geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a truncated 24-cell is a

uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons. There are 47 non-Prism (geometry), prism ...

(4-dimensional uniform

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

) formed as the

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

of the regular

24-cell In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octa ...

. There are two degrees of truncations, including a bitruncation.

Truncated 24-cell

The truncated 24-cell or truncated icositetrachoron is a uniform 4-dimensional polytope (or

), which is bounded by 48 cells: 24

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

s, and 24 truncated octahedra. Each vertex joins three truncated octahedra and one cube, in an equilateral triangular pyramid

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

Construction

The truncated 24-cell can be constructed from polytopes with three symmetry groups: *F₄ ,4,3 A

of the

. *B₄ ,3,4 A

cantitruncation In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each Flag (geometry), flag of the original polytope and a Facet (geometry), facet for each face of any dimension of the original pol ...

of the

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the ...

, with two families of truncated octahedral cells. *D₄ ^1,1,1 An omnitruncation of the demitesseract, with three families of truncated octahedral cells.

Zonotope

It is also a

zonotope In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

: it can be formed as the

Minkowski sum In geometry, the Minkowski sum of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...

of the six line segments connecting opposite pairs among the twelve permutations of the vector (+1,−1,0,0).

Cartesian coordinates

The

Cartesian coordinate In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

s of the vertices of a truncated 24-cell having edge length sqrt(2) are all coordinate permutations and sign combinations of: : (0,1,2,3) !×2³ = 192 vertices The dual configuration has coordinates at all coordinate permutation and signs of : (1,1,1,5) ×2⁴ = 64 vertices: (1,3,3,3) ×2⁴ = 64 vertices: (2,2,2,4) ×2⁴ = 64 vertices

Structure

The 24 cubical cells are joined via their square faces to the truncated octahedra; and the 24 truncated octahedra are joined to each other via their hexagonal faces.

Projections

The parallel projection of the truncated 24-cell into 3-dimensional space, truncated octahedron first, has the following layout: * The projection envelope is a truncated cuboctahedron. * Two of the truncated octahedra project onto a truncated octahedron lying in the center of the envelope. * Six cuboidal volumes join the square faces of this central truncated octahedron to the center of the octagonal faces of the great rhombicuboctahedron. These are the images of 12 of the cubical cells, a pair of cells to each image. * The 12 square faces of the great rhombicuboctahedron are the images of the remaining 12 cubes. * The 6 octagonal faces of the great rhombicuboctahedron are the images of 6 of the truncated octahedra. * The 8 (non-uniform) truncated octahedral volumes lying between the hexagonal faces of the projection envelope and the central truncated octahedron are the images of the remaining 16 truncated octahedra, a pair of cells to each image.

Images

Related polytopes

The convex hull of the truncated 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 480 cells: 48

s, 144

square antiprism In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even number, even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an ''anticube''. If all its faces are regular ...

s, 288

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

(as tetragonal disphenoids), and 384 vertices. Its vertex figure is a hexakis

triangular cupola In geometry, the triangular cupola is the cupola with hexagon as its base and triangle as its top. If the edges are equal in length, the triangular cupola is the Johnson solid. It can be seen as half a cuboctahedron. The triangular cupola can b ...

Bitruncatotetracontaoctachoron vertex figure

Vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

Bitruncated 24-cell

The bitruncated 24-cell. 48-cell, or tetracontoctachoron is a 4-dimensional uniform

(or

) derived from the

. E. L. Elte identified it in 1912 as a semiregular polytope. It is constructed by bitruncating the 24-cell (truncating at halfway to the depth which would yield the dual 24-cell). Being a uniform 4-polytope, it is

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...

. In addition, it is

cell-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruen ...

, consisting of 48

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangle (geometry), triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triak ...

s, and also

edge-transitive In geometry, a polytope (for example, a polygon or a polyhedron) or a Tessellation, tiling is isotoxal () or edge-transitive if its Symmetry, symmetries act Transitive group action, transitively on its Edge (geometry), edges. Informally, this mea ...

, with 3

s cells per edge and with one triangle and two octagons around each edge. The 48 cells of the bitruncated 24-cell correspond with the 24 cells and 24 vertices of the 24-cell. As such, the centers of the 48 cells form the

root system In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and ...

of type F₄. Its vertex figure is a '' tetragonal disphenoid'', a tetrahedron with 2 opposite edges length 1 and all 4 lateral edges length √(2+√2).

