Tetrahedron (EntityTopic, 18)
From Hi.gher. Space
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 can be defined using the coordinates:
(0, 0, 1);
(2sqrt(2)/3, 0, –1/3);
(-sqrt(2)/3, sqrt(6)/3, –1/3);
(-sqrt(2)/3, -sqrt(6)/3, –1/3);
Equations
- Variables:
l ⇒ length of edges of tetrahedron
- All points (x, y, z) that lie on the surface of a tetrahedron will satisfy the following equations:
Unknown
- All points (x, y, z) that lie on the edges of a tetrahedron will satisfy the following equations:
Unknown
- The hypervolumes of a tetrahedron are given by:
total edge length = 6l
surface area = sqrt(3)l2
volume = 12-1sqrt(2)l3
- The perpendicular height h of a tetrahedron is given by:
h = 3-1sqrt(6)l
- The planar cross-sections (n) of a tetrahedron are:
Unknown
Use
Tetrahedral cells are found in these tetrashapes on FGwiki:
- Hexacosichoron (600×, 100%)
- Hexadecachoron (8×, 100%)
- Pentachoron (5×, 100%)
- Square dipyramid (4×, 67%)
- Pentachoric hemicate (5×, 50%)
Simplices |
triangle • tetrahedron • pyrochoron • pyroteron • pyropeton |
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. 111 Triangular prism | 11. 12 Tetrahedron | 12. 4 Glome |
List of tapertopes |