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In four-dimensional
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a runcinated 5-cell is a convex uniform 4-polytope, being a
runcination In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers. It is a higher order truncatio ...
(a 3rd order truncation, up to face-planing) of the regular
5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
. There are 3 unique degrees of runcinations of the 5-cell, including with permutations, truncations, and cantellations.


Runcinated 5-cell

The runcinated 5-cell or small prismatodecachoron is constructed by expanding the
cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
of a
5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
radially and filling in the gaps with triangular
prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...
s (which are the face prisms and edge figures) and
tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
(cells of the dual 5-cell). It consists of 10 tetrahedra and 20 triangular prisms. The 10 tetrahedra correspond with the cells of a 5-cell and its dual. Topologically, under its highest symmetry,
3,3,3 In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...
, there is only one geometrical form, containing 10 tetrahedra and 20 uniform triangular prisms. The rectangles are always squares because the two pairs of edges correspond to the edges of the two sets of 5 regular tetrahedra each in dual orientation, which are made equal under extended symmetry.
E. L. Elte Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibor extermination camp, Sobibór) Em ...
identified it in 1912 as a semiregular polytope.


Alternative names

* Runcinated
5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
( Norman Johnson) * Runcinated pentachoron * Runcinated
4-simplex In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...
* Expanded 5-cell/4-simplex/pentachoron * Small prismatodecachoron (Acronym: Spid) (Jonathan Bowers)


Structure

Two of the ten tetrahedral cells meet at each vertex. The triangular prisms lie between them, joined to them by their triangular faces and to each other by their square faces. Each triangular prism is joined to its neighbouring triangular prisms in ''anti'' orientation (i.e., if edges A and B in the shared square face are joined to the triangular faces of one prism, then it is the other two edges that are joined to the triangular faces of the other prism); thus each pair of adjacent prisms, if rotated into the same
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
, would form a
gyrobifastigium In geometry, the gyrobifastigium is the 26th Johnson solid (). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile ...
.


Dissection

The ''runcinated 5-cell'' can be dissected by a central cuboctahedron into two
tetrahedral cupola In 4-dimensional geometry, the tetrahedral cupola is a polychoron bounded by one tetrahedron, a parallel cuboctahedron, connected by 10 triangular prisms, and 4 triangular pyramids.cuboctahedron being dissected by a central hexagon into two
triangular cupola In geometry, the triangular cupola is one of the Johnson solids (). It can be seen as half a cuboctahedron. Formulae The following formulae for the volume (V), the surface area (A) and the height (H) can be used if all faces are regular, ...
. :


Images


Coordinates

The Cartesian coordinates of the vertices of an origin-centered runcinated 5-cell with edge length 2 are: An alternate simpler set of coordinates can be made in 5-space, as 20 permutations of: : (0,1,1,1,2) This construction exists as one of 32 orthant facets of the runcinated 5-orthoplex. A second construction in 5-space, from the center of a rectified 5-orthoplex is given by coordinate permutations of: : (1,-1,0,0,0)


Root vectors

Its 20 vertices represent the root vectors of the simple Lie group A4. It is also the
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
for the
5-cell honeycomb In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1. Struct ...
in 4-space.


Cross-sections

The maximal cross-section of the runcinated 5-cell with a 3-dimensional
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
is a cuboctahedron. This cross-section divides the runcinated 5-cell into two tetrahedral hypercupolae consisting of 5 tetrahedra and 10 triangular prisms each.


Projections

The tetrahedron-first
orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...
of the runcinated 5-cell into 3-dimensional space has a cuboctahedral envelope. The structure of this projection is as follows: * The cuboctahedral envelope is divided internally as follows: :* Four flattened tetrahedra join 4 of the triangular faces of the cuboctahedron to a central tetrahedron. These are the images of 5 of the tetrahedral cells. :* The 6 square faces of the cuboctahedron are joined to the edges of the central tetrahedron via distorted triangular prisms. These are the images of 6 of the triangular prism cells. :* The other 4 triangular faces are joined to the central tetrahedron via 4 triangular prisms (distorted by projection). These are the images of another 4 of the triangular prism cells. :* This accounts for half of the runcinated 5-cell (5 tetrahedra and 10 triangular prisms), which may be thought of as the 'northern hemisphere'. * The other half, the 'southern hemisphere', corresponds to an isomorphic division of the cuboctahedron in dual orientation, in which the central tetrahedron is dual to the one in the first half. The triangular faces of the cuboctahedron join the triangular prisms in one hemisphere to the flattened tetrahedra in the other hemisphere, and vice versa. Thus, the southern hemisphere contains another 5 tetrahedra and another 10 triangular prisms, making the total of 10 tetrahedra and 20 triangular prisms.


