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93-424: In geometry , an octahedron ( pl. : octahedra or octahedrons ) is a polyhedron with eight faces. One special case is the regular octahedron , a Platonic solid composed of eight equilateral triangles , four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. Many types of irregular octahedra also exist, including both convex and non-convex shapes. A regular octahedron

186-486: A 2 ≈ 3.464 a 2 , V = 1 3 2 a 3 ≈ 0.471 a 3 . {\displaystyle {\begin{aligned}A&=2{\sqrt {3}}a^{2}&\approx 3.464a^{2},\\V&={\frac {1}{3}}{\sqrt {2}}a^{3}&\approx 0.471a^{3}.\end{aligned}}} The radius of a circumscribed sphere r u {\displaystyle r_{u}} (one that touches

279-406: A , r i = 6 6 a ≈ 0.408 a , r m = 1 2 a = 0.5 a . {\displaystyle r_{u}={\frac {\sqrt {2}}{2}}a\approx 0.707a,\qquad r_{i}={\frac {\sqrt {6}}{6}}a\approx 0.408a,\qquad r_{m}={\frac {1}{2}}a=0.5a.} The dihedral angle of a regular octahedron between two adjacent triangular faces

372-500: A , b , c ) {\displaystyle (a,b,c)} and radius r {\displaystyle r} is the set of all points ( x , y , z ) {\displaystyle (x,y,z)} such that | x − a | + | y − b | + | z − c | = r . {\displaystyle \left|x-a\right|+\left|y-b\right|+\left|z-c\right|=r.} . The skeleton of

465-411: A rectified tetrahedron – and can be called a tetratetrahedron . This can be shown by a 2-color face model. With this coloring, the octahedron has tetrahedral symmetry . Compare this truncation sequence between a tetrahedron and its dual: The above shapes may also be realized as slices orthogonal to the long diagonal of a tesseract . If this diagonal is oriented vertically with a height of 1, then

558-520: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as

651-418: A parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to

744-642: A rectified tetrahedron . These symmetries can be emphasized by different colorings of the faces. An octahedron can be any polyhedron with eight faces. In a previous example, the regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges. There are 257 topologically distinct convex octahedra, excluding mirror images. More specifically there are 2, 11, 42, 74, 76, 38, 14 for octahedra with 6 to 12 vertices respectively. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it

837-426: A regular compound . A regular icosahedron produced this way is called a snub octahedron . The regular octahedron can be considered as the antiprism , a prism like polyhedron in which lateral faces are replaced by alternating equilateral triangles. It is also called trigonal antiprism . Therefore, it has the property of quasiregular , a polyhedron in which two different polygonal faces are alternating and meet at

930-430: A square bifrustum . The octahedron can be generated as the case of a 3D superellipsoid with all exponent values set to 1. Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría )  'land measurement'; from γῆ ( gê )  'earth, land' and μέτρον ( métron )  'a measure') is a branch of mathematics concerned with properties of space such as

1023-425: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes

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1116-405: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,

1209-523: A decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In

1302-427: A graph partitioned into three independent sets each consisting of two opposite vertices. More generally, it is a Turán graph T 6 , 3 {\displaystyle T_{6,3}} . The octahedral graph is 4-connected , meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of

1395-440: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,

1488-428: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry

1581-451: A number of apparently different definitions, which are all equivalent in the most common cases. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle , regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in

1674-444: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received

1767-528: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to

1860-602: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies

1953-548: A regular octahedron can be represented as a graph according to Steinitz's theorem , provided the graph is planar —its edges of a graph are connected to every vertex without crossing other edges—and 3-connected graph —its edges remain connected whenever two of more three vertices of a graph are removed. Its graph called the octahedral graph , a Platonic graph . The octahedral graph can be considered as complete tripartite graph K 2 , 2 , 2 {\displaystyle K_{2,2,2}} ,

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2046-419: A regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids. One can also divide

2139-513: A regular polyhedron is a quadrirectangular irregular tetrahedron . The faces of the octahedron's characteristic tetrahedron lie in the octahedron's mirror planes of symmetry . The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess, among its mirror planes, some that do not pass through any of its faces. The octahedron's symmetry group

