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rectified 24-cell

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bgcolor=#e7dcc3 align=center colspan=3|Rectified 24-cell
bgcolor=#ffffff align=center colspan=3|280px
Schlegel diagram
8 of 24 cuboctahedral cells shown
bgcolor=#e7dcc3|Type

|colspan=2|Uniform 4-polytope

bgcolor=#e7dcc3|Schläfli symbols

|colspan=2|r{3,4,3} = \left\{\begin{array}{l}3\\4,3\end{array}\right\}
rr{3,3,4}=r\left\{\begin{array}{l}3\\3,4\end{array}\right\}
r{31,1,1} = r\left\{\begin{array}{l}3\\3\\3\end{array}\right\}

bgcolor=#e7dcc3|Coxeter diagrams

|colspan=2|{{CDD|node|3|node_1|4|node|3|node}}
{{CDD|node_1|3|node|3|node_1|4|node}}
{{CDD|node_1|3|node|split1|nodes_11}} or {{CDD|node|splitsplit1|branch3_11|node_1}}

bgcolor=#e7dcc3|Cells

|48

|24 3.4.3.4 20px
24 4.4.4 20px

bgcolor=#e7dcc3|Faces

|240

|96 {3}
144 {4}

bgcolor=#e7dcc3|Edges

|colspan=2|288

bgcolor=#e7dcc3|Vertices

|colspan=2|96

bgcolor=#e7dcc3|Vertex figure

|colspan=2|50px50px50px
Triangular prism

bgcolor=#e7dcc3|Symmetry groups

|colspan=2|F4 [3,4,3], order 1152
B4 [3,3,4], order 384
D4 [31,1,1], order 192

bgcolor=#e7dcc3|Properties

|colspan=2|convex, edge-transitive

bgcolor=#e7dcc3|Uniform index

|colspan=2|22 23 24

File:Rectified icositetrachoron net.png]]

In geometry, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by rectification of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.{{Sfn|Coxeter|1973|p=154|loc=§8.4}}

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC24.

It can also be considered a cantellated 16-cell with the lower symmetries B4 = [3,3,4]. B4 would lead to a bicoloring of the cuboctahedral cells into 8 and 16 each. It is also called a runcicantellated demitesseract in a D4 symmetry, giving 3 colors of cells, 8 for each.

Construction

The rectified 24-cell can be derived from the 24-cell by the process of rectification: the 24-cell is truncated at the midpoints. The vertices become cubes, while the octahedra become cuboctahedra.

Cartesian coordinates

A rectified 24-cell having an edge length of {{radic|2}} has vertices given by all permutations and sign permutations of the following Cartesian coordinates:

: (0,1,1,2) [4!/2!×23 = 96 vertices]

The dual configuration with edge length 2 has all coordinate and sign permutations of:

: (0,2,2,2) [4×23 = 32 vertices]

: (1,1,1,3) [4×24 = 64 vertices]

Images

{{24-cell_4-cube_Coxeter_plane_graphs|t1|100|t2}}

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!Stereographic projection

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Center of stereographic projection
with 96 triangular faces blue

Symmetry constructions

There are three different symmetry constructions of this polytope. The lowest {D}_4 construction can be doubled into {C}_4 by adding a mirror that maps the bifurcating nodes onto each other. {D}_4 can be mapped up to {F}_4 symmetry by adding two mirror that map all three end nodes together.

The vertex figure is a triangular prism, containing two cubes and three cuboctahedra. The three symmetries can be seen with 3 colored cuboctahedra in the lowest {D}_4 construction, and two colors (1:2 ratio) in {C}_4, and all identical cuboctahedra in {F}_4.

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!Coxeter group

!{F}_4 = [3,4,3]

!{C}_4 = [4,3,3]

!{D}_4 = [3,31,1]

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!Order

|1152

|384

|192

align=center

!Full
symmetry
group

|[3,4,3]

|[4,3,3]

|<[3,31,1]> = [4,3,3]
[3[31,1,1]] = [3,4,3]

align=center

!Coxeter diagram

|{{CDD|node|3|node_1|4|node|3|node}}

|{{CDD|node|4|node_1|3|node|3|node_1}}

|{{CDD|nodes_11|split2|node|3|node_1}}

Facets

|3: {{CDD|node|3|node_1|4|node}}
2: {{CDD|node_1|4|node|3|node}}

|2,2: {{CDD|node_1|3|node|3|node_1}}
2: {{CDD|node|4|node_1|2|node_1}}

|1,1,1: {{CDD|node_1|3|node|3|node_1}}
2: {{CDD|node_1|2|node_1|2|node_1}}

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!Vertex figure

|80px

|80px

|80px

Alternate names

  • Rectified 24-cell, Cantellated 16-cell (Norman Johnson)
  • Rectified icositetrachoron (Acronym rico) (George Olshevsky, Jonathan Bowers)
  • Cantellated hexadecachoron
  • Disicositetrachoron
  • Amboicositetrachoron (Neil Sloane & John Horton Conway)

Related polytopes

The convex hull of the rectified 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 192 cells: 48 cubes, 144 square antiprisms, and 192 vertices. Its vertex figure is a triangular bifrustum.

Related uniform polytopes

{{Demitesseract family}}

{{24-cell_family}}

The rectified 24-cell can also be derived as a cantellated 16-cell:

{{Tesseract family}}

Citations

{{RefList}}

References

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=Regular Polytopes (book) }}
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{isbn|978-1-56881-220-5}} (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • {{PolyCell | urlname = section2.html| title = 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 23}}
  • {{PolyCell | urlname = section3.html| title = 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 23}}
  • {{PolyCell | urlname = section7.html| title = 7. Uniform polychora derived from glomeric tetrahedron B4 - Model 23 }}
  • {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|o3x4o3o - rico}}

{{Polytopes}}

Category:Uniform 4-polytopes