Related skew polyhedron

The
regular skew polyhedron In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra ...
, , exists in 4-space with 6 squares around each vertex, in a zig-zagging nonplanar vertex figure. These square faces can be seen on the runcinated 5-cell, using all 60 edges and 20 vertices. The 40 triangular faces of the runcinated 5-cell can be seen as removed. The dual regular skew polyhedron, , is similarly related to the hexagonal faces of the
bitruncated 5-cell In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell. There are two degrees of truncations, including a bitruncation. Truncated 5-cell The truncated 5-cell, tr ...
.


Runcitruncated 5-cell

The runcitruncated
5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
or prismatorhombated pentachoron is composed of 60 vertices, 150 edges, 120 faces, and 30 cells. The cells are: 5
truncated tetrahedra In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron ...
, 10 hexagonal prisms, 10
triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...
s, and 5
cuboctahedra A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a t ...
. Each vertex is surrounded by five cells: one truncated tetrahedron, two hexagonal prisms, one triangular prism, and one cuboctahedron; the
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is a rectangular pyramid.


Alternative names

* Runcitruncated pentachoron * Runcitruncated
4-simplex In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...
* Diprismatodispentachoron * Prismatorhombated pentachoron (Acronym: prip) (Jonathan Bowers)


Images


Coordinates

The Cartesian coordinates of an origin-centered runcitruncated 5-cell having edge length 2 are: {, class="wikitable collapsible collapsed" ! colspan=2, Coordinates , - , :\left(\frac{7}{\sqrt{10,\ \sqrt{\frac{3}{2,\ \pm\sqrt{3},\ \pm1\right) :\left(\frac{7}{\sqrt{10,\ \sqrt{\frac{3}{2,\ 0,\ \pm2\right) :\left(\frac{7}{\sqrt{10,\ \frac{-1}{\sqrt{6,\ \frac{2}{\sqrt{3,\ \pm2\right) :\left(\frac{7}{\sqrt{10,\ \frac{-1}{\sqrt{6,\ \frac{-4}{\sqrt{3,\ 0\right) :\left(\frac{7}{\sqrt{10,\ \frac{-5}{\sqrt{6,\ \frac{1}{\sqrt{3,\ \pm1\right) :\left(\frac{7}{\sqrt{10,\ \frac{-5}{\sqrt{6,\ \frac{-2}{\sqrt{3,\ 0\right) :\left(\sqrt{\frac{2}{5,\ \pm\sqrt{6},\ \pm\sqrt{3},\ \pm1\right) :\left(\sqrt{\frac{2}{5,\ \pm\sqrt{6},\ 0,\ \pm2\right) :\left(\sqrt{\frac{2}{5,\ \sqrt{\frac{2}{3,\ \frac{5}{\sqrt{3,\ \pm1\right) , :\left(\sqrt{\frac{2}{5,\ \sqrt{\frac{2}{3,\ \frac{-1}{\sqrt{3,\ \pm3\right) :\left(\sqrt{\frac{2}{5,\ \sqrt{\frac{2}{3,\ \frac{-4}{\sqrt{3,\ \pm2\right) :\left(\sqrt{\frac{2}{5,\ -\sqrt{\frac{2}{3,\ \frac{4}{\sqrt{3,\ \pm2\right) :\left(\sqrt{\frac{2}{5,\ -\sqrt{\frac{2}{3,\ \frac{1}{\sqrt{3,\ \pm3\right) :\left(\sqrt{\frac{2}{5,\ -\sqrt{\frac{2}{3,\ \frac{-5}{\sqrt{3,\ \pm1\right) :\left(\frac{-3}{\sqrt{10,\ \frac{5}{\sqrt{6,\ \frac{2}{\sqrt{3,\ \pm2\right) :\left(\frac{-3}{\sqrt{10,\ \frac{5}{\sqrt{6,\ \frac{-4}{\sqrt{3,\ 0\right) :\left(\frac{-3}{\sqrt{10,\ \frac{1}{\sqrt{6,\ \frac{4}{\sqrt{3,\ \pm2\right) :\left(\frac{-3}{\sqrt{10,\ \frac{1}{\sqrt{6,\ \frac{1}{\sqrt{3,\ \pm3\right) , :\left(\frac{-3}{\sqrt{10,\ \frac{1}{\sqrt{6,\ \frac{-5}{\sqrt{3,\ \pm1\right) :\left(\frac{-3}{\sqrt{10,\ \frac{-7}{\sqrt{6,\ \frac{2}{\sqrt{3,\ 0\right) :\left(\frac{-3}{\sqrt{10,\ \frac{-7}{\sqrt{6,\ \frac{-1}{\sqrt{3,\ \pm1\right) :\left(-4\sqrt{\frac{2}{5,\ 2\sqrt{\frac{2}{3,\ \frac{1}{\sqrt{3,\ \pm1\right) :\left(-4\sqrt{\frac{2}{5,\ 2\sqrt{\frac{2}{3,\ \frac{-2}{\sqrt{3,\ 0\right) :\left(-4\sqrt{\frac{2}{5,\ 0,\ \pm\sqrt{3},\ \pm1\right) :\left(-4\sqrt{\frac{2}{5,\ 0,\ 0,\ \pm2\right) :\left(-4\sqrt{\frac{2}{5,\ -2\sqrt{\frac{2}{3,\ \frac{2}{\sqrt{3,\ 0\right) :\left(-4\sqrt{\frac{2}{5,\ -2\sqrt{\frac{2}{3,\ \frac{-1}{\sqrt{3,\ 1\right) The vertices can be more simply constructed on a
