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 ...

, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or

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 ...

), which is bounded by 48

cells Cell most often refers to: * Cell (biology), the functional basic unit of life * Cellphone, a phone connected to a cellular network * Clandestine cell, a penetration-resistant form of a secret or outlawed organization * Electrochemical cell, a d ...

: 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 cuboctahedra. It can be obtained by

rectification Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...

of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.

E. L. Elte Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) Emanuël Lodewijk 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 = ,3,4 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

: the 24-cell is truncated at the midpoints. The vertices become

s, while the

octahedra In geometry, an octahedron (: 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 i ...

become cuboctahedra.

Cartesian coordinates

A rectified 24-cell having an edge length of has vertices given by all permutations and sign permutations of the following

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: : (0,1,1,2) !/2!×2³ = 96 vertices The dual configuration with edge length 2 has all coordinate and sign permutations of: : (0,2,2,2) ×2³ = 32 vertices: (1,1,1,3) ×2⁴ = 64 vertices

Images

Symmetry constructions

There are three different symmetry constructions of this polytope. The lowest

_4

construction can be doubled into

_4

by adding a mirror that maps the bifurcating nodes onto each other.

_4

can be mapped up to

_4

symmetry by adding two mirror that map all three end nodes together. The

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 ...

is a

triangular prism In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...

, containing two cubes and three cuboctahedra. The three symmetries can be seen with 3 colored cuboctahedra in the lowest

_4

construction, and two colors (1:2 ratio) in

_4

, and all identical cuboctahedra in

_4

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

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, and 192 vertices. Its vertex figure is a

triangular bifrustum In geometry, the triangular bifrustum is the second in an infinite series of bifrustum polyhedra. It has 6 trapezoid and 2 triangle faces. It may also be called the truncated triangular bipyramid; however, that term is ambiguous, as it may also r ...

Related uniform polytopes

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

Citations

References

* T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900 * * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 26. pp. 409: Hemicubes: 1_n1) * Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) * ** ** * {{Polytopes Uniform 4-polytopes