Hexadecachoron

Tetrarss
A Tetrarss.
Geometry
Dimensionality	4D
Schläfli Symbol	{4,4,4}
Subfacets
Vertices	8 points
Edges	24 line segments
Faces	32 triangles
Cells	16 tetrahedrons

A hexadochoron (also called tetrarss or 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 = $\frac{1}{6}l^4$

See Also

	Zeroth	First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth	Eleventh	Twelfth	Thirteenth	Fourteenth	Fifteenth	Sixteenth	Seventeenth	Eighteenth	Nineteenth	Twentieth	Twenty-first	Twenty-second	Twenty-third	Twenty-fourth	Twenty-fifth	Twenty-sixth	Twenty-seventh	Twenty-eighth	Twenty-ninth	Thirtieth	Thirty-first	Thirty-second	Thirty-third	Thirty-fourth	Thirty-fifth	Thirty-sixth	Thirty-seventh	Thirty-eighth	Thirty-ninth	Fortieth
Simplex	Point	Line	Triangle	Tetrahedron	Pentachoron	Hexateron	Heptapeton	Octaexon	Enneazetton	Decayotton	Hendecaxennon	Dodecadakon	Tredecahendakon	Quattuordecadokon	Quindecatradakon	Sexdecateradakon	Septendecapetadakon	Octodecaexadakon	Novemdecazekadakon	Icosayokadakon	Henicosanekadakon	Doicosahotadakon	Treicosadakhotadakon	Quattuoricosaikhotadakon	Quinicosatrakhotadakon	Sexicosatekhotadakon	Septenicosapekhotadakon	Octoicosaexakhotadakon	Novemicosazekhotadakon	Triacontayokhotadakon	Hentriacontanekhotadakon	Dotriacontabotadakon	Tretriacontadakbotadakon	Quattuortriacontaikbotadakon	Quintriacontatrakbotadakon	Sextriacontatekbotadakon	Septentriacontapekbotadakon	Octotriacontaexakbotadakon	Novemtriacontazekbotadakon	Tetracontayokbotadakon	Untetracontanekbotadakon
Hypercube	Point	Line	Square	Cube	Tesseract	Penteract	Hexeract	Hepteract	Octeract	Enneract	Dekeract	Hendekeract	Dodekeract	Tridekeract	Tetradekeract	Pentadekeract	Hexadekeract	Heptadekeract	Octadekeract	Enneadekeract	Icoseract	Unicoseract	Doicoseract	Treicoseract	Quattuoricoseract	Quinicoseract	Sexicoseract	Septenicoseract	Octoicoseract	Novemicoseract	Triaconteract	Untriaconteract	Dotriaconteract	Tretriaconteract	Quattuortriaconteract	Quintriaconteract	Sextriaconteract	Septentriaconteract	Octotriaconteract	Novemtriaconteract	Tetraconteract
Cross	Point	Line	Triangle	Octahedron	Tetrarss	Pentarss	Hexarss	Heptarss	Octarss	Ennearss	Decarss	Hendecarss	Dodecarss	Tredecarss	Quattuordecarss	Quindecatrarss	Sexdecaterarss	Septendecapetarss	Octodecaexarss	Novemdecazettarss	Icosayottarss	Henicosaxennarss	Doicosadakarss	Treicosahendarss	Quattuoricosadokarss	Quinicosatradakarss	Sexicosatedakarss	Septenicosapedakarss	Octoicosaexdakarss	Novemicosazedakarss	Triacontayodakarss	Hentriacontanedakarss	Dotriacontaikarss	Tretriacontaikenarss	Quattuortriacontaicodarss	Quintriacontaictrarss	Sextriacontaicterarss	Septentriacontaikpetarss	Octotriacontaikectarss	Novemtriacontaiczetarss	Tetracontaicyotarss
Hypersphere	Point	Line	Circle	Sphere	Glome	Hyperglome	Hexaphere	Heptaphere	Octaphere	Enneaphere	Decaphere	Hendecaphere	Dodecaphere	Tredecaphere	Quattuordecaphere	Quindecatraphere	Sexdecateraphere	Septendecapetaphere	Octodecaexaphere	Novemdecazekaphere	Icosayokaphere	Henicosanekaphere	Doicosahotaphere	Treicosadakhotaphere	Quattuoricosaikhotaphere	Quinicosatrakhotaphere	Sexicosatekhotaphere	Septenicosapekhotaphere	Octoicosaexakhotaphere	Novemicosazekhotaphere	Triacontayokhotaphere	Hentriacontanekhotaphere	Dotriacontabotaphere	Tretriacontadakbotaphere	Quattuortriacontaikbotaphere	Quintriacontatrakbotaphere	Sextriacontatekbotaphere	Septentriacontapekbotaphere	Octotriacontaexakbotaphere	Novemtriacontazekbotaphere	Tetracontayokbotaphere