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. 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 · l²
volume = ^√2∕₁₂ · l³

• The perpendicular height h of a tetrahedron with side length l is given by:

h = ^√6∕₃ · l

Incidence matrix

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