: 8281: 8278: 8276: 8273: 8271: 8268: 8266: 8263: 8261: 8258: 8256: 8253: 8251: 8248: 8246: 8243: 8241: 8238: 8236: 8233: 8231: 8228: 8226: 8223: 8221: 8218: 8216: 8213: 8211: 8208: 8206: 8203: 8201: 8198: 8196: 8193: 8191: 8188: 8186: 8183: 8181: 8178: 8176: 8173: 8171: 8168: 8167: 8165: 8163: 8159: 8155: 8151: 8144: 8139: 8137: 8132: 8130: 8125: 8124: 8121: 8114: 8110: 8107: 8106: 8101: 8097: 8093: 8092: 8087: 8083: 8079: 8075: 8074: 8069: 8065: 8064: 8054: 8050: 8045: 8040: 8036: 8032: 8028: 8024: 8017: 8012: 8008: 8004: 7997: 7992: 7988: 7984: 7980: 7976: 7975: 7974:Gödel's Proof 7969: 7965: 7963:9780684865232 7959: 7954: 7953: 7946: 7942: 7941: 7936: 7932: 7928: 7924: 7921: 7920:0-486-60090-4 7917: 7913: 7912:0-486-60089-0 7909: 7905: 7904:0-486-60088-2 7901: 7894: 7893: 7887: 7883: 7879: 7874: 7870: 7866: 7862: 7858: 7856:0-486-20630-0 7852: 7848: 7844: 7843: 7838: 7834: 7833: 7822: 7821:1-56881-238-8 7818: 7812: 7804: 7798: 7794: 7790: 7783: 7776: 7772: 7766: 7762: 7758: 7751: 7743: 7741:0-19-850825-5 7737: 7733: 7729: 7722: 7714: 7712:0-201-50867-2 7708: 7704: 7703: 7695: 7687: 7685:9780471251835 7681: 7677: 7673: 7669: 7662: 7654: 7652:0-8218-2101-6 7648: 7644: 7640: 7636: 7629: 7621: 7620: 7612: 7604: 7602:0-486-41151-6 7598: 7594: 7590: 7583: 7575: 7574: 7566: 7558: 7556:0-471-25183-6 7552: 7548: 7544: 7540: 7532: 7530: 7520: 7511: 7502: 7493: 7484: 7475: 7466: 7457: 7455: 7446: 7440: 7436: 7435: 7427: 7419: 7417:0-486-43481-8 7413: 7409: 7408: 7401: 7395: 7389: 7385: 7384: 7375: 7373:0-8493-3493-4 7369: 7365: 7364: 7355: 7347: 7341: 7337: 7333: 7329: 7322: 7315: 7311: 7308: 7304: 7298: 7289: 7280: 7278: 7268: 7261: 7257: 7251: 7242: 7235: 7230: 7223: 7218: 7214: 7210: 7206: 7202: 7198: 7194: 7189: 7180: 7173: 7172:0-486-64725-0 7169: 7163: 7154: 7145: 7136: 7127: 7113:on 2019-06-18 7112: 7108: 7101: 7092: 7083: 7074: 7065: 7056: 7047: 7040: 7034: 7025: 7018: 7013: 7007: 6998: 6990: 6983: 6976: 6972: 6965: 6956: 6948: 6944: 6940: 6936: 6932: 6928: 6921: 6913: 6907: 6903: 6896: 6887: 6879: 6873: 6869: 6868: 6859: 6850: 6844:, p. 10. 6843: 6838: 6832:, p. 19. 6831: 6826: 6824: 6819: 6810: 6807: 6805: 6802: 6800: 6797: 6795: 6792: 6790: 6787: 6785: 6782: 6780: 6777: 6776: 6768: 6765: 6763: 6760: 6758: 6755: 6753: 6750: 6748: 6745: 6743: 6740: 6738: 6735: 6733: 6730: 6728: 6725: 6723: 6720: 6718: 6715: 6713: 6710: 6709: 6700: 6696: 6692: 6688: 6684: 6680: 6676: 6672: 6671:Alfred Tarski 6668: 6665: 6662: 6658: 6654: 6650: 6647: 6643: 6639: 6635: 6631: 6628: 6625: 6624:non-Euclidean 6621: 6616: 6615: 6611:, p. 208 6610: 6606: 6600: 6591: 6589: 6585: 6577: 6570: 6568: 6562: 6560: 6552: 6545: 6543: 6539: 6534: 6533:questions... 