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78-407: 4 (four) is a number, numeral and digit. It is the natural number following 3 and preceding 5. 4 is the smallest square number > 1, the smallest semiprime and composite number , and the 3rd highly composite number . The number 4 is considered unlucky in many East Asian cultures. Brahmic numerals represented 1, 2, and 3 with as many lines. 4 was simplified by joining its four lines into

156-432: A + b ) ( a − b ) {\displaystyle a^{2}-b^{2}=(a+b)(a-b)} This is the difference-of-squares formula , which can be useful for mental arithmetic: for example, 47 × 53 can be easily computed as 50 − 3 = 2500 − 9 = 2491 . A square number is also the sum of two consecutive triangular numbers . The sum of two consecutive square numbers is a centered square number . Every odd square

234-531: A perfect number . The sum of the n first square numbers is ∑ n = 0 N n 2 = 0 2 + 1 2 + 2 2 + 3 2 + 4 2 + ⋯ + N 2 = N ( N + 1 ) ( 2 N + 1 ) 6 . {\displaystyle \sum _{n=0}^{N}n^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+\cdots +N^{2}={\frac {N(N+1)(2N+1)}{6}}.} The first values of these sums,

312-450: A prime number has factors of only 1 and itself, and since m = 2 is the only non-zero value of m to give a factor of 1 on the right side of the equation above, it follows that 3 is the only prime number one less than a square ( 3 = 2 − 1 ). More generally, the difference of the squares of two numbers is the product of their sum and their difference. That is, a 2 − b 2 = (

390-576: A rectangle or oblong , kite , rhombus , and square . Four is the highest degree general polynomial equation for which there is a solution in radicals . The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors. Three colors are not, in general, sufficient to guarantee this. The largest planar complete graph has four vertices. A solid figure with four faces as well as four vertices

468-443: A square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 3 and can be written as 3 × 3 . The usual notation for the square of a number n is not the product n  ×  n , but the equivalent exponentiation n , usually pronounced as " n squared". The name square number comes from

546-439: A convex polyhedron can be obtained by the process of polar reciprocation . Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron. Abstract polyhedra also have duals, obtained by reversing the partial order defining the polyhedron to obtain its dual or opposite order . These have

624-498: A convex polyhedron, or more generally any simply connected polyhedron with surface a topological sphere, it always equals 2. For more complicated shapes, the Euler characteristic relates to the number of toroidal holes, handles or cross-caps in the surface and will be less than 2. All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For example,

702-455: A cross that looks like the modern plus sign. The Shunga would add a horizontal line on top of the digit, and the Kshatrapa and Pallava evolved the digit to a point where the speed of writing was a secondary concern. The Arabs ' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end

780-418: A list that includes many of these formulas.) Volumes of more complicated polyhedra may not have simple formulas. Volumes of such polyhedra may be computed by subdividing the polyhedron into smaller pieces (for example, by triangulation ). For example, the volume of a regular polyhedron can be computed by dividing it into congruent pyramids , with each pyramid having a face of the polyhedron as its base and

858-423: A polyhedron into a single number χ {\displaystyle \chi } defined by the formula The same formula is also used for the Euler characteristic of other kinds of topological surfaces. It is an invariant of the surface, meaning that when a single surface is subdivided into vertices, edges, and faces in more than one way, the Euler characteristic will be the same for these subdivisions. For

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936-399: A polyhedron is called its symmetry group . All the elements that can be superimposed on each other by symmetries are said to form a symmetry orbit . For example, all the faces of a cube lie in one orbit, while all the edges lie in another. If all the elements of a given dimension, say all the faces, lie in the same orbit, the figure is said to be transitive on that orbit. For example, a cube

1014-411: A polyhedron that can be constructed by attaching more elementary polyhedrons. For example, triaugmented triangular prism is a composite polyhedron since it can be constructed by attaching three equilateral square pyramids onto the square faces of a triangular prism ; the square pyramids and the triangular prism are elementary. A midsphere of a convex polyhedron is a sphere tangent to every edge of

