Tetrahedron (EntityTopic, 18)
From Hi.gher. Space
(Difference between revisions)
m (CSG -> SSC) |
Username5243 (Talk | contribs) (→Cartesian coordinates) |
||
(39 intermediate revisions not shown) | |||
Line 1: | Line 1: | ||
- | {{Shape|Tetrahedron| | + | <[#ontology [kind topic] [cats 3D Simplex Demihypercube] [alt [[freebase:07jht]] [[wikipedia:Tetrahedron]]]]> |
+ | {{STS Shape | ||
+ | | name=Tetrahedron | ||
+ | | image=<[#embed [hash 5JT5JWF1ADKFT7YS0W23RK4MZY] [width 150]]> | ||
+ | | dim=3 | ||
+ | | elements=4 [[triangle]]s, 6 [[digon]]s, 4 [[point]]s | ||
+ | | sym=[[Pyrohedral symmetry|T<sub>d</sub>, A<sub>3</sub>, [3,3], (*332)]] | ||
+ | | genus=0 | ||
+ | | ssc=xPP or {G3<sup>3</sup>} | ||
+ | | ssc2=Kt1 | ||
+ | | pv_circle=~0.1225 | ||
+ | | pv_square=⅓ | ||
+ | | extra={{STS Matrix| | ||
+ | 3 0 | ||
+ | 3 1 | ||
+ | 1 1}}{{STS Tapertope | ||
+ | | order=1, 2 | ||
+ | | notation=1<sup>2</sup> | ||
+ | | index=11 | ||
+ | }}{{STS Polytope | ||
+ | | flayout={{FLD|a3|er|e3}} | ||
+ | | petrie=4,0 | ||
+ | | dual=''Self-dual'' | ||
+ | | bowers=Tet | ||
+ | }}{{STS Uniform polytope | ||
+ | | wythoff=<nowiki>3 | 2 3 or | 2 2 2</nowiki> | ||
+ | | schlaefli={[[Triangle|3,]]3} or sr{2,2} | ||
+ | | dynkin=x3o3o | ||
+ | | conway=Y3 | ||
+ | | vfigure=Equilateral [[triangle]], edge 1 | ||
+ | | vlayout=[[Triangle|3]]<sup>3</sup> | ||
+ | }}}} | ||
+ | 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. | ||
- | A | + | == Cartesian coordinates == |
+ | A regular tetrahedron with edge length 2, centered at the origin, can be defined using the coordinates: | ||
+ | <blockquote>(√2/2, √2/2, √2/2);<br>(−√2/2, −√2/2, √2/2);<br>(−√2/2, √2/2, −√2/2);<br>(√2/2, −√2/2, −√2/2).</blockquote> | ||
+ | Alternatively, a regular tetrahedron with symmetry through the z-axis and edge length 2 can be defined using the coordinates: | ||
+ | <blockquote>(-1, -√3/3, -√6/6)<br>(1, -√3/3, -√6/6)<br>(0, 2√3/3, -√6/6)<br>(0, 0, √6/2)</blockquote> | ||
+ | Finally, a regular tetrahedron with edge length 1 and two opposite edges parallel to the axes can be defined using the coordinates: | ||
+ | <blockquote>({{Over|1|2}}, {{Over|√2|4}}, 0);<br>(−{{Over|1|2}}, {{Over|√2|4}}, 0);<br>(0, −{{Over|√2|4}}, {{Over|1|2}});<br>(0, −{{Over|√2|4}}, −{{Over|1|2}}).</blockquote> | ||
+ | *The cross sections of a tetrahedron parallel to an axis are a point that expands into a triangle. | ||
== Equations == | == Equations == | ||
- | + | *The [[hypervolume]]s of a tetrahedron with side length ''l'' 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 with side length ''l'' is given by: |
- | <blockquote>''h'' = | + | <blockquote>''h'' = <sup>√6</sup>∕<sub>3</sub> · ''l''</blockquote> |
- | + | <[#polytope [id 1]]> | |
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
+ | {{Simplices|3}} | ||
+ | {{Demihypercubes|3}} | ||
{{Trishapes}} | {{Trishapes}} | ||
- | {{ | + | {{Tapertope Nav|10|11|12|11<sup>1</sup><br>Triangular prism|1<sup>2</sup><br>Tetrahedron|4<br>Glome|hedra}} |
- | + | ||
- | + |
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:
(1∕2, √2∕4, 0);
(−1∕2, √2∕4, 0);
(0, −√2∕4, 1∕2);
(0, −√2∕4, −1∕2).
- 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 = √2∕12 · l3
- The perpendicular height h of a tetrahedron with side length l is given by:
h = √6∕3 · l
Incidence matrix
Dual: Self-dual
# | TXID | Va | Ea | 3a | Type | Name |
---|---|---|---|---|---|---|
0 | Va | = point | ; | |||
1 | Ea | 2 | = digon | ; | ||
2 | 3a | 3 | 3 | = base of pyramid: triangle | ; | |
3 | C1a | 4 | 6 | 4 | = tetrahedron | ; |
Usage as facets
- 16× 1-facets of a aerochoron
- 600× 1-facets of a hydrochoron
- pyramid: 5× 1-facets of a pyrochoron
- 9× 1-facets of a duotrianglone
- 5× 1-facets of a pyrorectichoron
- 8× 1-facets of a tetrahedral bipyramid
- prism: 2× 1-facets of a tetrahedral prism
- 12× 1-facets of a (dual of triangular diprism)
- 8× 1-facets of a octahedral pyramid
- 8× 1-facets of a square pyramid bipyramid
- 1× 1-facets of a K4.8
- 4× 1-facets of a K4.8 dual
- 4× 1-facets of a square biantiprismatic ring
- 40× 1-facets of a castellated rhodoperihedral prism
- 288× 1-facets of a (dual of truncated snub demitesseract)
- 288× 1-facets of a (dual of truncated snub demitesseract)
- 8× 1-facets of a D4.11
- 32× 1-facets of a D4.11
- 4× 1-facets of a (dual of bilunabirotunda pseudopyramid)
- 4× 1-facets of a bilunabirotunda pseudopyramid
- 2× 1-facets of a D4.16 (named cap tets)
- 6× 1-facets of a D4.16 (named sweep tets)
- 2× 1-facets of a D4.16 dual
- 4× 1-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 6× 1-facets of a triangular hebesphenorotunda pseudopyramid (named bola to roof)
- 3× 1-facets of a triangular hebesphenorotunda pseudopyramid (named lat to roof)
- 12× 1-facets of a triangular hebesphenorotundaeic rhombochoron
- 12× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named oogg)
- 24× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named bola to roof)
- 12× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named lat to roof)
- 12× 1-facets of a D4.7 (named roob)
- 12× 1-facets of a D4.7 (named rgcc)
- 24× 1-facets of a D4.7 (named rygc)
- 80× 2-facets of a aeroteron
- 15× 2-facets of a pyroteron
- 36× 2-facets of a (dual of triangular triprism)
- 24× 2-facets of a (dual of triangular triprism)
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 |