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
in 5-space, as the
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s of: : (0,1,1,2,3) This construction is from the positive orthant facet of the
runcitruncated 5-orthoplex In Five-dimensional space, five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order Truncation (geometry), truncation (runcination) of the regular 5-orthoplex. There are 8 runcinations of the 5-orthoplex wi ...
.


Omnitruncated 5-cell

{, class="wikitable" align="right" style="margin-left:10px" width="250" , - , bgcolor=#e7dcc3 align=center colspan=3, Omnitruncated 5-cell , - , bgcolor=#ffffff align=center colspan=3,
Schlegel diagram with half of the truncated octahedral cells shown. , - , bgcolor=#e7dcc3, Type , colspan=2, Uniform 4-polytope , - , bgcolor=#e7dcc3,
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
, colspan=2, t0,1,2,3{3,3,3} , - , bgcolor=#e7dcc3, Coxeter diagram , colspan=2, , - , bgcolor=#e7dcc3, Cells , 30 , 10 (4.6.6)
20 (4.4.6) , - , bgcolor=#e7dcc3, Faces , 150 , 90{4}
60{6} , - , bgcolor=#e7dcc3, Edges , colspan=2, 240 , - , bgcolor=#e7dcc3, Vertices , colspan=2, 120 , - , bgcolor=#e7dcc3,
Vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
, colspan=2,
Phyllic disphenoid In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
, - , bgcolor=#e7dcc3, Coxeter group , colspan=2,
Aut AUT may refer to the following. Locations *Austria (ISO 3166-1 country code) *Agongointo-Zoungoudo Underground Town, Benin *Aktio–Preveza Undersea Tunnel, Greece *Airstrip on Atauro Island, East Timor (IATA airport code) Organizations *Arriva ...
(A4), , order 240 , - , bgcolor=#e7dcc3, Properties , colspan=2, convex, isogonal, zonotope , - , bgcolor=#e7dcc3, Uniform index , colspan=2, '' 8'' 9 '' 10'' The omnitruncated 5-cell or great prismatodecachoron is composed of 120 vertices, 240 edges, 150 faces (90
squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
and 60
hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...
s), and 30 cells. The cells are: 10
truncated octahedra In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
, and 20 hexagonal prisms. Each vertex is surrounded by four cells: two truncated octahedra, and two hexagonal prisms, arranged in two phyllic disphenoidal
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s. Coxeter calls this Hinton's polytope after
C. H. Hinton Charles Howard Hinton (1853 – 30 April 1907) was a British mathematician and writer of science fiction works titled ''Scientific Romances''. He was interested in n-dimensional space, higher dimensions, particularly the Four-dimensional space, ...
, who described it in his book ''The Fourth Dimension'' in 1906. It forms a
uniform honeycomb In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of face ...
which Coxeter calls
Hinton's honeycomb In Four-dimensional space, four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb (geometry), honeycomb. It is composed of 5-cells and recti ...
.''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, , (The classification of Zonohededra, page 73)