1092-411: A polyhedron to create new faces—or facets—without creating any new vertices). A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face . The stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron. Polyhedra may be classified and are often named according to

1170-422: A polyhedron, an intermediate sphere in radius between the insphere and circumsphere , for polyhedra for which all three of these spheres exist. Every convex polyhedron is combinatorially equivalent to a canonical polyhedron , a polyhedron that has a midsphere whose center coincides with the centroid of the polyhedron. The shape of the canonical polyhedron (but not its scale or position) is uniquely determined by

1248-431: A shared edge) and that every vertex is incident to a single alternating cycle of edges and faces (disallowing shapes like the union of two cubes sharing only a single vertex). For polyhedra defined in these ways, the classification of manifolds implies that the topological type of the surface is completely determined by the combination of its Euler characteristic and orientability. For example, every polyhedron whose surface

1326-424: A single symmetry orbit: Some classes of polyhedra have only a single main axis of symmetry. These include the pyramids , bipyramids , trapezohedra , cupolae , as well as the semiregular prisms and antiprisms. Regular polyhedra are the most highly symmetrical. Altogether there are nine regular polyhedra: five convex and four star polyhedra. The five convex examples have been known since antiquity and are called

1404-418: A square number can end only with square digits (like in base 12, a prime number can end only with prime digits or 1), that is, 0, 1, 4 or 9, as follows: Similar rules can be given for other bases, or for earlier digits (the tens instead of the units digit, for example). All such rules can be proved by checking a fixed number of cases and using modular arithmetic . In general, if a prime   p divides

1482-642: A square number  m then the square of p must also divide m ; if p fails to divide ⁠ m / p ⁠ , then m is definitely not square. Repeating the divisions of the previous sentence, one concludes that every prime must divide a given perfect square an even number of times (including possibly 0 times). Thus, the number m is a square number if and only if, in its canonical representation , all exponents are even. Squarity testing can be used as alternative way in factorization of large numbers. Instead of testing for divisibility, test for squarity: for given m and some number  k , if k − m

1560-475: A triangular number minus eight. Since all triangular numbers have an odd factor, but no two values of 2 differ by an amount containing an odd factor, the only perfect square of the form 2 − 1 is 1, and the only perfect square of the form 2 + 1 is 9. Polyhedron A convex polyhedron is a polyhedron that bounds a convex set . Every convex polyhedron can be constructed as the convex hull of its vertices, and for every finite set of points, not all on

1638-630: Is n , with 0 = 0 being the zeroth one. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, for example, 4 9 = ( 2 3 ) 2 {\displaystyle \textstyle {\frac {4}{9}}=\left({\frac {2}{3}}\right)^{2}} . Starting with 1, there are ⌊ m ⌋ {\displaystyle \lfloor {\sqrt {m}}\rfloor } square numbers up to and including m , where

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1716-1625: Is a tetrahedron , which is the smallest possible number of faces and vertices a polyhedron can have. The regular tetrahedron, also called a 3- simplex , is the simplest Platonic solid . It has four regular triangles as faces that are themselves at dual positions with the vertices of another tetrahedron. The smallest non- cyclic group has four elements; it is the Klein four-group . A n alternating groups are not simple for values n {\displaystyle n} ≤ 4 {\displaystyle 4} . There are four Hopf fibrations of hyperspheres : S 0 ↪ S 1 → S 1 , S 1 ↪ S 3 → S 2 , S 3 ↪ S 7 → S 4 , S 7 ↪ S 15 → S 8 . {\displaystyle {\begin{aligned}S^{0}&\hookrightarrow S^{1}\to S^{1},\\S^{1}&\hookrightarrow S^{3}\to S^{2},\\S^{3}&\hookrightarrow S^{7}\to S^{4},\\S^{7}&\hookrightarrow S^{15}\to S^{8}.\\\end{aligned}}} They are defined as locally trivial fibrations that map f : S 2 n − 1 → S n {\displaystyle f:S^{2n-1}\rightarrow S^{n}} for values of n = 2 , 4 , 8 {\displaystyle n=2,4,8} (aside from

