Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Polyhedron
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== General characteristics == ===Number of faces=== Polyhedra may be classified and are often named according to 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.<ref>{{citation|title=The New Elements of Mathematics, Volume II: Algebra and Geometry|first=Charles S.|last=Peirce|author-link=Charles Sanders Peirce|editor-first=Carolyn|editor-last=Eisele|editor-link=Carolyn Eisele|year=1976|publisher=Mouton Publishers & Humanities Press|page=297|isbn=9783110818840 |url=https://books.google.com/books?id=Z3ldDwAAQBAJ&pg=PA297}}</ref> For a complete list of the Greek numeral prefixes see {{slink|Numeral prefix|Table of number prefixes in English}}, in the column for Greek cardinal numbers. The names of tetrahedra, hexahedra, octahedra (eight-sided polyhedra), dodecahedra (twelve-sided polyhedra), and icosahedra (twenty-sided polyhedra) are sometimes used without additional qualification to refer to the [[Platonic solid]]s, and sometimes used to refer more generally to polyhedra with the given number of sides without any assumption of symmetry.<ref>{{citation|orig-date=1996|year=2020|title=Crystal Structures: Patterns and Symmetry|first1=Michael|last1=O'Keefe|first2=Bruce G.|last2=Hyde|publisher=Dover Publications|page=134|isbn=9780486836546 |url=https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA134}}</ref> ===Topological classification=== {{main|Toroidal polyhedron|Euler characteristic}} [[File:Tetrahemihexahedron rotation.gif|thumb|The [[tetrahemihexahedron]], a non-orientable self-intersecting polyhedron with four triangular faces (red) and three square faces (yellow). As with a [[Möbius strip]] or [[Klein bottle]], a continuous path along the surface of this polyhedron can reach the point on the opposite side of the surface from its starting point, making it impossible to separate the surface into an inside and an outside. (Topologically, this polyhedron is a [[real projective plane]].)]] Some polyhedra have two distinct sides to their surface. For example, 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 [[Orientability|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 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 is orientable or non-orientable by considering a topological [[cell complex]] with the same incidences between its vertices, edges, and faces.<ref name=ringel>{{citation | last = Ringel | first = Gerhard | contribution = Classification of surfaces | doi = 10.1007/978-3-642-65759-7_3 | pages = 34–53 | publisher = Springer | title = Map Color Theorem | year = 1974| isbn = 978-3-642-65761-0 }}</ref> A more subtle distinction between polyhedron surfaces is given by their [[Euler characteristic]], which combines the numbers of vertices <math>V</math>, edges <math>E</math>, and faces <math>F</math> of a polyhedron into a single number <math>\chi</math> defined by the formula :<math>\chi=V-E+F.\ </math> 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 a convex polyhedron, or more generally any simply connected polyhedron with the surface of a topological sphere, it always equals 2. For more complicated shapes, the Euler characteristic relates to the number of [[toroid]]al holes, handles or [[cross-cap]]s in the surface and will be less than 2.<ref>{{citation | last = Richeson | first = David S. | author-link = David Richeson | isbn = 978-0-691-12677-7 | location = Princeton, NJ | mr = 2440945 | publisher = Princeton University Press | title = Euler's Gem: The polyhedron formula and the birth of topology | title-link = Euler's Gem | year = 2008}}, pp. 157, 180.</ref> All polyhedra with odd-numbered Euler characteristics are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For example, the one-holed [[toroid]] and the [[Klein bottle]] both have <math>\chi = 0</math>, with the first being orientable and the other not.<ref name=ringel/> 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 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 is an orientable manifold and whose Euler characteristic is 2 must be a topological sphere.<ref name=ringel/> A [[toroidal polyhedron]] is a polyhedron whose Euler characteristic is less than or equal to 0, or equivalently whose [[Genus (mathematics)|genus]] is 1 or greater. Topologically, the surfaces of such polyhedra are [[torus]] surfaces having one or more holes through the middle.<ref>{{citation|first=B. M.|last=Stewart|title=Adventures Among the Toroids: A Study of Orientable Polyhedra with Regular Faces|title-link= Adventures Among the Toroids |edition=2nd|year=1980|isbn=978-0-686-11936-4|publisher=B. M. Stewart}}.