Tetrahedron (EntityTopic, 18)

From Hi.gher. Space

(Difference between revisions)
m
(→Cartesian coordinates)
 
(25 intermediate revisions not shown)
Line 1: Line 1:
 +
<[#ontology [kind topic] [cats 3D Simplex Demihypercube] [alt [[freebase:07jht]] [[wikipedia:Tetrahedron]]]]>
{{STS Shape
{{STS Shape
| name=Tetrahedron
| name=Tetrahedron
-
| image=http://img123.imageshack.us/img123/704/tetrahedron5jf.png
+
| image=<[#embed [hash 5JT5JWF1ADKFT7YS0W23RK4MZY] [width 150]]>
| dim=3
| dim=3
-
| elements=4, 6, 4
+
| 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
| genus=0
| ssc=xPP or {G3<sup>3</sup>}
| ssc=xPP or {G3<sup>3</sup>}
Line 17: Line 19:
| index=11
| index=11
}}{{STS Polytope
}}{{STS Polytope
-
| flayout={{FLD|a3|end|e3}}
+
| flayout={{FLD|a3|er|e3}}
| petrie=4,0
| petrie=4,0
| dual=''Self-dual''
| dual=''Self-dual''
 +
| bowers=Tet
}}{{STS Uniform polytope
}}{{STS Uniform polytope
| 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}
 +
| 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>
-
| bowers=Tet
 
-
| kana=ﾃ
 
}}}}
}}}}
 +
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 '''tetrahedron''' is a special case of a [[pyramid]] where the base is a [[triangle]].
+
== 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 ==
-
*Variables:
+
*The [[hypervolume]]s of a tetrahedron with side length ''l'' are given by:
-
<blockquote>''l'' ⇒ length of edges of tetrahedron</blockquote>
+
-
 
+
-
*All points (''x'', ''y'', ''z'') that lie on the surface of a tetrahedron will satisfy the following equations:
+
-
<blockquote>''Unknown''</blockquote>
+
-
 
+
-
*All points (''x'', ''y'', ''z'') that lie on the edges of a tetrahedron will satisfy the following equations:
+
-
<blockquote>''Unknown''</blockquote>
+
-
 
+
-
*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 = sqrt(3)''l''<sup>2</sup><br>
+
surface area = √3 &middot; ''l''<sup>2</sup><br>
-
volume = 12<sup>-1</sup>sqrt(2)''l''<sup>3</sup></blockquote>
+
volume = <sup>√2</sup>∕<sub>12</sub> &middot; ''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'' = 3<sup>-1</sup>sqrt(6)''l''</blockquote>
+
<blockquote>''h'' = <sup>√6</sup>∕<sub>3</sub> &middot; ''l''</blockquote>
-
*The [[planar]] [[cross-section]]s (''n'') of a tetrahedron are:
+
<[#polytope [id 1]]>
-
<blockquote>''Unknown''</blockquote>
+
-
== Use ==
+
{{Simplices|3}}
-
Tetrahedral cells are found in these tetrashapes on FGwiki:
+
{{Demihypercubes|3}}
-
*[[Hexacosichoron]] (600×, 100%)
+
-
*[[Hexadecachoron]] (8×, 100%)
+
-
*[[Pentachoron]] (5×, 100%)
+
-
*[[Square dipyramid]] (4×, 67%)
+
-
*[[Pentachoric hemicate]] (5×, 50%)
+
-
 
+
-
{{Simplices}}
+
{{Trishapes}}
{{Trishapes}}
{{Tapertope Nav|10|11|12|11<sup>1</sup><br>Triangular prism|1<sup>2</sup><br>Tetrahedron|4<br>Glome|hedra}}
{{Tapertope Nav|10|11|12|11<sup>1</sup><br>Triangular prism|1<sup>2</sup><br>Tetrahedron|4<br>Glome|hedra}}
-
 
-
[[Category:Regular polyhedra]]
 

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

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

Usage as facets


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
Retrieved from "http://hi.gher.space/wiki/Tetrahedron"