Tetrahedron (EntityTopic, 18)
From Hi.gher. Space
(Difference between revisions)
m |
(fix equations) |
||
| Line 41: | Line 41: | ||
*Variables: | *Variables: | ||
<blockquote>''l'' ⇒ length of edges of tetrahedron</blockquote> | <blockquote>''l'' ⇒ length of edges of tetrahedron</blockquote> | ||
| - | |||
| - | |||
| - | |||
| - | |||
| - | |||
| - | |||
*The [[hypervolume]]s of a tetrahedron are given by: | *The [[hypervolume]]s of a tetrahedron are given by: | ||
<blockquote>total edge length = 6''l''<br> | <blockquote>total edge length = 6''l''<br> | ||
| - | surface area = | + | surface area = √3 · ''l''<sup>2</sup><br> |
| - | volume = | + | volume = <sup>√2</sup>∕<sub>12</sub> · ''l''<sup>3</sup></blockquote> |
*The perpendicular height ''h'' of a tetrahedron is given by: | *The perpendicular height ''h'' of a tetrahedron is given by: | ||
<blockquote>''h'' = 3<sup>-1</sup>sqrt(6)''l''</blockquote> | <blockquote>''h'' = 3<sup>-1</sup>sqrt(6)''l''</blockquote> | ||
| - | |||
| - | |||
| - | |||
== Use == | == Use == | ||
Revision as of 01:29, 18 November 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
- The hypervolumes of a tetrahedron are given by:
total edge length = 6l
surface area = √3 · l2
volume = √2∕12 · l3
- The perpendicular height h of a tetrahedron is given by:
h = 3-1sqrt(6)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%)
| 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 | ||