1794-450: Is a polyhedron that forms a convex set as a solid. That being said, it is a three-dimensional solid whose every line segment connects two of its points lies its interior or on its boundary ; none of its faces are coplanar (they do not share the same plane) and none of its edges are collinear (they are not segments of the same line). A convex polyhedron can also be defined as a bounded intersection of finitely many half-spaces , or as

1872-456: Is a polyhedron with symmetries acting transitively on its faces. Their topology can be represented by a face configuration . All 5 Platonic solids and 13 Catalan solids are isohedra, as well as the infinite families of trapezohedra and bipyramids . Some definitions of isohedra allow geometric variations including concave and self-intersecting forms. Many of the symmetries or point groups in three dimensions are named after polyhedra having

1950-422: Is also a centered octagonal number . Another property of a square number is that (except 0) it has an odd number of positive divisors, while other natural numbers have an even number of positive divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs. Lagrange's four-square theorem states that any positive integer can be written as

2028-409: Is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a centered octagonal number . The difference between any two odd perfect squares is a multiple of 8. The difference between 1 and any higher odd perfect square always is eight times a triangular number, while the difference between 9 and any higher odd perfect square is eight times

2106-459: Is always the product of m − 1 {\displaystyle m-1} and m + 1 ; {\displaystyle m+1;} that is, m 2 − 1 = ( m − 1 ) ( m + 1 ) . {\displaystyle m^{2}-1=(m-1)(m+1).} For example, since 7 = 49 , one has 6 × 8 = 48 {\displaystyle 6\times 8=48} . Since

2184-435: Is an orientable manifold and whose Euler characteristic is 2 must be a topological sphere. A toroidal polyhedron is a polyhedron whose Euler characteristic is less than or equal to 0, or equivalently whose genus is 1 or greater. Topologically, the surfaces of such polyhedra are torus surfaces having one or more holes through the middle. For every convex polyhedron, there exists a dual polyhedron having The dual of

2262-466: Is called a lattice polyhedron or integral polyhedron . The Ehrhart polynomial of a lattice polyhedron counts how many points with integer coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polynomials lies at the intersection of combinatorics and commutative algebra . There is a far-reaching equivalence between lattice polyhedra and certain algebraic varieties called toric varieties . This

2340-460: Is common instead to slice the polyhedron by a small sphere centered at the vertex. Again, this produces a shape for the vertex figure that is invariant up to scaling. All of these choices lead to vertex figures with the same combinatorial structure, for the polyhedra to which they can be applied, but they may give them different geometric shapes. The surface area of a polyhedron is the sum of areas of its faces, for definitions of polyhedra for which

2418-414: Is face-transitive, while a truncated cube has two symmetry orbits of faces. The same abstract structure may support more or less symmetric geometric polyhedra. But where a polyhedral name is given, such as icosidodecahedron , the most symmetrical geometry is often implied. There are several types of highly symmetric polyhedron, classified by which kind of element – faces, edges, or vertices – belong to

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2496-483: Is known as the bellows theorem. A polyhedral compound is made of two or more polyhedra sharing a common centre. Symmetrical compounds often share the same vertices as other well-known polyhedra and may often also be formed by stellation. Some are listed in the list of Wenninger polyhedron models . An orthogonal polyhedron is one all of whose edges are parallel to axes of a Cartesian coordinate system. This implies that all faces meet at right angles , but this condition

2574-474: Is no universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra ) or include shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not manifolds ). As Branko Grünbaum observed, "The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... at each stage ...

