In 4-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 ...

, the tesseract 16-cell compound is a

polytope compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connec ...

composed of a regular

tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of e ...

and its dual, the regular

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 ...

. Its

convex hull In geometry, the convex hull or 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 ...

is the regular

24-cell In 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 "octahedral complex"), icosatetrahedroid, o ...

, which is self-dual. A '' compound polytope'' is a figure that is composed of several polytopes sharing a common center. The outer vertices of a compound can be connected to form a

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...

called its

. The compound is a

facetting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. New edges of a faceted polyhedron may be cre ...

of the convex hull. In 4-polytope compounds constructed as dual pairs, cells and vertices swap positions and faces and edges swap positions. Because of this the number of cells and vertices are equal, as are faces and edges. Mid-edges of the tesseract cross mid-face in the 16-cell, and vice versa. The tesseract 16-cell compound can be seen as the 4-dimensional analogue of a

compound of cube and octahedron The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound. Construction The 14 Cartesian coordinates of the vertices of the compound are. : 6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2) : ...

. It is one of four compound polytopes which are obtained by combining a regular convex 4-polytope with its dual; the other three being the

compound of two 5-cells 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 ...

, compound of two 24-cells and

compound of 120-cell and 600-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...

Construction

The 24

Cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

s of the vertices of the compound are: : 8: (±2, 0, 0, 0), ( 0, ±2, 0, 0), ( 0, 0, ±2, 0), ( 0, 0, 0, ±2) : 16: ( ±1, ±1, ±1, ±1) These are the first two vertex sets of the

stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specif ...

s of a 16-cell.The Stellated Forms of the Sixteen-Cell
B. L. Chilton The American Mathematical Monthly Vol. 74, No. 4 (Apr., 1967), pp. 372–378

Faceting the 24-cell

The

of the tesseract 16-cell compound is the regular

. This makes the compound a

faceting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. New edges of a faceted polyhedron may be ...

of the 24-cell. The 24-cell is a '' rectified 16-cell'' and also the convex hull of a compound of three 16-cells. The tesseract is the convex hull of a compound of two 16-cells. Thus the tesseract 16-cell compound is a lower-symmetry form of the 24-cell, which is the whole package (the F₄ symmetry group). The intersection of the tesseract and 16-cell compound is the uniform

bitruncated tesseract In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a bitruncation, and a tritruncation, which creates the ''truncated 16-cell''. Truncated tesse ...

: = ∩ .

References

* * {{mathworld, title=16-Cell, urlname=16-Cell

External links

Compound of the Tesseract and Its Dual, the 16-CellRotating Compound of the Tesseract and Its Dual
Polyhedral compounds

Construction

Faceting the 24-cell

See also

References

External links