6532: 6531:psychological 6528: 6524: 6520: 6518: 6514: 6510: 6505: 6503: 6499: 6493: 6491: 6487: 6483: 6473: 6471: 6467: 6456: 6452: 6448: 6437: 6428: 6424: 6421:This section 6419: 6416: 6412: 6411: 6406:Logical basis 6403: 6401: 6397: 6392: 6390: 6385: 6376: 6374: 6370: 6366: 6362: 6358: 6354: 6350: 6345: 6343: 6338: 6336: 6332: 6328: 6324: 6319: 6316: 6301: 6299: 6295: 6290: 6284: 6282: 6278: 6273: 6271: 6267: 6263: 6259: 6255: 6250: 6240: 6238: 6234: 6229: 6227: 6223: 6222:non-Euclidean 6219: 6215: 6211: 6207: 6200: 6195: 6186: 6184: 6180: 6176: 6172: 6168: 6164: 6160: 6156: 6151: 6149: 6145: 6141: 6135: 6125: 6123: 6119: 6115: 6111: 6107: 6102: 6100: 6096: 6074: 6071: 6065: 6056: 6039: 6024: 6021: 6015: 6006: 5989: 5974: 5966: 5951: 5943: 5928: 5925: 5919: 5916: 5905: 5890: 5887: 5882: 5877: 5871: 5867: 5860: 5852: 5848: 5836: 5833: 5832: 5816: 5813: 5809: 5800: 5793: 5789: 5785: 5778: 5773: 5770: 5767: 5760: 5756: 5748: 5733: 5730: 5726: 5717: 5713: 5709: 5704: 5697: 5693: 5685: 5670: 5667: 5663: 5657: 5653: 5646: 5642: 5634: 5619: 5611: 5596: 5593: 5589: 5583: 5580: 5574: 5570: 5562: 5547: 5544: 5540: 5534: 5528: 5523: 5516: 5512: 5504: 5501: 5500: 5484: 5481: 5477: 5468: 5464: 5460: 5453: 5448: 5445: 5442: 5435: 5431: 5423: 5408: 5405: 5401: 5392: 5388: 5384: 5379: 5372: 5368: 5360: 5345: 5342: 5338: 5331: 5328: 5321: 5317: 5309: 5294: 5286: 5271: 5268: 5264: 5257: 5254: 5247: 5243: 5235: 5220: 5217: 5213: 5207: 5201: 5196: 5189: 5185: 5177: 5174: 5173: 5157: 5154: 5147: 5142: 5138: 5125: 5110: 5107: 5100: 5095: 5091: 5078: 5063: 5060: 5053: 5050: 5038: 5020: 5017: 5006: 4988: 4985: 4974: 4956: 4953: 4942: 4940:Short radius 4939: 4938: 4922: 4919: 4910: 4903: 4899: 4894: 4883: 4868: 4865: 4859: 4856: 4845: 4830: 4822: 4807: 4799: 4784: 4781: 4776: 4766: 4751: 4748: 4741: 4738: 4726: 4723: 4722: 4706: 4698: 4683: 4675: 4660: 4652: 4637: 4629: 4614: 4606: 4591: 4583: 4580: 4579: 4575: 4571: 4568: 4564: 4561: 4557: 4554: 4550: 4547: 4543: 4540: 4536: 4534: 4531: 4530: 4526: 4522: 4519: 4515: 4512: 4508: 4505: 4502: 4498: 4496: 4494: 4491: 4490: 4487:10 600-cells 4486: 4483: 4480: 4477: 4474: 4471: 4468: 4467: 4464: 4460: 4458: 4454: 4452: 4448: 4446: 4442: 4440: 4439:8-tetrahedron 4436: 4434: 4433:5-tetrahedron 4430: 4428: 4425: 4424: 4420: 4417: 4415:24 octahedra 4414: 4411: 4408: 4406:5 tetrahedra 4405: 4402: 4401: 4397: 4394: 4392:96 triangles 4391: 4388: 4386:32 triangles 4385: 4383:10 triangles 4382: 4379: 4378: 4374: 4371: 4368: 4365: 4362: 4359: 4357: 4354: 4353: 4349: 4346: 4343: 4340: 4338:8 octahedral 4337: 4334: 4331: 4330: 4326: 4322: 4319: 4315: 4312: 4308: 4305: 4301: 4298: 4294: 4291: 4287: 4284: 4283: 4190: 4098: 4006: 3914: 3822: 3730: 3727: 3726: 3686: 