Tetrahedron (EntityTopic, 18)

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Revision as of 22:00, 12 March 2011


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
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
Unknown

Use

Tetrahedral cells are found in these tetrashapes on FGwiki:


Simplices
triangletetrahedronpyrochoronpyroteronpyropeton


Demihypercubes
tetrahedronaerochorondemipenteractdemihexeract


Notable Trishapes
Regular: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind


10. 111
Triangular prism
11. 12
Tetrahedron
12. 4
Glome
List of tapertopes