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71-440: 29 is the number of days February has on a leap year . 29 is the tenth prime number . 29 is the fifth primorial prime , like its twin prime 31 . 29 is the smallest positive whole number that cannot be made from the numbers { 1 , 2 , 3 , 4 } {\displaystyle \{1,2,3,4\}} , using each digit exactly once and using only addition, subtraction, multiplication, and division. None of

142-476: A polyhedron ( pl. : polyhedra or polyhedrons ; from Greek πολύ (poly-)  'many' and ἕδρον (-hedron)  'base, seat') is a three-dimensional figure with flat polygonal faces , straight edges and sharp corners or vertices . 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

213-400: A Monday, it occurs as part of a common year starting on Friday , in which February 1st is a Monday and the 28th is a Sunday; the most recent occurrence was 2021 , and the next one will be 2027 . In countries that start their week on a Sunday, it occurs in a common year starting on Thursday ; the most recent occurrence was 2015 and the next occurrence will be 2026 . The pattern is broken by

284-439: 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

355-498: 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,

426-491: A fundamental polyhedron , and the highest dimension that holds arithmetic discrete groups of reflections with noncompact unbounded fundamental polyhedra. February February is the second month of the year in the Julian and Gregorian calendars . The month has 28 days in common years and 29 in leap years , with the 29th day being called the leap day . February is the third and last month of meteorological winter in

497-418: 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

568-423: 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

639-399: 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

710-411: 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

781-411: 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

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852-422: 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

923-586: A proposal was put forward in Kmetijske in rokodelske novice by the Slovene Society of Ljubljana to call this month talnik (related to ice melting), but it did not stick. The idea was proposed by a priest, Blaž Potočnik. Another name of February in Slovene was vesnar , after the mythological character Vesna . Having only 28 days in common years, February is the only month of the year that can pass without

994-431: 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

1065-537: A single full moon . Using Coordinated Universal Time as the basis for determining the date and time of a full moon, this last happened in 2018 and will next happen in 2037. The same is true regarding a new moon : again using Coordinated Universal Time as the basis, this last happened in 2014 and will next happen in 2033. February is also the only month of the calendar that, at intervals alternating between one of six years and two of eleven years, has exactly four full 7-day weeks . In countries that start their week on

1136-424: 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

1207-741: A skipped leap year, but no leap year has been skipped since 1900 and no others will be skipped until 2100. February meteor showers include the Alpha Centaurids (appearing in early February), the March Virginids (lasting from February 14 to April 25, peaking around March 20), the Delta Cancrids (appearing December 14 to February 14, peaking on January 17), the Omicron Centaurids (late January through February, peaking in mid-February), Theta Centaurids (January 23 – March 12, only visible in

1278-450: 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

1349-456: 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

1420-477: Is also the twenty-fifth even, square -free sphenic number with three distinct prime numbers p × q × r {\displaystyle p\times q\times r} as factors, and the fifteenth such that p + q + r + 1 {\displaystyle p+q+r+1} is prime (where in its case, 2 + 5 + 29 + 1 = 37 ). The 29th dimension is the highest dimension for compact hyperbolic Coxeter polytopes that are bounded by

1491-435: 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

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1562-505: Is called únor , meaning month of submerging (of river ice). In Slovene , February is traditionally called svečan , related to icicles or Candlemas . This name originates from sičan , written as svičan in the New Carniolan Almanac from 1775 and changed to its final form by Franc Metelko in his New Almanac from 1824. The name was also spelled sečan , meaning "the month of cutting down of trees". In 1848,

1633-466: 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

1704-460: 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

1775-414: 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

1846-483: 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

1917-474: 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 ...

1988-437: 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

2059-722: Is the amethyst , which symbolizes piety, humility, spiritual wisdom, and sincerity. This list does not necessarily imply either official status nor general observance. (All Baha'i, Islamic, and Jewish observances begin at the sundown prior to the date listed, and end at sundown of the date in question unless otherwise noted.) First Saturday First Sunday First Week of February (first Monday, ending on Sunday) First Monday First Friday Second Saturday Second Sunday Second Monday Second Tuesday Week of February 22 Third Monday Third Thursday Third Friday Last Friday Last Saturday Last day of February Polyhedron In geometry ,

2130-456: 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

2201-506: 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

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2272-619: The Northern Hemisphere . In the Southern Hemisphere , February is the third and last month of meteorological summer (being the seasonal equivalent of what is August in the Northern Hemisphere). "February" is pronounced in several different ways. The beginning of the word is commonly pronounced either as / ˈ f ɛ b j u -/ FEB -yoo- or / ˈ f ɛ b r u -/ FEB -roo- ; many people drop