2652-437: Is orientable or non-orientable by considering a topological cell complex with the same incidences between its vertices, edges, and faces. A more subtle distinction between polyhedron surfaces is given by their Euler characteristic , which combines the numbers of vertices V {\displaystyle V} , edges E {\displaystyle E} , and faces F {\displaystyle F} of

2730-678: Is seen with an open top: [REDACTED] . Television stations that operate on channel 4 have occasionally made use of another variation of the "open 4", with the open portion being on the side, rather than the top. This version resembles the Canadian Aboriginal syllabics letter ᔦ. The magnetic ink character recognition "CMC-7" font also uses this variety of "4". There are four elementary arithmetic operations in mathematics: addition ( + ), subtraction ( − ), multiplication ( × ), and division ( ÷ ). Lagrange's four-square theorem states that every positive integer can be written as

2808-491: Is the square of an integer  n then k − n divides m . (This is an application of the factorization of a difference of two squares .) For example, 100 − 9991 is the square of 3, so consequently 100 − 3 divides 9991. This test is deterministic for odd divisors in the range from k − n to k + n where k covers some range of natural numbers k ≥ m . {\displaystyle k\geq {\sqrt {m}}.} A square number cannot be

2886-456: Is weaker: Jessen's icosahedron has faces meeting at right angles, but does not have axis-parallel edges. Aside from the rectangular cuboids , orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons, also known as rectilinear polygons . Orthogonal polyhedra are used in computational geometry , where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding

2964-506: The Minkowski sums of line segments, and include several important space-filling polyhedra. A space-filling polyhedron packs with copies of itself to fill space. Such a close-packing or space-filling is often called a tessellation of space or a honeycomb. Space-filling polyhedra must have a Dehn invariant equal to zero. Some honeycombs involve more than one kind of polyhedron. A convex polyhedron in which all vertices have integer coordinates

3042-509: The Platonic solids . These are the triangular pyramid or tetrahedron , cube , octahedron , dodecahedron and icosahedron : There are also four regular star polyhedra, known as the Kepler–Poinsot polyhedra after their discoverers. The dual of a regular polyhedron is also regular. Uniform polyhedra are vertex-transitive and every face is a regular polygon . They may be subdivided into

3120-712: The convex hull of finitely many points, in either case, restricted to intersections or hulls that have nonzero volume. Important classes of convex polyhedra include the family of prismatoid , the Platonic solids , the Archimedean solids and their duals the Catalan solids , and the regular polygonal faces polyhedron. The prismatoids are the polyhedron whose vertices lie on two parallel planes and their faces are likely to be trapezoids and triangles. Examples of prismatoids are pyramids , wedges , parallelipipeds , prisms , antiprisms , cupolas , and frustums . The Platonic solids are

3198-449: The deltahedron (whose faces are all equilateral triangles and Johnson solids (whose faces are arbitrary regular polygons). The convex polyhedron can be categorized into elementary polyhedron or composite polyhedron. An elementary polyhedron is a convex regular-faced polyhedron that cannot be produced into two or more polyhedrons by slicing it with a plane. Quite opposite to a composite polyhedron, it can be alternatively defined as

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3276-404: The n th square number can be computed from the previous square by n = ( n − 1) + ( n − 1) + n = ( n − 1) + (2 n − 1) . Alternatively, the n th square number can be calculated from the previous two by doubling the ( n  − 1) th square, subtracting the ( n  − 2) th square number, and adding 2, because n = 2( n − 1) − ( n − 2) + 2 . For example, The square minus one of a number m

3354-409: The real number system , square numbers are non-negative . A non-negative integer is a square number when its square root is again an integer. For example, 9 = 3 , {\displaystyle {\sqrt {9}}=3,} so 9 is a square number. A positive integer that has no square divisors except 1 is called square-free . For a non-negative integer n , the n th square number

3432-575: The regular , quasi-regular , or semi-regular , and may be convex or starry. The duals of the uniform polyhedra have irregular faces but are face-transitive , and every vertex figure is a regular polygon. A uniform polyhedron has the same symmetry orbits as its dual, with the faces and vertices simply swapped over. The duals of the convex Archimedean polyhedra are sometimes called the Catalan solids . The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are convex or not. An isohedron

3510-485: The square pyramidal numbers , are: (sequence A000330 in the OEIS ) 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201... The sum of the first odd integers, beginning with one, is a perfect square: 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, etc. This explains Galileo's law of odd numbers : if a body falling from rest covers one unit of distance in

