Difference between revisions of "Tetrahedron"
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Regular tetrahedra, in which all edges have equal [[length]] and all faces are [[congruent]] [[equilateral triangle]]s, are one of the five types of [[Platonic solid]]s. | Regular tetrahedra, in which all edges have equal [[length]] and all faces are [[congruent]] [[equilateral triangle]]s, are one of the five types of [[Platonic solid]]s. | ||
| + | |||
| + | <asy> | ||
| + | import three; | ||
| + | currentprojection = orthographic(-1.2,-0.2,0.4); | ||
| + | triple[] P = {(0,0,(2/3)^.5),(3^(-0.5),0,0),(-1/2/3^.5,1/2,0),(-1/2/3^.5,-1/2,0)}; | ||
| + | void drawFrontFace(int x, int y, int z){ draw(P[x] -- P[y] -- P[z] -- cycle, linewidth(0.7)); | ||
| + | /* fill(P[x] -- P[y] -- P[z] -- cycle, rgb(0.7,0.7,0.7)); */ | ||
| + | } | ||
| + | void drawBackFace(int x, int y, int z){ draw(P[x] -- P[y] -- P[z] -- cycle, linetype("2 6")); | ||
| + | } | ||
| + | drawFrontFace(0,3,2);drawBackFace(0,1,3);drawBackFace(0,2,3);drawBackFace(1,2,3); | ||
| + | </asy> | ||
The [[polyhedral dual]] of a tetrahedron is another tetrahedron. | The [[polyhedral dual]] of a tetrahedron is another tetrahedron. | ||
Revision as of 22:08, 6 December 2016
The tetrahedron (plural tetrahedra) or triangular pyramid is the simplest polyhedron. Tetrahedra have four vertices, four triangular faces and six edges. Three faces and three edges meet at each vertex.
Any four points chosen in space will be the vertices of a tetrahedron as long as they do not all lie on a single plane.
Regular tetrahedra, in which all edges have equal length and all faces are congruent equilateral triangles, are one of the five types of Platonic solids.
![[asy] import three; currentprojection = orthographic(-1.2,-0.2,0.4); triple[] P = {(0,0,(2/3)^.5),(3^(-0.5),0,0),(-1/2/3^.5,1/2,0),(-1/2/3^.5,-1/2,0)}; void drawFrontFace(int x, int y, int z){ draw(P[x] -- P[y] -- P[z] -- cycle, linewidth(0.7)); /* fill(P[x] -- P[y] -- P[z] -- cycle, rgb(0.7,0.7,0.7)); */ } void drawBackFace(int x, int y, int z){ draw(P[x] -- P[y] -- P[z] -- cycle, linetype("2 6")); } drawFrontFace(0,3,2);drawBackFace(0,1,3);drawBackFace(0,2,3);drawBackFace(1,2,3); [/asy]](http://latex.artofproblemsolving.com/5/8/a/58a906ea42c6080e76d6bfbd11d822a0315523cd.png)
The polyhedral dual of a tetrahedron is another tetrahedron.
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