Cantellated 24-cell
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24-cell

Cantellated 24-cell

Cantitruncated 24-cell
Orthogonal projections in F4 Coxeter plane

In four-dimensional geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a cantellated 24-cell is a convex uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....

, being a cantellation (a 2nd order truncation) of the regular 24-cell.

There are 2 unique degrees of runcinations of the 24-cell including with permutations truncations.

Cantellated 24-cell

Cantellated 24-cell
Type Uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....

Schläfli symbol t0,2{3,4,3}
Coxeter-Dynkin diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Cells 144 24 (3.4.4.4)
24 (3.4.3.4)
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...



96 (3.4.4)
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....

Faces 720 288 triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

s
432 square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

s
Edges 864
Vertices 288
Vertex figure
Vertex figure
In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...


Irreg. triangular prism
Symmetry group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

F4, [3,4,3]
Properties convex
Convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

Uniform index 24
Truncated 24-cell
In geometry, a truncated 5-cell is a uniform polychoron formed as the truncation of the regular 5-cell.There are two degrees of trunctions, including a bitruncation.- Truncated 5-cell:...

25 26
Runcinated 24-cell
In four-dimensional geometry, a runcinated 24-cell is a convex uniform polychoron, being a runcination of the regular 24-cell....


The cantellated 24-cell is a uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....

.

The boundary of the cantellated 24-cell is composed of
24 truncated octahedral
Truncated octahedron
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....

 cells,
24 cuboctahedral
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

 cells and 96 triangular prism
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....

s.
Together they have 288 triangular faces, 432 square faces, 864 edges, and 288 vertices.

Construction

When the cantellation
Cantellation (geometry)
In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex. The operation also applies to regular tilings and honeycombs...

 process is applied to 24-cell,
each of the 24 octahedra
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

 becomes a small rhombicuboctahedron.
In addition however, since each octahedra
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

's edge was previously shared with two
other octahedra
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

, the separating edges form the three parallel edges of a
triangular prism
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....

 - 96 triangular prisms, since the 24-cell contains 96 edges.
Further, since each vertex was previously shared with 12 faces,
the vertex would split into 12 (24*12=288) new vertices.
Each group of 12 new vertices forms a cuboctahedron
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

.

Coordinates

The Cartesian coordinates of the vertices of the cantellated 24-cell having edge length 2 are all permutations of coordinates and sign of:
The permutations of the second set of coordinates coincide with the vertices of an inscribed runcitruncated tesseract.

The dual configuration has all permutations and signs of:

Structure

The 24 small rhombicuboctahedra are joined to each other via their triangular faces, to the cuboctahedra
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

 via their axial square faces, and to the triangular prism
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....

s via their off-axial square faces. The cuboctahedra are joined to the triangular prisms via their triangular faces. Each triangular prism is joined to two cuboctahedra at its two ends.

Images

Schlegel diagrams

Schlegel diagram

Showing 24 cuboctahedra
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

.

Showing 96 triangular prism
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....

s.

Cantitruncated 24-cell

Cantitruncated 24-cell

Schlegel diagram, centered on truncated cuboctahedron
Type Uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....

Schläfli symbol t0,1,2{3,4,3}
Coxeter-Dynkin diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Cells 144 24 4.6.8 
96 4.4.3
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....

 
24 3.8.8
Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....

 
Faces 720 192{3}
288{4}
96{6}
144{8}
Edges 1152
Vertices 576
Vertex figure
Vertex figure
In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...


sphenoid
Sphenoid
Sphenoid may refer to:* In anatomy, the sphenoid bone* In geometry, a tetrahedron with mirror symmetry...

Symmetry group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

F4, [3,4,3]
Properties convex
Convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

Uniform index 27 28 29

The cantitruncated 24-cell is a uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....

 derived from the 24-cell. It is bounded by 24 great rhombicuboctahedra corresponding with the cells of a 24-cell, 24 truncated cube
Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....

s corresponding with the cells of the dual 24-cell, and 96 triangular prism
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....

s corresponding with the edges of the first 24-cell.

Coordinates

The Cartesian coordinates of a cantitruncated 24-cell having edge length 2 are all permutations of coordinates and sign of:
(1,1+√2,1+2√2,3+3√2)
(0,2+√2,2+2√2,2+3√2)


The dual configuration has coordinates as all permutations and signs of:
(1,1+√2,1+√2,5+2√2)
(1,3+√2,3+√2,3+2√2)
(2,2+√2,2+√2,4+2√2)

Projections

Stereographic projection
Stereographic projection
The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...

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