3588-406: The tetrahemihexahedron , it is not possible to colour the two sides of each face with two different colours so that adjacent faces have consistent colours. In this case the polyhedron is said to be non-orientable. For polyhedra with self-crossing faces, it may not be clear what it means for adjacent faces to be consistently coloured, but for these polyhedra it is still possible to determine whether it

3666-399: The area of a face is well-defined. The geodesic distance between any two points on the surface of a polyhedron measures the length of the shortest curve that connects the two points, remaining within the surface. By Alexandrov's uniqueness theorem , every convex polyhedron is uniquely determined by the metric space of geodesic distances on its surface. However, non-convex polyhedra can have

3744-416: The associated symmetry. These include: Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. Examples include the snub cuboctahedron and snub icosidodecahedron . A zonohedron is a convex polyhedron in which every face is a polygon that is symmetric under rotations through 180°. Zonohedra can also be characterized as

3822-439: The centre of the polyhedron as its apex. In general, it can be derived from the divergence theorem that the volume of a polyhedral solid is given by 1 3 | ∑ F ( Q F ⋅ N F ) area ⁡ ( F ) | , {\displaystyle {\frac {1}{3}}\left|\sum _{F}(Q_{F}\cdot N_{F})\operatorname {area} (F)\right|,} where

3900-458: The column for Greek cardinal numbers. The names of tetrahedra, hexahedra, octahedra (8-sided polyhedra), dodecahedra (12-sided polyhedra), and icosahedra (20-sided polyhedra) are sometimes used without additional qualification to refer to the Platonic solids , and sometimes used to refer more generally to polyhedra with the given number of sides without any assumption of symmetry. Some polyhedra have two distinct sides to their surface. For example,

3978-411: The combinatorial structure of the given polyhedron. Some polyhedrons do not have the property of convexity, and they are called non-convex polyhedrons . Such polyhedrons are star polyhedrons and Kepler–Poinsot polyhedrons , which constructed by either stellation (process of extending the faces—within their planes—so that they meet) or faceting (whose process of removing parts of

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4056-432: The expression ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } represents the floor of the number  x . The squares (sequence A000290 in the OEIS ) smaller than 60  = 3600 are: The difference between any perfect square and its predecessor is given by the identity n − ( n − 1) = 2 n − 1 . Equivalently, it is possible to count square numbers by adding together

4134-413: The first arbitrary time interval, it covers 3, 5, 7, etc., units of distance in subsequent time intervals of the same length. From s = u t + 1 2 a t 2 {\displaystyle s=ut+{\tfrac {1}{2}}at^{2}} , for u = 0 and constant a (acceleration due to gravity without air resistance); so s is proportional to t , and the distance from

4212-456: The five ancientness polyhedrons— tetrahedron , octahedron , icosahedron , cube , and dodecahedron —classified by Plato in his Timaeus whose connecting four classical elements of nature. The Archimedean solids are the class of thirteen polyhedrons whose faces are all regular polygons and whose vertices are symmetric to each other; their dual polyhedrons are Catalan solids . The class of regular polygonal faces polyhedron are

4290-552: The inside and outside of a convex polyhedron paper model can each be given a different colour (although the inside colour will be hidden from view). These polyhedra are orientable . The same is true for non-convex polyhedra without self-crossings. Some non-convex self-crossing polyhedra can be coloured in the same way but have regions turned "inside out" so that both colours appear on the outside in different places; these are still considered to be orientable. However, for some other self-crossing polyhedra with simple-polygon faces, such as

4368-780: The last square, the last square's root, and the current root, that is, n = ( n − 1) + ( n − 1) + n . The number m is a square number if and only if one can arrange m points in a square: The expression for the n th square number is n . This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (shown in magenta). The formula follows: n 2 = ∑ k = 1 n ( 2 k − 1 ) . {\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1).} For example, 5 = 25 = 1 + 3 + 5 + 7 + 9 . There are several recursive methods for computing square numbers. For example,