3647: 3608: 3569: 3530: 3491: 3489: 3486: 3485: 3481: 3478: 3475: 3472: 3469: 3466: 3464: 3461: 3460: 3457: 3454: 3448: 3445: 3443: 3440: 3434: 3431: 3429: 3424: 3421: 3419: 3416: 3410: 3407: 3405: 3402: 3396: 3393: 3391: 3388: 3382: 3379: 3376: 3375: 3372: 3364: 3358: 3356: 3348: 3342: 3340: 3337: 3336: 3333: 3329: 3326: 3324: 3320: 3304: 3296: 3292: 3288: 3284: 3280: 3277:. He defined 3276: 3272: 3268: 3263: 3261: 3257: 3253: 3249: 3245: 3244:Arthur Cayley 3241: 3237: 3233: 3230:In the 1840s 3223: 3221: 3217: 3208: 3199: 3197: 3193: 3189: 3185: 3183: 3179: 3175: 3170: 3165: 3153: 3148: 3144: 3142: 3138: 3134: 3129: 3127: 3123: 3119: 3115: 3110: 3108: 3104: 3103: 3097: 3090: 3083: 3079: 3072: 3065: 3041: 3031: 3027: 3023: 3018: 3014: 3007: 3002: 2992: 2988: 2984: 2979: 2975: 2966: 2958: 2955: 2943: 2942: 2941: 2940:The equation 2938: 2935: 2930: 2928: 2924: 2920: 2915: 2913: 2909: 2897: 2892: 2888: 2875: 2871: 2869: 2854: 2846: 2836: 2832: 2825:Later history 2822: 2820: 2817:, but can be 2816: 2812: 2808: 2803: 2801: 2789: 2784: 2777: 2772: 2765: 2760: 2759: 2758: 2756: 2752: 2748: 2743: 2741: 2737: 2733: 2729: 2725: 2713: 2708: 2704: 2700: 2694: 2689: 2686: 2679: 2674: 2673: 2672: 2661: 2657: 2653: 2649: 2645: 2641: 2640:curve fitting 2637: 2633: 2632:design matrix 2629: 2625: 2621: 2617: 2614: 2613: 2608: 2601: 2597: 2593: 2589: 2585: 2581: 2578: 2577: 2569: 2564: 2557: 2553: 2550: 2546: 2542: 2538: 2534: 2530: 2529:inviscid flow 2526: 2522: 2519: 2518: 2514: 2509: 2502: 2498: 2494: 2490: 2486: 2483: 2482: 2472: 2465: 2461: 2457: 2453: 2449: 2446: 2445: 2435: 2430: 2422: 2418: 2414: 2411: 2410: 2406: 2402: 2398: 2394: 2390: 2386: 2382: 2378: 2374: 2370: 2367: 2366: 2362: 2361:manufacturing 2358: 2354: 2351: 2350: 2340: 2333: 2329: 2325: 2324:astrodynamics 2321: 2317: 2313: 2309: 2305: 2301: 2298: 2297: 2292: 2285: 2281: 2277: 2273: 2269: 2266: 2262: 2258: 2254: 2251: 2250: 2247: 2242: 2235: 2231: 2227: 2223: 2219: 2216: 2215: 2205: 2198: 2194: 2190: 2186: 2182: 2178: 2174: 2171: 2170: 2165: 2158: 2154: 2150: 2146: 2142: 2138: 2134: 2131: 2130: 2125: 2118: 2114: 2110: 2106: 2102: 2099: 2098: 2093: 2086: 2082: 2078: 2074: 2070: 2067: 2066: 2055: 2053: 2049: 2045: 2041: 2033: 2029: 2025: 2021: 2016: 2012: 2009: 2005: 2001: 1996: 1993: 1989: 1985: 1981: 1971: 1969: 1951: 1947: 1943: 1940: 1918: 1914: 1910: 1907: 1893: 1890: 1886: 1877: 1875: 1865: 1863: 1859: 1848: 1839: 1830: 1827: 1823: 1819: 1815: 1814:pons asinorum 1808:Pons asinorum 1801: 1800: 1793: 1788: 1784: 1780: 1776: 1772: 1771: 1763: 1758: 1754: 1747: 1742: 1738: 1734: 1733: 1732:pons asinorum 1725: 1720: 1719: 1713: 1711: 1710:line segments 1707: 1703: 1698: 1696: 1692: 1682: 1680: 1679:supplementary 1675: 1673: 1672:complementary 1663: 1662:A, B, and C. 1649: 1647: 1642: 1640: 1636: 1631: 1630: 1625: 1621: 1617: 1616: 1601: 1596: 1592: 1586: 1582: 1581: 1580: 1578: 1574: 1569: 1567: 1560: 1550: 1548: 1540: 1537: 1534: 1531: 1528: 1524: 1520: 1519: 1518: 1516: 1511: 1505: 1502: 1498: 1495: 1491: 1488: 1484: 1482:to any point. 