</ref> One of the notable example is the [[Szilassi polyhedron]], which has the geometrically ralizes the [[Heawood map]]. ===Duality=== {{main|Dual polyhedron}} [[File:Dual Cube-Octahedron.svg|thumb|180px|The octahedron is dual to the cube]] For every convex polyhedron, there exists a dual polyhedron having * faces in place of the original's vertices and vice versa, and * the same number of edges. The dual of a convex polyhedron can be obtained by the process of [[Dual polyhedron#Polar reciprocation|polar reciprocation]].<ref>{{citation | last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy | last2 = Rollett | first2 = A.P. | edition = 2nd | location = Oxford | mr = 0124167 | publisher = Clarendon Press | title = Mathematical models | title-link = Mathematical Models (Cundy and Rollett) | year = 1961 | contribution = 3.2 Duality | pages = 78–79}}.<!-- Describes only duality by polar reciprocation through the midsphere --></ref> 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.<ref>{{citation | last1 = Grünbaum | first1 = B. | author1-link = Branko Grünbaum | last2 = Shephard | first2 = G.C. | author2-link = Geoffrey Colin Shephard | doi = 10.1112/blms/1.3.257 | journal = [[Bulletin of the London Mathematical Society]] | mr = 0250188 | pages = 257–300 | title = Convex polytopes | url = http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf | volume = 1 | issue = 3 | year = 1969 | access-date = 2017-02-21 | archive-url = https://web.archive.org/web/20170222114014/http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf | archive-date = 2017-02-22 }}. See in particular the bottom of page 260.</ref> Abstract polyhedra also have duals, obtained by reversing the [[partial order]] defining the polyhedron to obtain its [[Duality (order theory)|dual or opposite order]].<ref name=grunbaum-same/> These have 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.<ref name=acoptic/> ===Vertex figures=== {{Main|Vertex figure}} For every vertex one can define a [[vertex figure]], which describes 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.<ref name=cromwell/> For the [[Platonic solid]]s and other highly-symmetric polyhedra, this slice may be chosen to pass through the midpoints of each edge incident to the vertex,<ref>{{citation| first = H. S. M. | last = Coxeter | author-link = Harold Scott MacDonald Coxeter|title=Regular Polytopes|title-link=Regular Polytopes (book)|publisher=Methuen|year=1947|page=[https://books.google.com/books?id=iWvXsVInpgMC&pg=PA16 16]}}</ref> 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 the vertex from the other vertices.<ref>{{citation | last = Barnette | first = David | journal = Pacific Journal of Mathematics | mr = 328773 | pages = 349–354 | title = A proof of the lower bound conjecture for convex polytopes | url = https://projecteuclid.org/euclid.pjm/1102946311 | volume = 46 | year = 1973| issue = 2 | doi = 10.2140/pjm.1973.46.349 | doi-access = free }}</ref> 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;<ref>{{citation | last = Luotoniemi | first = Taneli | editor1-last = Swart | editor1-first = David | editor2-last = Séquin | editor2-first = Carlo H. | editor3-last = Fenyvesi | editor3-first = Kristóf | contribution = Crooked houses: Visualizing the polychora with hyperbolic patchwork | contribution-url = https://archive.bridgesmathart.org/2017/bridges2017-17.html | isbn = 978-1-938664-22-9 | location = Phoenix, Arizona | pages = 17–24 | publisher = Tessellations Publishing | title = Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture | year = 2017}}</ref> 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 is common instead to slice the polyhedron by a small sphere centered at the vertex.<ref>{{citation | date = January 1930 | doi = 10.1098/rsta.1930.0009 | first = H. S. M. | last = Coxeter | author-link = Harold Scott MacDonald Coxeter | issue = 670–680 | journal = Philosophical Transactions of the Royal Society of London, Series A | pages = 329–425 | publisher = The Royal Society | title = The polytopes with regular-prismatic vertex figures | volume = 229| bibcode = 1930RSPTA.229..329C }}</ref> 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. ===Surface area and lines inside polyhedra === The [[surface area]] of a polyhedron is the sum of the areas of its faces, for definitions of polyhedra for which 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 the same surface distances as each other, or the same as certain convex polyhedra.<ref>{{citation | last = Hartshorne | first = Robin | author-link = Robin Hartshorne | contribution = Example 44.2.3, the "punched-in icosahedron" | doi = 10.1007/978-0-387-22676-7 | isbn = 0-387-98650-2 | mr = 1761093 | page = 442 | publisher = Springer-Verlag, New York | series = Undergraduate Texts in Mathematics | title = Geometry: Euclid and beyond | year = 2000}}</ref> When segment lines connect two vertices that are not in the same face, they form the [[diagonal|diagonal lines]].