Tetrahedron (EntityTopic, 18)
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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)
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 l are given by:
total edge length = 6l
surface area = √3 · l^{2}
volume = ^{√2}∕_{12} · l^{3}
- The perpendicular height h of a tetrahedron with side length l is 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 Trishapes | |
Regular: | tetrahedron • cube • octahedron • dodecahedron • icosahedron |
Direct 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 prism | 11. 1^{2} Tetrahedron | 12. 4 Glome |
List of tapertopes |