1481: 1477: 1476:straight line 1473: 1472: 1468: 1467: 1466: 1464: 1460: 1455: 1453: 1449: 1445: 1435: 1426: 1424: 1420: 1415: 1413: 1409: 1405: 1404:prime numbers 1401: 1400:number theory 1396: 1394: 1388: 1386: 1381: 1378: 1372: 1365: 1359: 1357: 1353: 1350:, which uses 1349: 1345: 1341: 1336: 1334: 1330: 1326: 1323:'s theory of 1322: 1318: 1315: 1311: 1306: 1304: 1303:number theory 1300: 1296: 1292: 1288: 1284: 1280: 1276: 1272: 1268: 1263: 1261: 1260: 1255: 1251: 1247: 1243: 1239: 1238: 1233: 1229: 1226: 1222: 1211: 1206: 1204: 1199: 1197: 1192: 1191: 1189: 1188: 1177: 1174: 1172: 1169: 1168: 1167: 1166: 1162: 1161: 1156: 1153: 1151: 1148: 1146: 1143: 1141: 1138: 1136: 1133: 1131: 1128: 1126: 1123: 1121: 1118: 1116: 1113: 1111: 1108: 1106: 1103: 1102: 1101: 1100: 1096: 1095: 1090: 1087: 1085: 1082: 1080: 1077: 1075: 1072: 1070: 1067: 1065: 1062: 1060: 1057: 1055: 1052: 1051: 1050: 1049: 1045: 1044: 1039: 1036: 1034: 1031: 1029: 1026: 1024: 1021: 1019: 1016: 1014: 1011: 1009: 1006: 1004: 1001: 999: 996: 994: 991: 989: 986: 984: 981: 980: 979: 978: 974: 973: 968: 965: 963: 960: 958: 955: 953: 950: 948: 945: 943: 940: 938: 935: 934: 933: 932: 929: 926: 925: 915: 914: 907: 904: 902: 899: 897: 894: 892: 889: 887: 884: 882: 879: 877: 874: 872: 869: 867: 864: 862: 859: 857: 854: 852: 849: 847: 844: 842: 839: 837: 834: 832: 829: 827: 824: 822: 819: 817: 814: 812: 809: 807: 804: 802: 799: 797: 794: 792: 789: 787: 784: 782: 779: 777: 774: 772: 769: 767: 764: 762: 759: 757: 754: 752: 749: 747: 744: 742: 739: 737: 734: 732: 729: 727: 724: 722: 719: 717: 714: 712: 709: 707: 704: 703: 695: 694: 691: 688: 687: 680: 677: 675: 672: 671: 666: 660: 659: 652: 649: 647: 644: 642: 639: 637: 634: 632: 629: 627: 624: 622: 619: 617: 614: 610: 607: 606: 605: 602: 601: 598: 595: 594: 590: 584: 583: 572: 569: 567: 566:Circumference 564: 562: 559: 558: 557: 556: 553: 550: 549: 544: 541: 539: 536: 535: 534: 533: 530: 529:Quadrilateral 527: 526: 521: 518: 516: 513: 511: 508: 506: 503: 502: 501: 500: 497: 496:Parallelogram 494: 493: 488: 485: 483: 480: 478: 475: 474: 473: 472: 469: 466: 465: 460: 457: 455: 452: 450: 447: 446: 445: 444: 438: 432: 431: 424: 421: 417: 414: 412: 409: 408: 407: 404: 403: 399: 393: 392: 385: 382: 381: 377: 371: 370: 363: 360: 358: 355: 353: 350: 349: 346: 343: 341: 338: 335: 334:Perpendicular 331: 330:Orthogonality 