2232-586: A right triangle with edges 1 3 {\displaystyle {\sqrt {\tfrac {1}{3}}}} , 1 {\displaystyle 1} , 2 3 {\displaystyle {\sqrt {\tfrac {2}{3}}}} , and a right triangle with edges 4 3 {\displaystyle {\sqrt {\tfrac {4}{3}}}} , 2 {\displaystyle {\sqrt {2}}} , 2 3 {\displaystyle {\sqrt {\tfrac {2}{3}}}} . There are 3 uniform colorings of

2325-427: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert , in his work on creating

2418-600: A technical sense a type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c.  1900 , with

2511-518: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in

2604-494: A theory of ratios that avoided the problem of incommensurable magnitudes , which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of

2697-456: A vertex. Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space . This and the regular tessellation of cubes are the only such uniform honeycombs in 3-dimensional space. The uniform tetrahemihexahedron is a tetrahedral symmetry faceting of the regular octahedron, sharing edge and vertex arrangement . It has four of the triangular faces, and 3 central squares. A regular octahedron

2790-411: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing

2883-404: Is vertex-transitive , i.e. isogonal. The Catalan solids , the bipyramids , and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids , prisms , and antiprisms , respectively. The Platonic solids , which are either self-dual or dual with another Platonic solid, are vertex-, edge-, and face-transitive (i.e. isogonal, isotoxal, and isohedral). A form that

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2976-608: Is 109.47°. This can be obtained from the dihedral angle of an equilateral square pyramid: its dihedral angle between two adjacent triangular faces is the dihedral angle of an equilateral square pyramid between two adjacent triangular faces, and its dihedral angle between two adjacent triangular faces on the edge in which two equilateral square pyramids are attached is twice the dihedral angle of an equilateral square pyramid between its triangular face and its square base. An octahedron with edge length 2 {\displaystyle {\sqrt {2}}} can be placed with its center at

3069-559: Is a 3-ball in the Manhattan ( ℓ 1 ) metric . Like all regular convex polytopes, the octahedron can be dissected into an integral number of disjoint orthoschemes , all of the same shape characteristic of the polytope. A polytope's characteristic orthoscheme is a fundamental property because the polytope is generated by reflections in the facets of its orthoscheme. The orthoscheme occurs in two chiral forms which are mirror images of each other. The characteristic orthoscheme of

3162-540: Is a bipyramid constructed by attaching two square pyramids base-to-base. These pyramids cover their square bases, so the resulting polyhedron has eight triangular faces. A square bipyramid is said to be right if the square pyramids are symmetrically regular and both of their apices are on the line passing through the base's center; otherwise, it is oblique. The resulting bipyramid has three-dimensional point group of dihedral group D 4 h {\displaystyle D_{4\mathrm {h} }} of sixteen:

3255-400: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in

3348-406: Is a regular octahedron. The surface area A {\displaystyle A} of a regular octahedron can be ascertained by summing all of its eight equilateral triangles, whereas its volume V {\displaystyle V} is twice the volume of a square pyramid; if the edge length is a {\displaystyle a} , A = 2 3

3441-413: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood

3534-645: Is an isozonohedron but not an isohedron. A polyhedron (or polytope in general) is k -isohedral if it contains k faces within its symmetry fundamental domains. Similarly, a k -isohedral tiling has k separate symmetry orbits (it may contain m different face shapes, for m = k , or only for some m < k ). ("1-isohedral" is the same as "isohedral".) A monohedral polyhedron or monohedral tiling ( m = 1) has congruent faces, either directly or reflectively, which occur in one or more symmetry positions. An m -hedral polyhedron or tiling has m different face shapes (" dihedral ", " trihedral "... are

3627-409: Is defined. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are

3720-510: Is denoted B 3 . The octahedron and its dual polytope , the cube , have the same symmetry group but different characteristic tetrahedra. The characteristic tetrahedron of the regular octahedron can be found by a canonical dissection of the regular octahedron [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] which subdivides it into 48 of these characteristic orthoschemes [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] surrounding

3813-405: Is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) Some of the polyhedrons do have eight faces aside from being square bipyramids in the following: The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with

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3906-398: Is isohedral, has regular vertices, and is also edge-transitive (i.e. isotoxal) is said to be a quasiregular dual. Some theorists regard these figures as truly quasiregular because they share the same symmetries, but this is not generally accepted. A polyhedron which is isohedral and isogonal is said to be noble . Not all isozonohedra are isohedral. For example, a rhombic icosahedron