<ref name=pb>{{citation | last1 = Posamentier | first1 = Alfred S. | last2 = Bannister | first2 = Robert L. | year = 2014 | edition = 2nd | title = Geometry, Its Elements and Structure: Second Edition | url = https://books.google.com/books?id=XktMBAAAQBAJ&pg=PA543 | page = 543 | publisher = Dover Publications | isbn = 978-0-486-49267-4 }}</ref> However, not all polyhedra have diagonal lines, as in the family of [[Pyramid (geometry)|pyramids]],{{cn|date=February 2025}} [[Schönhardt polyhedron]] in which three diagonal lines lies entirely outside of it, and [[Császár polyhedron]] has no diagonal lines (rather, every pair of vertices is connected by an edge).<ref name=bagemihl>{{citation | last = Bagemihl | first = F. | authorlink = Frederick Bagemihl | journal = [[American Mathematical Monthly]] | pages = 411–413 | title = On indecomposable polyhedra | volume = 55 | year = 1948 | doi = 10.2307/2306130 | issue = 7 | jstor = 2306130}}</ref> ===Volume=== 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 [[parallelepiped]]s can easily be expressed in terms of their edge lengths or other coordinates. (See [[volume#Formulas|Volume § Volume formulas]] for a list that includes many of these formulas.) Volumes of more complicated polyhedra may not have simple formulas. The volumes of such polyhedra may be computed by subdividing the polyhedron into smaller pieces (for example, by [[point-set triangulation|triangulation]]). For example, the [[Platonic solid#Radii, area, and volume|volume of a Platonic solid]] can be computed by dividing it into congruent [[Pyramid (geometry)|pyramids]], with each pyramid having a face of the polyhedron as its base and 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 <math display="block"> \frac{1}{3} \left| \sum_F (Q_F \cdot N_F) \operatorname{area}(F) \right|, </math> where the sum is over faces <math> F </math> of the polyhedron, <math> Q_F </math> is an arbitrary point on face <math> F </math>, <math> N_F </math> is the [[unit vector]] perpendicular to <math> F </math> pointing outside the solid, and the multiplication dot is the [[dot product]].<ref>{{citation |last=Goldman |first=Ronald N.|author-link=Ron Goldman (mathematician) |editor-last=Arvo |editor-first=James |title=Graphic Gems Package: Graphics Gems II |publisher=Academic Press |year=1991 |pages=170–171 |chapter=Chapter IV.1: Area of planar polygons and volume of polyhedra}}</ref> 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 [[algorithm]]s to determine the volume in these cases.<ref>{{citation | last1 = Büeler | first1 = B. | last2 = Enge | first2 = A. | last3 = Fukuda | first3 = K. | doi = 10.1007/978-3-0348-8438-9_6 | chapter = Exact Volume Computation for Polytopes: A Practical Study | title = Polytopes — Combinatorics and Computation | pages = 131–154 | year = 2000 | isbn = 978-3-7643-6351-2 | citeseerx = 10.1.1.39.7700 }}</ref> ===Dehn invariant=== {{main|Dehn invariant}} In two dimensions, the [[Bolyai–Gerwien theorem]] asserts that any polygon may be transformed into any other polygon of the same area by [[Dissection problem|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 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 the same volumes and Dehn invariants can be cut up and reassembled into each other.<ref>{{citation |last=Sydler |first=J.-P. | author-link = Jean-Pierre Sydler |title=Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions |journal=[[Commentarii Mathematici Helvetici|Comment. Math. Helv.]]|language=fr |volume=40 |year=1965 |pages=43–80 |doi= 10.1007/bf02564364| mr = 0192407|s2cid=123317371 | url = https://eudml.org/doc/139296 }}</ref> The Dehn invariant is not a number, but a [[Vector (mathematics)|vector]] in an infinite-dimensional vector space, determined from the lengths and [[dihedral angle]]s of a polyhedron's edges.<ref>{{SpringerEOM|first=M.|last=Hazewinkel|title=Dehn invariant|id=Dehn_invariant&oldid=35803}}</ref> Another of Hilbert's problems, [[Hilbert's 18th problem|Hilbert's eighteenth problem]], concerns (among other things) polyhedra that [[Honeycomb (geometry)|tile space]]. Every such polyhedron must have Dehn invariant zero.<ref>{{citation | last = Debrunner | first = Hans E. | doi = 10.1007/BF01235384 | issue = 6 | journal = [[Archiv der Mathematik]] | language = de | mr = 604258 | pages = 583–587 | title = Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln | volume = 35 | year = 1980| s2cid = 121301319 }}.</ref> The Dehn invariant has also been connected to [[flexible polyhedron|flexible polyhedra]] by the strong bellows theorem, which states that the Dehn invariant of any flexible polyhedron remains invariant as it flexes.<ref>{{citation | last = Alexandrov | first = Victor | arxiv = 0901.2989 | doi = 10.1007/s00022-011-0061-7 | issue = 1–2 | journal = Journal of Geometry | mr = 2823098 | pages = 1–13 | title = The Dehn invariants of the Bricard octahedra | volume = 99 | year = 2010| citeseerx = 10.1.1.243.7674 | s2cid = 17515249 }}.</ref>
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)