4446-440: The literature on higher-dimensional geometry uses the term "polyhedron" to mean something else: not a three-dimensional polytope, but a shape that is different from a polytope in some way. For instance, some sources define a convex polyhedron to be the intersection of finitely many half-spaces , and a polytope to be a bounded polyhedron. The remainder of this article considers only three-dimensional polyhedra. A convex polyhedron

4524-565: The local structure of the polyhedron around the vertex. Precise definitions vary, but a vertex figure can be thought of as the polygon exposed where a slice through the polyhedron cuts off a vertex. For the Platonic solids and other highly-symmetric polyhedra, this slice may be chosen to pass through the midpoints of each edge incident to the vertex, but other polyhedra may not have a plane through these points. For convex polyhedra, and more generally for polyhedra whose vertices are in convex position , this slice can be chosen as any plane separating

4602-443: The name of the shape. The unit of area is defined as the area of a unit square ( 1 × 1 ). Hence, a square with side length n has area n . If a square number is represented by n points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of n ; thus, square numbers are a type of figurate numbers (other examples being cube numbers and triangular numbers ). In

4680-566: The number of faces. The naming system is based on Classical Greek, and combines a prefix counting the faces with the suffix "hedron", meaning "base" or "seat" and referring to the faces. For example a tetrahedron is a polyhedron with four faces, a pentahedron is a polyhedron with five faces, a hexahedron is a polyhedron with six faces, etc. For a complete list of the Greek numeral prefixes see Numeral prefix § Table of number prefixes in English , in

4758-434: The one-holed toroid and the Klein bottle both have χ = 0 {\displaystyle \chi =0} , with the first being orientable and the other not. For many (but not all) ways of defining polyhedra, the surface of the polyhedron is required to be a manifold . This means that every edge is part of the boundary of exactly two faces (disallowing shapes like the union of two cubes that meet only along

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4836-486: The polyhedron as a solid, whether they describe it as a surface, or whether they describe it more abstractly based on its incidence geometry . In all of these definitions, a polyhedron is typically understood as a three-dimensional example of the more general polytope in any number of dimensions. For example, a polygon has a two-dimensional body and no faces, while a 4-polytope has a four-dimensional body and an additional set of three-dimensional "cells". However, some of

4914-430: The same Euler characteristic and orientability as the initial polyhedron. However, this form of duality does not describe the shape of a dual polyhedron, but only its combinatorial structure. For some definitions of non-convex geometric polyhedra, there exist polyhedra whose abstract duals cannot be realized as geometric polyhedra under the same definition. For every vertex one can define a vertex figure , which describes

4992-601: The same plane, the convex hull is a convex polyhedron. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a generalization of a 2-dimensional polygon and a 3-dimensional specialization of a polytope , a more general concept in any number of dimensions . Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others, and there

5070-442: The same surface distances as each other, or the same as certain convex polyhedra. Polyhedral solids have an associated quantity called volume that measures how much space they occupy. Simple families of solids may have simple formulas for their volumes; for example, the volumes of pyramids, prisms, and parallelepipeds can easily be expressed in terms of their edge lengths or other coordinates. (See Volume § Volume formulas for

5148-465: The same volume that cannot be cut into smaller polyhedra and reassembled into each other. To prove this Dehn discovered another value associated with a polyhedron, the Dehn invariant , such that two polyhedra can only be dissected into each other when they have the same volume and the same Dehn invariant. It was later proven by Sydler that this is the only obstacle to dissection: every two Euclidean polyhedra with

5226-494: The same volumes and Dehn invariants can be cut up and reassembled into each other. The Dehn invariant is not a number, but a vector in an infinite-dimensional vector space, determined from the lengths and dihedral angles of a polyhedron's edges. Another of Hilbert's problems, Hilbert's 18th problem , concerns (among other things) polyhedra that tile space . Every such polyhedron must have Dehn invariant zero. The Dehn invariant has also been connected to flexible polyhedra by