3999-437: Is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given

4092-415: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study

4185-524: Is one of the Platonic solids , a set of polyhedrons whose faces are congruent regular polygons and the same number of faces meet at each vertex. This ancient set of polyhedrons was named after Plato who, in his Timaeus dialogue, related these solids to nature. One of them, the regular octahedron, represented the classical element of wind . Following its attribution with nature by Plato, Johannes Kepler in his Harmonices Mundi sketched each of

4278-418: Is one of the eight convex deltahedra because all of the faces are equilateral triangles . It is a composite polyhedron made by attaching two equilateral square pyramids . Its dual polyhedron is the cube , and they have the same three-dimensional symmetry groups , the octahedral symmetry O h {\displaystyle \mathrm {O} _{\mathrm {h} }} . The regular octahedron

4371-482: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,

4464-426: Is the three-dimensional case of the more general concept of a cross-polytope . A regular octahedron is an octahedron that is a regular polyhedron . All the faces of a regular octahedron are equilateral triangles of the same size, and exactly four triangles meet at each vertex. A regular octahedron is convex, meaning that for any two points within it, the line segment connecting them lies entirely within it. It

4557-753: The Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs

4650-690: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c.  1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated

4743-523: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c.  1230  – c.  1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first

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4836-518: The Oxford Calculators , including the mean speed theorem , by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with

4929-509: The Riemann surface , and Henri Poincaré , the founder of algebraic topology and the geometric theory of dynamical systems . As a consequence of these major changes in the conception of geometry, the concept of " space " became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of

5022-744: The characteristic angles 𝟀, 𝝉, 𝟁), plus 2 {\displaystyle {\sqrt {2}}} , 1 {\displaystyle 1} , 2 3 {\displaystyle {\sqrt {\tfrac {2}{3}}}} (edges that are the characteristic radii of the octahedron). The 3-edge path along orthogonal edges of the orthoscheme is 1 {\displaystyle 1} , 1 3 {\displaystyle {\sqrt {\tfrac {1}{3}}}} , 2 3 {\displaystyle {\sqrt {\tfrac {2}{3}}}} , first from an octahedron vertex to an octahedron edge center, then turning 90° to an octahedron face center, then turning 90° to

5115-399: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry,

5208-425: The maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the pentagonal dipyramid , the snub disphenoid , and an irregular polyhedron with 12 vertices and 20 triangular faces. The interior of the compound of two dual tetrahedra is an octahedron, and this compound—called the stella octangula —is its first and only stellation . Correspondingly,

5301-455: The stellated octahedron . The octahedron is one of a family of uniform polyhedra related to the cube. It is also one of the simplest examples of a hypersimplex , a polytope formed by certain intersections of a hypercube with a hyperplane . The octahedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3, n }, continuing into the hyperbolic plane . The regular octahedron can also be considered

5394-631: The 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis , and introducing

5487-496: The 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into

5580-474: The 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry. Points are generally considered fundamental objects for building geometry. They may be defined by

5673-515: The Platonic solids. In his Mysterium Cosmographicum , Kepler also proposed the Solar System by using the Platonic solids setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from the innermost to the outermost: regular octahedron, regular icosahedron , regular dodecahedron , regular tetrahedron , and cube . Many octahedra of interest are square bipyramids . A square bipyramid

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5766-590: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in

5859-416: The appearance is symmetrical by rotating around the axis of symmetry that passing through apices and base's center vertically, and it has mirror symmetry relative to any bisector of the base; it is also symmetrical by reflecting it across a horizontal plane. Therefore, this square bipyramid is face-transitive or isohedral. If the edges of a square bipyramid are all equal in length, then that square bipyramid

5952-412: The concept of angle and distance, finite geometry that omits continuity , and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry

6045-513: The contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes ( c.  287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of

6138-468: The distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model

6231-401: The edges of an octahedron in the ratio of the golden mean to define the vertices of a regular icosahedron . This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. Five octahedra define any given icosahedron in this fashion, and together they define

6324-502: The features of it: A space frame of alternating tetrahedra and half-octahedra derived from the Tetrahedral-octahedral honeycomb was invented by Buckminster Fuller in the 1950s. It is commonly regarded as the strongest building structure for resisting cantilever stresses. A regular octahedron can be augmented into a tetrahedron by adding 4 tetrahedra on alternated faces. Adding tetrahedra to all 8 faces creates