Alternative names

* Bitruncated 24-cell ( Norman W. Johnson) * 48-cell as a

4-polytope * Bitruncated icositetrachoron * Bitruncated polyoctahedron * Tetracontaoctachoron (Cont) (Jonathan Bowers)

Structure

The truncated cubes are joined to each other via their octagonal faces in ''anti'' orientation; i. e., two adjoining truncated cubes are rotated 45 degrees relative to each other so that no two triangular faces share an edge. The sequence of truncated cubes joined to each other via opposite octagonal faces form a cycle of 8. Each truncated cube belongs to 3 such cycles. On the other hand, the sequence of truncated cubes joined to each other via opposite triangular faces form a cycle of 6. Each truncated cube belongs to 4 such cycles. Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

, dividing the full group order of a subgroup order by removing one mirror at a time. Edges exist at 4 symmetry positions. Squares exist at 3 positions, hexagons 2 positions, and octagons one. Finally the 4 types of cells exist centered on the 4 corners of the fundamental simplex.

Coordinates

The

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

Projections

Projection to 2 dimensions

Projection to 3 dimensions

Related regular skew polyhedron

The regular skew polyhedron, , exists in 4-space with 4 octagonal around each vertex, in a zig-zagging nonplanar vertex figure. These octagonal faces can be seen on the bitruncated 24-cell, using all 576 edges and 288 vertices. The 192 triangular faces of the bitruncated 24-cell can be seen as removed. The dual regular skew polyhedron, , is similarly related to the square faces of the runcinated 24-cell.

Disphenoidal 288-cell

The disphenoidal 288-cell is the dual of the bitruncated 24-cell. It is a 4-dimensional

(or polychoron) derived from the

. It is constructed by doubling and rotating the 24-cell, then constructing the

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

. Being the dual of a uniform polychoron, it is

, consisting of 288 congruent tetragonal disphenoids. In addition, it is

under the group Aut(F₄).

Images

Geometry

The vertices of the 288-cell are precisely the 24 Hurwitz unit quaternions with norm squared 1, united with the 24 vertices of the dual 24-cell with norm squared 2, projected to the unit

3-sphere In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ...

. These 48 vertices correspond to the binary octahedral group 2O or <2,3,4>, order 48. Thus, the 288-cell is the only non-regular 4-polytope which is the convex hull of a quaternionic group, disregarding the infinitely many dicyclic (same as binary dihedral) groups; the regular ones are the

(≘ 2T or <2,3,3>, order 24) and the

600-cell In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...

(≘ 2I or <2,3,5>, order 120). (The

corresponds to the binary dihedral group 2D₂ or <2,2,2>, order 16.) The inscribed 3-sphere has radius 1/2+/4 ≈ 0.853553 and touches the 288-cell at the centers of the 288 tetrahedra which are the vertices of the dual bitruncated 24-cell. The vertices can be coloured in 2 colours, say red and yellow, with the 24 Hurwitz units in red and the 24 duals in yellow, the yellow

being congruent to the red one. Thus the product of 2 equally coloured quaternions is red and the product of 2 in mixed colours is yellow. Placing a fixed red vertex at the north pole (1,0,0,0), there are 6 yellow vertices in the next deeper “latitude” at (/2,x,y,z), followed by 8 red vertices in the latitude at (1/2,x,y,z). The complete coordinates are given as linear combinations of the quaternionic units which at the same time can be taken as the elements of the group The next deeper latitude is the equator hyperplane intersecting the 3-sphere in a 2-sphere which is populated by 6 red and 12 yellow vertices. Layer ''2'' is a 2-sphere circumscribing a regular octahedron whose edges have length 1. A tetrahedron with vertex north pole has 1 of these edges as long edge whose 2 vertices are connected by short edges to the north pole. Another long edge runs from the north pole into layer ''1'' and 2 short edges from there into layer ''2''. There are 192 long edges with length 1 connecting equal colours and 144 short edges with length ≈ 0.765367 connecting mixed colours. 192*2/48 = 8 long and 144*2/48 = 6 short, that is together 14 edges meet at any vertex. The 576 faces are

isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides ...

with 1 long and 2 short edges, all congruent. The angles at the base are arccos(/4) ≈ 49.210°. 576*3/48 = 36 faces meet at a vertex, 576*1/192 = 3 at a long edge, and 576*2/144 = 8 at a short one. The 288 cells are tetrahedra with 4 short edges and 2 antipodal and perpendicular long edges, one of which connects 2 red and the other 2 yellow vertices. All the cells are congruent. 288*4/48 = 24 cells meet at a vertex. 288*2/192 = 3 cells meet at a long edge, 288*4/144 = 8 at a short one. 288*4/576 = 2 cells meet at a triangle.

Related polytopes

B₄ family of uniform polytopes: F₄ family of uniform polytopes:

References

* H.S.M. Coxeter: ** 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. (1966) * x3x4o3o=x3x3x4o - tico, o3x4x3o - cont * {{PolyCell , urlname = section3.html, title = 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 24, 27 Uniform 4-polytopes