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.

Cartesian coordinates

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

(1, 1, 1);
(−1, −1, 1);
(−1, 1, −1);
(1, −1, −1).

Alternatively, a regular tetrahedron with symmetry through the z-axis and edge length 2√63 can be defined using the coordinates:

(0, 0, 1);
(2√23, 0, –1/3);
(−√23, √63, –1/3);
(−√23, −√63, –1/3).

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

(12, √24, 0);
(−12, √24, 0);
(0, −√24, 12);
(0, −√24, −12).

Equations

• The hypervolumes of a tetrahedron with side length l are given by:
total edge length = 6l
surface area = √3 · l2
volume = √212 · l3
• The perpendicular height h of a tetrahedron with side length l is given by:
h = √63 · l

Incidence matrix

Dual: Self-dual

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

Usage as facets

 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. 111Triangular prism 11. 12Tetrahedron 12. 4Glome List of tapertopes