328: 326: 323: 321: 318: 316: 313: 312: 309: 306: 305: 304: 294: 291: 290: 285: 284: 275: 272: 271: 270: 267: 265: 262: 260: 257: 255: 254:Computational 252: 250: 247: 243: 240: 239: 238: 235: 233: 230: 228: 225: 221: 218: 216: 213: 211: 208: 207: 206: 203: 199: 196: 194: 191: 190: 189: 186: 184: 181: 179: 176: 174: 171: 169: 166: 164: 161: 157: 154: 150: 147: 146: 145: 142: 141: 140: 139:Non-Euclidean 137: 135: 132: 131: 127: 121: 120: 113: 109: 106: 104: 101: 100: 98: 97: 93: 89: 85: 80: 76: 75: 72: 69: 68: 62: 58: 54: 50: 49: 44: 39: 33: 19: 10185:Trigonometry 10076: 9999: 9987: 9975: 9956: 9889:Optimization 9751:Differential 9684: 9675:Differential 9642:Order theory 9637:Graph theory 9541:Group theory 9380: 9367: 9209:Thomas Heath 9200: 9083: 9067:Law of sines 8923: 8855:Golden ratio 8849: 8720: 8713: 8704: 8698:(Theodosius) 8694: 8686: 8679: 8672: 8665: 8656: 8646: 8640:(Hipparchus) 8636: 8626: 8618: 8611: 8602: 8592: 8584: 8579:(Apollonius) 8575: 8567: 8555: 8530:Zeno of Elea 8290:Eratosthenes 8280:Dionysodorus 8104: 8089: 8071: 8026: 8022: 8006: 8002: 7986: 7973: 7951: 7938: 7891: 7877: 7868: 7841: 7823:. Pp. 25–26. 7811: 7792: 7782: 7774: 7760: 7750: 7731: 7721: 7701: 7694: 7675: 7671: 7661: 7642: 7638: 7628: 7618: 7611: 7592: 7582: 7572: 7565: 7546: 7542: 7519: 7510: 7501: 7492: 7483: 7474: 7465: 7433: 7426: 7406: 7399: 7382: 7362: 7354: 7335: 7331: 7321: 7297: 7288: 7267: 7259: 7250: 7241: 7229: 7221: 7188: 7179: 7162: 7153: 7148:Eves (1963). 7144: 7135: 7126: 7115:. Retrieved 7111:the original 7100: 7091: 7082: 7073: 7064: 7055: 7046: 7033: 7024: 7016: 7006: 6997: 6988: 6982: 6964: 6955: 6930: 6926: 6920: 6901: 6895: 6886: 6866: 6858: 6849: 6837: 6747:Metric space 6674: 6660: 6656: 6653:real numbers 6641: 6637: 6633: 6608: 6605:George Pólya 6598: 6581: 6575: 6566: 6564: 6556: 6550: 6541: 6537: 6535: 6530: 6526: 6522: 6521: 6516: 6512: 6508: 6506: 6501: 6497: 6495: 6479: 6463: 6431: 6427:adding to it 6422: 6393: 6386: 6382: 6346: 6339: 6320: 6312: 6285: 6274: 6253: 6246: 6230: 6204: 6171:Moritz Pasch 6167:completeness 6158: 6154: 6152: 6140:János Bolyai 6137: 6103: 6092: 4724:Edge length 4581:Long radius 4484:25 24-cells 4451:6-octahedron 3453:dodecahedron 3450: 3436: 3426: 3412: 3398: 3384: 3278: 3264: 3250:. These are 3229: 3213: 3202:19th century 3186: 3166: 3162: 3159:18th century 3130: 3125: 3121: 3117: 3113: 3111: 3099: 3092: 3085: 3081: 3074: 3067: 3063: 3061: 2939: 2931: 2926: 2922: 2916: 2906: 2872: 2851: 2804: 2800:architecture 2797: 2744: 2721: 2670: 2615: 2579: 2549:irrotational 2521:Field Theory 2520: 2491:- Euclidean 2484: 2447: 2437:3D CAD Model 2412: 2368: 2352: 2299: 2261:aerodynamics 2252: 2246:Oscillations 2244:Vibration - 2236:effectively. 