Triangular prism (EntityTopic, 20)

From Hi.gher. Space

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<[#ontology [kind topic] [cats Triangle Prism 3D Tapertope Uniform Polytope] [alt [[freebase:087c5y]] [[wikipedia:Triangular_prism]]]]>
{{STS Shape
{{STS Shape
| dim=3
| dim=3
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| elements=5, 9, 6
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| elements=2 [[triangle]]s, 3 [[square]]s, 9 [[digon]]s, 6 [[point]]s
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| sym=[[Pyroprismatic symmetry|D<sub>3h</sub>]]
| genus=0
| genus=0
| ssc=[G3x]
| ssc=[G3x]
| ssc2=+G3
| ssc2=+G3
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| extra={{STS Rotope
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| extra={{STS Matrix|
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| attrib=pure
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<i>2 2</i> <s>0 0</s>
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| notation=1<sup>1</sup>1 x<sup>y</sup>z
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3 0 3 0
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| index=8
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1 1 1 1}}{{STS Tapertope
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| order=2, 1
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| notation=11<sup>1</sup>
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| index=10
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}}{{STS Polytope
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| flayout={{FLD|a3|i|a2|er}}
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| dual=[[Triangular bipyramid]]
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| 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}
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| dynkin=x2x3o
| conway=P3
| conway=P3
| vlayout=[[Triangle|3]]⋅[[Square|4]]<sup>2</sup>
| vlayout=[[Triangle|3]]⋅[[Square|4]]<sup>2</sup>
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| vfigure=Isosceles [[triangle]]
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| vfigure=Isosceles [[triangle]], edges 1, √2, √2
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| dual=[[Triangular bipyramid]]
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}}}}
}}}}
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The triangular prism is the prism whose base is a triangle. If the sides are squares, it is a uniform polyhedron.
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==Coordinates==
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The coordinates of a triangular prism of side 2 are:
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<blockquote>(±1, -√3/3, ±1)<br>(0, 2√3/3, ±1)</blockquote>
== Equations ==
== Equations ==
*Variables:
*Variables:
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*The [[hypervolume]]s of a triangular prism are given by:
*The [[hypervolume]]s of a triangular prism are given by:
<blockquote>total edge length = 9''l''<br>
<blockquote>total edge length = 9''l''<br>
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surface area = ''l''<sup>2</sup>(3 + 2<sup>-1</sup>√3)<br>
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surface area = (3 + {{Over|√3|2}}) {{DotHV}}<br>
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volume = ''l''<sup>3</sup>4<sup>-1</sup>√3</blockquote>
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volume = {{Over|√3|4}} {{DotHV|3}}</blockquote>
*The [[planar]] [[cross-section]]s (''n'') of a triangular prism are:
*The [[planar]] [[cross-section]]s (''n'') of a triangular prism are:
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<blockquote>''Unknown''</blockquote>
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<blockquote>[!x,!y] ⇒ [[square]]
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<br clear="all"><br>
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[!z] ⇒ [[triangle]]</blockquote>
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<[#polytope [id 32]]>
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{{Polyhedra}}
{{Polyhedra}}
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{{Rotope Nav|7|8|9|(III)<br>Sphere|I'I<br>Triangular prism|<nowiki>I''</nowiki><br>Tetrahedron|hedra}}
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{{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}}
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[[Category:Uniform prismahedra]]
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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
total edge length = 9l
surface area = (3 + √32) · l2
volume = √34 · l3
[!x,!y] ⇒ square [!z] ⇒ triangle

Incidence matrix

Dual: triangular bipyramid

#TXIDVaEaEb4a3aTypeName
0 Va = point ;
1 Ea 2 = digon ;
2 Eb 2 = digon ;
3 4a 422 = square ;
4 3a 303 = base of prism: triangle ;
5 C1a 63632 = triangular prism ;

Usage as facets


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


9. [11]1
Square pyramid
10. 111
Triangular prism
11. 12
Tetrahedron
List of tapertopes