octahedron#Tetratetrahedron

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In geometry, an octahedron ({{plural form}}: octahedra or octahedrons) is any 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. Many types of irregular octahedra also exist, including both convex and non-convex shapes.

Combinatorially equivalent to the regular octahedron

File:Br2-anim.gif with an antiparallelogram as its equator. The axis of symmetry passes through the plane of the antiparallelogram.]]

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 the features of it:

Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral.
Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all three quadrilaterals are planar squares.
Schönhardt polyhedron, a non-convex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.
Bricard octahedron, a non-convex self-crossing flexible polyhedron

Other convex polyhedra

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.{{Cite web |url=http://www.uwgb.edu/dutchs/symmetry/polynum0.htm |title=Enumeration of Polyhedra |access-date=2 May 2006 |archive-url=https://web.archive.org/web/20111010185122/http://www.uwgb.edu/dutchs/symmetry/polynum0.htm |archive-date=10 October 2011 |url-status=dead }} 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.{{cite web| url = http://www.numericana.com/data/polycount.htm| title = Counting polyhedra}}{{cite web |url=http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm |title=Polyhedra with 8 Faces and 6-8 Vertices |access-date=14 August 2016 |url-status=dead |archive-url=https://web.archive.org/web/20141117072140/http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm |archive-date=17 November 2014 }} (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)

Notable eight-sided convex polyhedra include:

File:Hexagonal Prism.svg | Hexagonal prism: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges. With all faces regular and all vertices symmetric to each other, this is a uniform polyhedron.

File:Truncatedtetrahedron.jpg | Truncated tetrahedron: The four faces from the tetrahedron are truncated to become regular hexagons, and there are four more equilateral triangle faces where each tetrahedron vertex was truncated. As a uniform polyhedron that is not a prism or antiprism, this is an Archimedean solid.{{r|kuchel|berman}}

File:Gyrobifastigium.png | Gyrobifastigium: Two uniform triangular prisms glued over one of their square sides so that no triangle shares an edge with another triangle. As a polyhedron whose faces are regular polygons, it is a Johnson solid.{{r|berman}} Its dual polyhedron is also an octahedron.

File:Augmented triangular prism.png | Augmented triangular prism: The result of gluing a triangular prism to a square pyramid, this has six equilateral triangle faces and two square faces. It is also a Johnson solid.{{r|berman}}

File:Triangular cupola.png | Triangular cupola: Another Johnson solid, this has one regular hexagon face, three square faces, and four equilateral triangle faces.{{r|berman}}

File:Heptagonal pyramid.png | Heptagonal pyramid: One face is a heptagon (usually regular), and the remaining seven faces are triangles (usually isosceles).{{r|humble}} It is not possible for the triangular faces to all be equilateral. It is self-dual.

File:Tetragonal trapezohedron.png | Tetragonal trapezohedron: The eight faces are congruent kites.

File:Dual elongated triangular dipyramid.png | Triangular bifrustum: The dual of an elongated triangular bipyramid (a Johnson solid), this can be realized with six isosceles trapezoid faces and two equilateral triangle faces.

File:Triangular truncated trapezohedron.png | Truncated triangular trapezohedron, also called Dürer's solid: Obtained by truncating two opposite corners of a cube or rhombohedron, this has six pentagon faces and two triangle faces.{{citation|last1=Futamura|first1=F.|author1-link=Fumiko Futamura|first2=M.|last2=Frantz|first3=A.|last3=Crannell|author3-link= Annalisa Crannell |title=The cross ratio as a shape parameter for Dürer's solid|journal=Journal of Mathematics and the Arts|volume=8|issue=3–4|year=2014|pages=111–119|doi=10.1080/17513472.2014.974483|arxiv=1405.6481|s2cid=120958490}}

References

{{reflist|refs=

{{cite journal

| last = Berman | first = Martin

| doi = 10.1016/0016-0032(71)90071-8

| journal = Journal of the Franklin Institute

| mr = 290245

| pages = 329–352

| title = Regular-faced convex polyhedra

| volume = 291

| year = 1971| issue = 5

}}

{{cite book

| last = Humble | first = Steve

| year = 2016

| title = The Experimenter's A-Z of Mathematics: Math Activities with Computer Support

| page = 23

| publisher = Taylor & Francis

| isbn = 978-1-134-13953-8

| url = https://books.google.com/books?id=S-80DwAAQBAJ&pg=PA23

}}

{{cite journal

| last = Kuchel | first = Philip W.

| year = 2012

| title = 96.45 Can you 'bend' a truncated truncated tetrahedron?

| journal = The Mathematical Gazette

| volume = 96 | issue = 536 | pages = 317–323

| doi = 10.1017/S0025557200004666

| jstor = 23248575

}}

Category:Polyhedra