Tetrahedron (EntityTopic, 18)

From Hi.gher. Space

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| image=<[#embed [hash 5JT5JWF1ADKFT7YS0W23RK4MZY] [width 150]]>
| image=<[#embed [hash 5JT5JWF1ADKFT7YS0W23RK4MZY] [width 150]]>
| dim=3
| dim=3
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| elements=4, 6, 4
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| elements=4 [[triangle]]s, 6 [[digon]]s, 4 [[point]]s
| sym=[[Pyrohedral symmetry|T<sub>d</sub>, A<sub>3</sub>, [3,3], (*332)]]
| sym=[[Pyrohedral symmetry|T<sub>d</sub>, A<sub>3</sub>, [3,3], (*332)]]
| genus=0
| genus=0
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| wythoff=<nowiki>3 | 2 3 or | 2 2 2</nowiki>
| wythoff=<nowiki>3 | 2 3 or | 2 2 2</nowiki>
| schlaefli={[[Triangle|3,]]3} or sr{2,2}
| schlaefli={[[Triangle|3,]]3} or sr{2,2}
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| dynkin=x3o3o
| conway=Y3
| conway=Y3
| vfigure=Equilateral [[triangle]], edge 1
| vfigure=Equilateral [[triangle]], edge 1
| vlayout=[[Triangle|3]]<sup>3</sup>
| vlayout=[[Triangle|3]]<sup>3</sup>
}}}}
}}}}
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A '''tetrahedron''' is the three-dimensional [[simplex]]. It is a special case of a [[pyramid]] where the base is a [[triangle]].
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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 ==
== Cartesian coordinates ==
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A regular tetrahedron with edge length 2√2, centered at the origin, can be defined using the coordinates:
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A regular tetrahedron with edge length 2, centered at the origin, can be defined using the coordinates:
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<blockquote>(1, 1, 1);<br>(−1, −1, 1);<br>(−1, 1, −1);<br>(1, −1, −1).</blockquote>
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<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>
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Alternatively, a regular tetrahedron with symmetry through the z-axis and edge length 2{{Over|√6|3}} can be defined using the coordinates:
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Alternatively, a regular tetrahedron with symmetry through the z-axis and edge length 2 can be defined using the coordinates:
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<blockquote>(0, 0, 1);<br>(2{{Over|√2|3}}, 0, –1/3);<br>(−{{Over|√2|3}}, {{Over|√6|3}}, –1/3);<br>(−{{Over|√2|3}}, −{{Over|√6|3}}, –1/3).</blockquote>
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<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:
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>
<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>
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*The cross sections of a tetrahedron parallel to an axis are a point that expands into a triangle.
== Equations ==
== Equations ==

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

#TXIDVaEa3aTypeName
0 Va = point ;
1 Ea 2 = digon ;
2 3a 33 = base of pyramid: triangle ;
3 C1a 464 = tetrahedron ;

Usage as facets


Simplices
triangletetrahedronpyrochoronpyroteronpyropeton


Demihypercubes
tetrahedronaerochorondemipenteractdemihexeract


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


10. 111
Triangular prism
11. 12
Tetrahedron
12. 4
Glome
List of tapertopes