Triangular prism (EntityTopic, 20)
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{{STS Shape | {{STS Shape | ||
| dim=3 | | dim=3 | ||
| - | | elements= | + | | elements=2 [[triangle]]s, 3 [[square]]s, 9 [[digon]]s, 6 [[point]]s |
| + | | sym=[[Pyroprismatic symmetry|D<sub>3h</sub>]] | ||
| genus=0 | | genus=0 | ||
| ssc=[G3x] | | ssc=[G3x] | ||
| Line 16: | Line 17: | ||
| flayout={{FLD|a3|i|a2|er}} | | flayout={{FLD|a3|i|a2|er}} | ||
| dual=[[Triangular bipyramid]] | | dual=[[Triangular bipyramid]] | ||
| + | | bowers=Trip | ||
}}{{STS Uniform polytope | }}{{STS Uniform polytope | ||
| wythoff=<nowiki>2 3 | 2</nowiki> | | wythoff=<nowiki>2 3 | 2</nowiki> | ||
| schlaefli=t{2,3} | | schlaefli=t{2,3} | ||
| + | | dynkin=x2x3o | ||
| conway=P3 | | conway=P3 | ||
| vlayout=[[Triangle|3]]⋅[[Square|4]]<sup>2</sup> | | vlayout=[[Triangle|3]]⋅[[Square|4]]<sup>2</sup> | ||
| - | | vfigure=Isosceles [[triangle]] | + | | vfigure=Isosceles [[triangle]], edges 1, √2, √2 |
}}}} | }}}} | ||
| + | The triangular prism is the prism whose base is a triangle. If the sides are squares, it is a uniform polyhedron. | ||
| + | ==Coordinates== | ||
| + | The coordinates of a triangular prism of side 2 are: | ||
| + | <blockquote>(±1, -√3/3, ±1)<br>(0, 2√3/3, ±1)</blockquote> | ||
== Equations == | == Equations == | ||
*Variables: | *Variables: | ||
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*The [[planar]] [[cross-section]]s (''n'') of a triangular prism are: | *The [[planar]] [[cross-section]]s (''n'') of a triangular prism are: | ||
| - | <blockquote> | + | <blockquote>[!x,!y] ⇒ [[square]] |
| - | < | + | [!z] ⇒ [[triangle]]</blockquote> |
| + | |||
| + | <[#polytope [id 32]]> | ||
| + | |||
{{Polyhedra}} | {{Polyhedra}} | ||
{{Tapertope Nav|9|10|11|[11]<sup>1</sup><br>Square pyramid|11<sup>1</sup><br>Triangular prism|1<sup>2</sup><br>Tetrahedron|hedra}} | {{Tapertope Nav|9|10|11|[11]<sup>1</sup><br>Square pyramid|11<sup>1</sup><br>Triangular prism|1<sup>2</sup><br>Tetrahedron|hedra}} | ||
| - | |||
Latest revision as of 14:45, 26 March 2017
The triangular prism is the prism whose base is a triangle. If the sides are squares, it is a uniform polyhedron.
Coordinates
The coordinates of a triangular prism of side 2 are:
(±1, -√3/3, ±1)
(0, 2√3/3, ±1)
Equations
- Variables:
l ⇒ length of edges of triangular prism
- The hypervolumes of a triangular prism are given by:
total edge length = 9l
surface area = (3 + √3∕2) · l2
volume = √3∕4 · l3
- The planar cross-sections (n) of a triangular prism are:
[!x,!y] ⇒ square [!z] ⇒ triangle
Incidence matrix
Dual: triangular bipyramid
| # | TXID | Va | Ea | Eb | 4a | 3a | Type | Name |
|---|---|---|---|---|---|---|---|---|
| 0 | Va | = point | ; | |||||
| 1 | Ea | 2 | = digon | ; | ||||
| 2 | Eb | 2 | = digon | ; | ||||
| 3 | 4a | 4 | 2 | 2 | = square | ; | ||
| 4 | 3a | 3 | 0 | 3 | = base of prism: triangle | ; | ||
| 5 | C1a | 6 | 3 | 6 | 3 | 2 | = triangular prism | ; |
Usage as facets
- 6× 1-facets of a duotrianglinder
- 8× 1-facets of a octahedral prism
- 4× 1-facets of a tetrahedral prism
- prism: 4× 1-facets of a triangular diprism
- 18× 1-facets of a (dual of triangular octagoltriate) (named squashed prisms)
- 6× 1-facets of a triangular octagoltriate (named prisms)
- 4× 1-facets of a square pyramid prism
- 4× 1-facets of a gyrated octahedral prism (named in-out)
- 4× 1-facets of a gyrated octahedral prism (named walls)
- 8× 1-facets of a K4.13 dual
- 2× 1-facets of a K4.8
- 2× 1-facets of a bilunabirotunda pseudopyramid
- 1× 1-facets of a D4.16
- 4× 1-facets of a bitrigonal diminished pyrocantichoron
- 16× 1-facets of a (4D analog of J37)
- 8× 1-facets of a (4D analog of J37)
- 8× 1-facets of a (4D analog of J37)
- 3× 1-facets of a triangular hebesphenorotunda pseudopyramid (named square to roof)
- 6× 1-facets of a triangular hebesphenorotundaeic rhombochoron
- 6× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named oooogg)
- 12× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named square to roof)
- 6× 1-facets of a D4.7
- 12× 2-facets of a triangular triprism
| 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 |
| 9. [11]1 Square pyramid | 10. 111 Triangular prism | 11. 12 Tetrahedron |
| List of tapertopes | ||
