# Tetrahedron (EntityTopic, 18)

(Difference between revisions)
 Revision as of 14:03, 26 March 2017 (view source)← Older edit Latest revision as of 14:05, 26 March 2017 (view source) (→Cartesian coordinates) Line 37: Line 37:
(√2/2, √2/2, √2/2);
(−√2/2, −√2/2, √2/2);
(−√2/2, √2/2, −√2/2);
(√2/2, −√2/2, −√2/2).
(√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: 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)
(-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: Finally, a regular tetrahedron with edge length 1 and two opposite edges parallel to the axes can be defined using the coordinates:
({{Over|1|2}}, {{Over|√2|4}}, 0);
(−{{Over|1|2}}, {{Over|√2|4}}, 0);
(0, −{{Over|√2|4}}, {{Over|1|2}});
(0, −{{Over|√2|4}}, −{{Over|1|2}}).
({{Over|1|2}}, {{Over|√2|4}}, 0);
(−{{Over|1|2}}, {{Over|√2|4}}, 0);
(0, −{{Over|√2|4}}, {{Over|1|2}});
(0, −{{Over|√2|4}}, −{{Over|1|2}}).

## Latest revision as of 14:05, 26 March 2017

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:

(12, √24, 0);
(−12, √24, 0);
(0, −√24, 12);
(0, −√24, −12).
• 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 = √212 · l3
• The perpendicular height h of a tetrahedron with side length l is given by:
h = √63 · l

## Incidence matrix

Dual: Self-dual

 # TXID Type Name Va Ea 3a 0 Va = point ; 1 Ea 2 = digon ; 2 3a 3 3 = base of pyramid: triangle ; 3 C1a 4 6 4 = tetrahedron ;

## Usage as facets

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