2217: 2183:properties. 2172: 2132: 2100: 2068: 2037: 1997: 1977: 1899: 1883: 1871: 1854: 1845: 1825: 1821: 1817: 1811: 1797: 1782: 1778: 1774: 1768: 1752: 1736: 1730: 1704:(infinite), 1699: 1688: 1676: 1669: 1660: 1643: 1627: 1615:constructive 1613: 1611: 1599: 1590: 1570: 1565: 1562: 1544: 1514: 1512: 1509: 1501:right angles 1458: 1456: 1441: 1416: 1397: 1389: 1384: 1382: 1376: 1374: 1363: 1337: 1307: 1294: 1270: 1269:begins with 1266: 1264: 1257: 1246:propositions 1235: 1220: 1219: 1038:Parameshvara 851:Parameshvara 621:Dodecahedron 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821:Lobachevsky 801:Jyeṣṭhadeva 751:Brahmagupta 679:Hypersphere 651:Tetrahedron 626:Icosahedron 198:Diophantine 10266:Categories 10169:Riemannian 10164:Projective 10149:Symplectic 10144:Hyperbolic 10077:Euclidean 9829:Statistics 9708:Arithmetic 9670:Arithmetic 9536:Elementary 9503:Set theory 9328:Babylonian 9228:arithmetic 9194:History of 9023:Apollonius 8708:(Menelaus) 8667:On Spirals 8586:Catoptrics 8525:Xenocrates 8520:Thymaridas 8505:Theodosius 8490:Theaetetus 8470:Simplicius 8460:Pythagoras 8445:Posidonius 8430:Philonides 8390:Nicomachus 8385:Metrodorus 8375:Menaechmus 8330:Hipparchus 8320:Heliodorus 8270:Diophantus 8255:Democritus 8235:Chrysippus 8205:Archimedes 8200:Apollonius 8170:Anaxagoras 8162:(timeline) 7830:References 7672:Cited work 7197:quaternion 7117:2013-12-29 7105:Tom Hull. 6695:arithmetic 6683:set theory 6675:elementary 6645:postulate. 6507:Then, the 6502:conditions 6445:See also: 6349:Otto Stolz 6318:infinite. 6262:congruence 6214:space-time 6208:theory of 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10154:Spherical 9756:Geometric 9746:Algebraic 9685:Euclidean 9660:Algebraic 9556:Universal 8789:Inscribed 8549:Treatises 8540:Zenodorus 8500:Theodorus 8475:Sosigenes 8420:Philolaus 8405:Oenopides 8400:Nicoteles 8395:Nicomedes 8355:Hypsicles 8250:Ctesibius 8240:Cleomedes 8225:Callippus 8210:Autolycus 8195:Aristotle 8175:Anthemius 8096:EMS Press 8078:EMS Press 6842:Eves 1963 6830:Eves 1963 6588:formalism 6527:empirical 6434:June 2010 6258:isotropic 6159:Elements, 6072:≈ 6057:× 6022:≈ 6007:× 5926:≈ 5888:≈ 5814:≈ 5790:ϕ 5731:≈ 5714:ϕ 5668:≈ 5594:≈ 5545:≈ 5482:≈ 5465:ϕ 5406:≈ 5389:ϕ 5343:≈ 5269:≈ 5218:≈ 5155:≈ 5139:ϕ 5108:≈ 5092:ϕ 5061:≈ 4920:≈ 4900:ϕ 4866:≈ 4860:ϕ 4782:≈ 4749:≈ 4332:Vertices 3428:24-point 3418:16-point 3295:polyhedra 3283:polytopes 3248:octonions 3024:− 2985:− 2919:Cartesian 2724:surveying 2397:airplanes 2381:cylinders 2320:satellite 2222:Vibration 2040:congruent 2008:rectangle 1944:∝ 1911:∝ 1603:triangle. 1499:That all 1478:from any 1140:Minkowski 1059:Descartes 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