Alternative names

* Omnitruncated
5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
* Omnitruncated
pentachoron In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...
* Omnitruncated
4-simplex In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...
* Great prismatodecachoron (Acronym: gippid) (Jonathan Bowers) * Hinton's polytope ( Coxeter)


Images

{, class=wikitable , + Net ,
Omnitruncated 5-cell ,
Dual to omnitruncated 5-cell


Perspective projections

{, class=wikitable width=600 , + , - align=center ,
Perspective Schlegel diagram
Centered on
truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
,
Stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...


Permutohedron

Just as the
truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
is the
permutohedron In mathematics, the permutohedron of order ''n'' is an (''n'' − 1)-dimensional polytope embedded in an ''n''-dimensional space. Its vertex coordinates (labels) are the permutations of the first ''n'' natural numbers. The edges ident ...
of order 4, the omnitruncated 5-cell is the permutohedron of order 5. The omnitruncated 5-cell is a zonotope, the Minkowski sum of five line segments parallel to the five lines through the origin and the five vertices of the 5-cell.


Tessellations

The
omnitruncated 5-cell honeycomb In Four-dimensional space, four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb (geometry), honeycomb. It is composed of 5-cells and recti ...
can tessellate 4-dimensional space by translational copies of this cell, each with 3 hypercells around each face. This honeycomb's Coxeter diagram is . Unlike the analogous honeycomb in three dimensions, the
bitruncated cubic honeycomb The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has 4 truncated octahedra around each vertex. Being composed entirely of ...
which has three different Coxeter group Wythoff constructions, this honeycomb has only one such construction.


Symmetry

The omnitruncated 5-cell has extended pentachoric symmetry, , order 240. The
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
of the omnitruncated 5-cell represents the Goursat tetrahedron of the ,3,3 Coxeter group. The extended symmetry comes from a 2-fold rotation across the middle order-3 branch, and is represented more explicitly as +[3,3,3. :