2343-509: 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

2414-712: 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

2485-449: 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

2556-575: 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

2627-406: 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

2698-610: The Middle Ages, when the numbered Anno Domini year began on March 25 or December 25, the second month was February whenever all twelve months were displayed in order. The Gregorian calendar reforms made slight changes to the system for determining which years were leap years, but also contained a 29-day February. Historical names for February include the Old English terms Solmonath (mud month) and Kale-monath (named for cabbage ) as well as Charlemagne 's designation Hornung. In Finnish,

2769-636: The UK. The Roman month Februarius was named after the Latin term februum , which means "purification", via the purification ritual Februa held on February 15 (full moon) in the old lunar Roman calendar . January and February were the last two months to be added to the Roman calendar, since the Romans originally considered winter a monthless period. They were added by Numa Pompilius about 713 BC. February remained

2840-399: 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

2911-416: 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

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2982-439: 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

3053-458: 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,

3124-411: 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

3195-450: The first "r", replacing it with / j / , as if it were spelled "Febuary". This comes about by analogy with "January" ( / ˈ dʒ æ n . j u -/ ), as well as by a dissimilation effect whereby having two "r"s close to each other causes one to change. The ending of the word is pronounced /- ɛr i / -⁠err-ee in the US and /- ər i / -⁠ər-ee in

3266-422: The first twenty-nine natural numbers have more than two different prime factors (in other words, this is the longest such consecutive sequence; the first sphenic number or triprime, 30 is the product of the first three primes 2 , 3 , and 5 ). 29 is also, On the other hand, 29 represents the sum of the first cluster of consecutive semiprimes with distinct prime factors ( 14 , 15 ). These two numbers are

3337-456: 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

3408-552: 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

3479-792: The last month of the calendar year until the time of the decemvirs ( c.  450 BC ), when it became the second month. At certain times February was truncated to 23 or 24 days, and a 27-day intercalary month, Intercalaris , was occasionally inserted immediately after February to realign the year with the seasons . February observances in Ancient Rome included Amburbium (precise date unknown), Sementivae (February 2), Februa (February 13–15), Lupercalia (February 13–15), Parentalia (February 13–22), Quirinalia (February 17), Feralia (February 21), Caristia (February 22), Terminalia (February 23), Regifugium (February 24), and Agonium Martiale (February 27). These days do not correspond to

3550-440: 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

3621-565: 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

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3692-414: The modern Gregorian calendar. Under the reforms that instituted the Julian calendar , Intercalaris was abolished, leap years occurred regularly every fourth year, and in leap years February gained a 29th day. Thereafter, it remained the second month of the calendar year, meaning the order that months are displayed (January, February, March, ..., December) within a year-at-a-glance calendar. Even during

3763-411: The month is called helmikuu , meaning "month of the pearl"; when snow melts on tree branches, it forms droplets, and as these freeze again, they are like pearls of ice. In Polish and Ukrainian , respectively, the month is called luty or лютий ( lyutiy ), meaning the month of ice or hard frost. In Macedonian the month is sečko ( сечко ), meaning month of cutting (wood). In Czech, it

3834-566: 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

3905-481: 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

3976-467: The only numbers whose arithmetic mean of divisors is the first perfect number and unitary perfect number , 6 (that is also the smallest semiprime with distinct factors). The pair (14, 15) is also the first floor and ceiling values of imaginary parts of non-trivial zeroes in the Riemann zeta function , ζ . {\displaystyle \zeta .} 29 is the largest prime factor of

4047-486: 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

4118-430: 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

4189-601: 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

4260-442: 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

4331-465: 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

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4402-494: 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

4473-426: The set of the first fifteen integers, or equivalently, the first two-hundred and ninety integers. Alternatively, a more precise version states that an integer quadratic matrix represents all positive integers when it contains the set of twenty-nine integers between 1 and 290 : The largest member 290 is the product between 29 and its index in the sequence of prime numbers , 10 . The largest member in this sequence

4544-420: The smallest number with an abundancy index of 3, It is also the largest prime factor of the smallest abundant number not divisible by the first even (of only one) and odd primes, 5391411025 = 5 × 7 × 11 × 13 × 17 × 19 × 23 × 29. Both of these numbers are divisible by consecutive prime numbers ending in 29. The 15 and 290 theorems describes integer-quadratic matrices that describe all positive integers , by

4615-450: The southern hemisphere), Eta Virginids (February 24 and March 27, peaking around March 18), and Pi Virginids (February 13 and April 8, peaking between March 3 and March 9). The zodiac signs of February are Aquarius (until February 18) and Pisces (February 19 onward). Its birth flowers are the violet ( Viola ), the common primrose ( Primula vulgaris ), and the Iris . Its birthstone

4686-491: 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

4757-450: 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

4828-420: 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

4899-475: 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

4970-474: 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

5041-520: 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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