Tetrahedron (EntityTopic, 18)

From Hi.gher. Space

Revision as of 12:25, 8 February 2014

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^√6∕₃ can be defined using the coordinates:

(0, 0, 1);
(2^√2∕₃, 0, –1/3);
(−^√2∕₃, ^√6∕₃, –1/3);
(−^√2∕₃, −^√6∕₃, –1/3).

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).

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

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%)