5304-844: The starting point are consecutive squares for integer values of time elapsed. The sum of the n first cubes is the square of the sum of the n first positive integers; this is Nicomachus's theorem . All fourth powers, sixth powers, eighth powers and so on are perfect squares. A unique relationship with triangular numbers T n {\displaystyle T_{n}} is: ( T n ) 2 + ( T n + 1 ) 2 = T ( n + 1 ) 2 {\displaystyle (T_{n})^{2}+(T_{n+1})^{2}=T_{(n+1)^{2}}} Squares of even numbers are even, and are divisible by 4, since (2 n ) = 4 n . Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2 n + 1) = 4 n ( n + 1) + 1 , and n ( n + 1)

5382-491: The strong bellows theorem, which states that the Dehn invariant of any flexible polyhedron remains invariant as it flexes. Many of the most studied polyhedra are highly symmetrical , that is, their appearance is unchanged by some reflection or rotation of space. Each such symmetry may change the location of a given vertex, face, or edge, but the set of all vertices (likewise faces, edges) is unchanged. The collection of symmetries of

5460-450: The sum is over faces F of the polyhedron, Q F is an arbitrary point on face F , N F is the unit vector perpendicular to F pointing outside the solid, and the multiplication dot is the dot product . In higher dimensions, volume computation may be challenging, in part because of the difficulty of listing the faces of a convex polyhedron specified only by its vertices, and there exist specialized algorithms to determine

5538-419: The sum of at most four squares . Four is one of four all-Harshad numbers . Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4 x = y 2 − z 2 {\displaystyle 4x=y^{2}-z^{2}} . A four-sided plane figure is a quadrilateral or quadrangle, sometimes also called a tetragon . It can be further classified as

5616-420: The sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form 4 (8 m + 7) . A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form 4 k + 3 . This is generalized by Waring's problem . In base 10 , a square number can end only with digits 0, 1, 4, 5, 6 or 9, as follows: In base 12 ,

5694-492: The trivial fibration mapping between two points and a circle ). In Knuth's up-arrow notation , 2 + 2 = 2 × 2 = 2 2 = 2 ↑ ↑ 2 = 2 ↑ ↑ ↑ 2 = . . . = 4 {\displaystyle 2+2=2\times 2=2^{2}=2\uparrow \uparrow 2=2\uparrow \uparrow \uparrow 2=\;...\;=4} , and so forth, for any number of up arrows. Square number In mathematics ,

5772-420: The vertex from the other vertices. When the polyhedron has a center of symmetry, it is standard to choose this plane to be perpendicular to the line through the given vertex and the center; with this choice, the shape of the vertex figure is determined up to scaling. When the vertices of a polyhedron are not in convex position, there will not always be a plane separating each vertex from the rest. In this case, it

5850-475: The volume in these cases. In two dimensions, the Bolyai–Gerwien theorem asserts that any polygon may be transformed into any other polygon of the same area by cutting it up into finitely many polygonal pieces and rearranging them . The analogous question for polyhedra was the subject of Hilbert's third problem . Max Dehn solved this problem by showing that, unlike in the 2-D case, there exist polyhedra of

5928-474: The writers failed to define what are the polyhedra". Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be described by its vertices (corner points), edges (line segments connecting certain pairs of vertices), faces (two-dimensional polygons ), and that it sometimes can be said to have a particular three-dimensional interior volume . One can distinguish among these different definitions according to whether they describe

6006-522: Was finished off with a curve. The Europeans dropped the finishing curve and gradually made the digit less cursive, ending up with a digit very close to the original Brahmin cross. While the shape of the character for the digit 4 has an ascender in most modern typefaces , in typefaces with text figures the glyph usually has a descender , as, for example, in [REDACTED] . On the seven-segment displays of pocket calculators and digital watches, as well as certain optical character recognition fonts, 4

6084-520: Was used by Stanley to prove the Dehn–Sommerville equations for simplicial polytopes . It is possible for some polyhedra to change their overall shape, while keeping the shapes of their faces the same, by varying the angles of their edges. A polyhedron that can do this is called a flexible polyhedron. By Cauchy's rigidity theorem , flexible polyhedra must be non-convex. The volume of a flexible polyhedron must remain constant as it flexes; this result

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