Tetrahedron (EntityTopic, 18)

From Hi.gher. Space

Revision as of 18:43, 24 November 2009

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%)