Coordinates

The Cartesian coordinates of the vertices of an origin-centered omnitruncated 5-cell having edge length 2 are: {, class=wikitable , :\left(\pm\sqrt{10},\ \pm\sqrt{6},\ \pm\sqrt{3},\ \pm1\right) :\left(\pm\sqrt{10},\ \pm\sqrt{6},\ 0,\ \pm2\right) :\pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3,\ \frac{5}{\sqrt{3,\ \pm1\right) :\pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3,\ \frac{-1}{\sqrt{3,\ \pm3\right) :\pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3,\ \frac{-4}{\sqrt{3,\ \pm2\right) :\left(\sqrt{\frac{5}{2,\ 3\sqrt{\frac{3}{2,\ \pm\sqrt{3},\ \pm1\right) :\left(-\sqrt{\frac{5}{2,\ -3\sqrt{\frac{3}{2,\ \pm\sqrt{3},\ \pm1\right) :\left(\sqrt{\frac{5}{2,\ 3\sqrt{\frac{3}{2,\ 0,\ \pm2\right) :\left(-\sqrt{\frac{5}{2,\ -3\sqrt{\frac{3}{2,\ 0,\ \pm2\right) , :\pm\left(\sqrt{\frac{5}{2,\ \frac{1}{\sqrt{6,\ \frac{7}{\sqrt{3,\ \pm1\right) :\pm\left(\sqrt{\frac{5}{2,\ \frac{1}{\sqrt{6,\ \frac{-2}{\sqrt{3,\ \pm4\right) :\pm\left(\sqrt{\frac{5}{2,\ \frac{1}{\sqrt{6,\ \frac{-5}{\sqrt{3,\ \pm3\right) :\pm\left(\sqrt{\frac{5}{2,\ -\sqrt{\frac{3}{2,\ \pm2\sqrt{3},\ \pm2\right) :\pm\left(\sqrt{\frac{5}{2,\ -\sqrt{\frac{3}{2,\ 0,\ \pm4\right) :\pm\left(\sqrt{\frac{5}{2,\ \frac{-7}{\sqrt{6,\ \frac{5}{\sqrt{3,\ \pm1\right) :\pm\left(\sqrt{\frac{5}{2,\ \frac{-7}{\sqrt{6,\ \frac{-1}{\sqrt{3,\ \pm3\right) , :\pm\left(\sqrt{\frac{5}{2,\ \frac{-7}{\sqrt{6,\ \frac{-4}{\sqrt{3,\ \pm2\right) :\pm\left(0,\ 4\sqrt{\frac{2}{3,\ \frac{5}{\sqrt{3,\ \pm1\right) :\pm\left(0,\ 4\sqrt{\frac{2}{3,\ \frac{-1}{\sqrt{3,\ \pm3\right) :\pm\left(0,\ 4\sqrt{\frac{2}{3,\ \frac{-4}{\sqrt{3,\ \pm2\right) :\pm\left(0,\ 2\sqrt{\frac{2}{3,\ \frac{7}{\sqrt{3,\ \pm1\right) :\pm\left(0,\ 2\sqrt{\frac{2}{3,\ \frac{-2}{\sqrt{3,\ \pm4\right) :\pm\left(0,\ 2\sqrt{\frac{2}{3,\ \frac{-5}{\sqrt{3,\ \pm3\right) These vertices can be more simply obtained in 5-space as the 120
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s of (0,1,2,3,4). This construction is from the positive orthant facet of the runcicantitruncated 5-orthoplex, t0,1,2,3{3,3,3,4}, .


Related polytopes

Nonuniform variants with ,3,3symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra on each other to produce a nonuniform polychoron with 10
truncated octahedra In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
, two types of 40 hexagonal prisms (20 ditrigonal prisms and 20 ditrigonal trapezoprisms), two kinds of 90 rectangular trapezoprisms (30 with ''D2d'' symmetry and 60 with ''C2v'' symmetry), and 240 vertices. Its vertex figure is an irregular
triangular bipyramid In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, i ...
.
Vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
This polychoron can then be alternated to produce another nonuniform polychoron with 10
icosahedra In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
, two types of 40 octahedra (20 with ''S6'' symmetry and 20 with ''D3'' symmetry), three kinds of 210
tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
(30 tetragonal disphenoids, 60 phyllic disphenoids, and 120 irregular tetrahedra), and 120 vertices. It has a symmetry of
3,3,3 In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...
sup>+], order 120.
Vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...


Full snub 5-cell

The full snub 5-cell or omnisnub 5-cell, defined as an Alternation (geometry), alternation of the omnitruncated 5-cell, cannot be made uniform, but it can be given Coxeter diagram , and
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
+, order 120, and constructed from 90 cells: 10
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
s, 20 octahedrons, and 60
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
s filling the gaps at the deleted vertices. It has 300 faces (triangles), 270 edges, and 60 vertices. Topologically, under its highest symmetry, +, the 10 icosahedra have ''T'' (chiral tetrahedral) symmetry, while the 20 octahedra have ''D3'' symmetry and the 60 tetrahedra have ''C2'' symmetry.


Related polytopes

These polytopes are a part of a family of 9 Uniform 4-polytope constructed from the ,3,3 Coxeter group.


Notes


References

*
H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
: ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 ** Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. * * o3x3x3o - spid, x3x3o3x - prip, x3x3x3x - gippid {{Polytopes 4-polytopes