Hexadecachoron

Hexadecachoron
Geometry
Dimensionality	4
Schläfli Symbol	'"`UNIQ--postMath-00000001-QINU`"'
Subfacets
Vertices	8 points
Edges	24 line segments
Faces	32 triangles
Cells	16 tetrahedra
Tera	1 hexadecachoron
Hypervolumes
Vertex Count	'"`UNIQ--postMath-00000002-QINU`"'
Edges Length	'"`UNIQ--postMath-00000003-QINU`"'
Surface Area	'"`UNIQ--postMath-00000004-QINU`"'
Surcell Volume	'"`UNIQ--postMath-00000005-QINU`"'
Surteron Bulk	'"`UNIQ--postMath-00000006-QINU`"'

Vertex-first projection of a hexadecachoron. In this projection, eight cells are obscured.

A hexadecachoron (also called tetracross) is the four-dimensional cross polytope and the dual of a tesseract. It is also called a 16-cell, due to its 16 tetrahedral cells. It has the schläfli symbol $\{3, 3, 4\}$, meaning that 4 tetrahedra meet at each edge. Its Bowers acronym is "hex".

Properties

The hexadecachoron is the 4-D demihypercube, meaning it can be constructed by taking half the vertices of a tesseract. It is also a bipyramid of an octahedron, or a square duopyramid. Eight tetrahedra join at each vertex.

Structure

When seen from one of its cells, the hexadecachoron first has one cell, then four attached to its faces. The next six share an edge with the top cell and are perpendicular to it. The remaining four side cells share only a vertex with the top cell, and are themselves joined to the final cell, in dual orientation to the top cell.

In vertex-first orientation, it is just two octahedral pyramids joined together.

 Hypervolumes

• vertex count = $8$
• edge length = $24 l$
• surface area = $8{ l }^{ 2 }\sqrt { 3 }$
• surcell volume = $\frac { 4\sqrt { 2 } }{ 3 } { l }^{ 3 }$
• surteron bulk = $\frac{1}{6}l^4$

Subfacets

• 8 points (0D)
• 24 line segments (1D)
• 32 triangles (2D)
• 16 tetrahedra (3D)

See also

	Zeroth	First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
Simplex	Point	Line	Triangle	Tetrahedron	Pentachoron	Hexateron	Heptapeton	Octaexon	Enneazetton	Decayotton	Hendecaxennon
Hypercube	Point	Line	Square	Cube	Tesseract	Penteract	Hexeract	Hepteract	Octeract	Enneract	Dekeract
Cross	Point	Line	Square	Octahedron	Hexadecachoron	Pentarss	Hexarss	Heptarss	Octarss	Ennearss	Decarss
Hypersphere	Point	Line	Circle	Sphere	Glome	Hyperglome	Hexaphere	Heptaphere	Octaphere	Enneaphere	Decaphere