# Tetrahedron (EntityTopic, 18)

### From Hi.gher. Space

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== Cartesian coordinates == | == Cartesian coordinates == | ||

- | A regular tetrahedron with edge length | + | A regular tetrahedron with edge length 2, centered at the origin, can be defined using the coordinates: |

- | <blockquote>( | + | <blockquote>(√2/2, √2/2, √2/2);<br>(−√2/2, −√2/2, √2/2);<br>(−√2/2, √2/2, −√2/2);<br>(√2/2, −√2/2, −√2/2).</blockquote> |

- | Alternatively, a regular tetrahedron with symmetry through the z-axis and edge length 2 | + | Alternatively, a regular tetrahedron with symmetry through the z-axis and edge length 2 can be defined using the coordinates: |

- | <blockquote>( | + | <blockquote>(-1, -√3/3, -√6/6)<br>(1, -√3/3, -√6/6)<br>(0, 2√3/3, -√6/6)<br>(0, 0, √6/2)</blockquote |

Finally, a regular tetrahedron with edge length 1 and two opposite edges parallel to the axes can be defined using the coordinates: | Finally, a regular tetrahedron with edge length 1 and two opposite edges parallel to the axes can be defined using the coordinates: | ||

<blockquote>({{Over|1|2}}, {{Over|√2|4}}, 0);<br>(−{{Over|1|2}}, {{Over|√2|4}}, 0);<br>(0, −{{Over|√2|4}}, {{Over|1|2}});<br>(0, −{{Over|√2|4}}, −{{Over|1|2}}).</blockquote> | <blockquote>({{Over|1|2}}, {{Over|√2|4}}, 0);<br>(−{{Over|1|2}}, {{Over|√2|4}}, 0);<br>(0, −{{Over|√2|4}}, {{Over|1|2}});<br>(0, −{{Over|√2|4}}, −{{Over|1|2}}).</blockquote> |

## Revision as of 14:03, 26 March 2017

A **tetrahedron** is the three-dimensional simplex. It is a special case of a pyramid where the base is a triangle. it is also the 3-D demicube. It is one of the five Platonic solids, containing four triangles joined three to a vertex.

## Cartesian coordinates

A regular tetrahedron with edge length 2, centered at the origin, can be defined using the coordinates:

(√2/2, √2/2, √2/2);

(−√2/2, −√2/2, √2/2);

(−√2/2, √2/2, −√2/2);

(√2/2, −√2/2, −√2/2).

Alternatively, a regular tetrahedron with symmetry through the z-axis and edge length 2 can be defined using the coordinates:

(-1, -√3/3, -√6/6)

(1, -√3/3, -√6/6)

(0, 2√3/3, -√6/6)

(0, 0, √6/2)</blockquote Finally, a regular tetrahedron with edge length 1 and two opposite edges parallel to the axes can be defined using the coordinates:(^{1}∕_{2},^{√2}∕_{4}, 0);

(−^{1}∕_{2},^{√2}∕_{4}, 0);

(0, −^{√2}∕_{4},^{1}∕_{2});

(0, −^{√2}∕_{4}, −^{1}∕_{2}).

- The cross sections of a tetrahedron parallel to an axis are a point that expands into a triangle.
## Equations

- The hypervolumes of a tetrahedron with side length
lare given by:total edge length = 6l

surface area = √3 ·l^{2}

volume =^{√2}∕_{12}·l^{3}

- The perpendicular height
hof a tetrahedron with side lengthlis given by:h=^{√6}∕_{3}·l## Incidence matrix

Dual:

Self-dual

# TXID Va Ea 3a Type Name 0 Va = point ; 1 Ea 2 = digon ; 2 3a 3 3 = base of pyramid: triangle; 3 C1a 4 6 4 = tetrahedron; ## Usage as facets

- 16× 1-facets of a aerochoron
- 600× 1-facets of a hydrochoron
pyramid: 5× 1-facets of a pyrochoron- 9× 1-facets of a duotrianglone
- 5× 1-facets of a pyrorectichoron
- 8× 1-facets of a tetrahedral bipyramid
prism: 2× 1-facets of a tetrahedral prism- 12× 1-facets of a (dual of triangular diprism)
- 8× 1-facets of a octahedral pyramid
- 8× 1-facets of a square pyramid bipyramid
- 1× 1-facets of a K4.8
- 4× 1-facets of a K4.8 dual
- 4× 1-facets of a square biantiprismatic ring
- 40× 1-facets of a castellated rhodoperihedral prism
- 288× 1-facets of a (dual of truncated snub demitesseract)
- 288× 1-facets of a (dual of truncated snub demitesseract)
- 8× 1-facets of a D4.11
- 32× 1-facets of a D4.11
- 4× 1-facets of a (dual of bilunabirotunda pseudopyramid)
- 4× 1-facets of a bilunabirotunda pseudopyramid
- 2× 1-facets of a D4.16 (named
cap tets)- 6× 1-facets of a D4.16 (named
sweep tets)- 2× 1-facets of a D4.16 dual
- 4× 1-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 6× 1-facets of a triangular hebesphenorotunda pseudopyramid (named
bola to roof)- 3× 1-facets of a triangular hebesphenorotunda pseudopyramid (named
lat to roof)- 12× 1-facets of a triangular hebesphenorotundaeic rhombochoron
- 12× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named
oogg)- 24× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named
bola to roof)- 12× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named
lat to roof)- 12× 1-facets of a D4.7 (named
roob)- 12× 1-facets of a D4.7 (named
rgcc)- 24× 1-facets of a D4.7 (named
rygc)- 80× 2-facets of a aeroteron
- 15× 2-facets of a pyroteron
- 36× 2-facets of a (dual of triangular triprism)
- 24× 2-facets of a (dual of triangular triprism)

Simplices triangle • tetrahedron• pyrochoron • pyroteron • pyropeton

Demihypercubes tetrahedron• aerochoron • demipenteract • demihexeract

Notable TrishapesRegular:tetrahedron• cube • octahedron • dodecahedron • icosahedronDirect truncates:tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate Mesotruncates:stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron Snubs:snub staurohedron • snub rhodohedron Curved:sphere • torus • cylinder • cone • frustum • crind

10. 11 ^{1}

Triangular prism11. 1^{2}

Tetrahedron12. 4

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