6417-428: The field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits

6510-489: The first five slices above occur at heights r , ⁠ 3 / 8 ⁠ , ⁠ 1 / 2 ⁠ , ⁠ 5 / 8 ⁠ , and s , where r is any number in the range 0 < r ≤ ⁠ 1 / 4 ⁠ , and s is any number in the range ⁠ 3 / 4 ⁠ ≤ s < 1 . The octahedron as a tetratetrahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3. n ), progressing from tilings of

6603-520: The first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem . Pythagoras established the Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c.  355 BC ) developed the method of exhaustion , which allowed the calculation of areas and volumes of curvilinear figures, as well as

6696-526: The former in topology and geometric group theory , the latter in Lie theory and Riemannian geometry . A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between

6789-598: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning

6882-537: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin ) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including

6975-552: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived

7068-411: The most influential books ever written. Euclid introduced certain axioms , or postulates , expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of

7161-429: The multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. In differential geometry,

7254-500: The nature of geometric structures modelled on, or arising out of, the complex plane . Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables , and has found applications to string theory and mirror symmetry . Face-transitive Isohedral polyhedra are called isohedra . They can be described by their face configuration . An isohedron has an even number of faces. The dual of an isohedral polyhedron

7347-408: The octahedron at all vertices), the radius of an inscribed sphere r i {\displaystyle r_{i}} (one that tangent to each of the octahedron's faces), and the radius of a midsphere r m {\displaystyle r_{m}} (one that touches the middle of each edge), are: r u = 2 2 a ≈ 0.707

7440-414: The octahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a 90-60-30 triangle which is one-sixth of an octahedron face. The three faces interior to the octahedron are: a 45-90-45 triangle with edges 1 {\displaystyle 1} , 2 {\displaystyle {\sqrt {2}}} , 1 {\displaystyle 1} ,

7533-737: The octahedron's center. Three left-handed orthoschemes and three right-handed orthoschemes meet in each of the octahedron's eight faces, the six orthoschemes collectively forming a trirectangular tetrahedron : a triangular pyramid with the octahedron face as its equilateral base, and its cube-cornered apex at the center of the octahedron. If the octahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths 4 3 {\displaystyle {\sqrt {\tfrac {4}{3}}}} , 1 {\displaystyle 1} , 1 3 {\displaystyle {\sqrt {\tfrac {1}{3}}}} around its exterior right-triangle face (the edges opposite

7626-415: The octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111. The octahedron's symmetry group is O h , of order 48, the three dimensional hyperoctahedral group . This group's subgroups include D 3d (order 12), the symmetry group of a triangular antiprism ; D 4h (order 16), the symmetry group of a square bipyramid ; and T d (order 24), the symmetry group of

7719-441: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of

7812-490: The origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are: ( ± 1 , 0 , 0 ) , ( 0 , ± 1 , 0 ) , ( 0 , 0 , ± 1 ) . {\displaystyle (\pm 1,0,0),\qquad (0,\pm 1,0),\qquad (0,0,\pm 1).} In three dimensional space , the octahedron with center coordinates (

7905-514: The physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem , a problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During

7998-407: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of

8091-482: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself axiomatically defined. With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that

8184-556: The same as "2-hedral", "3-hedral"... respectively). Here are some examples of k -isohedral polyhedra and tilings, with their faces colored by their k symmetry positions: A cell-transitive or isochoric figure is an n - polytope ( n ≥ 4) or n - honeycomb ( n ≥ 3) that has its cells congruent and transitive with each others. In 3 dimensions, the catoptric honeycombs , duals to the uniform honeycombs, are isochoric. In 4 dimensions, isochoric polytopes have been enumerated up to 20 cells. A facet-transitive or isotopic figure

8277-554: The same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . A solid

8370-410: The sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of * n 32 all of these tilings are Wythoff constructions within a fundamental domain of symmetry, with generator points at the right angle corner of the domain. As a trigonal antiprism , the octahedron is related to the hexagonal dihedral symmetry family. Truncation of two opposite vertices results in

8463-589: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry

8556-409: The theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and

8649-596: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in

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