A tetrahedron is a 3-dimensional simplex. Its Bowers acronym is "tet". Under the elemental naming scheme it is called a pyrohedron.
Variants[]
The tetrahedron can be seen as a triangle pyramid where all the sides are equal. It is also a line antiprism.
Properties[]
Two tetrahedra can be inscribed in a cube such that no vertex is used twice and the edges are all diagonals of the cube's faces.
The dual of a tetrahedron is another tetrahedron. The dual of a shape is what you get when you turn the faces into vertices, and connect vertices.
The tetrahedron is part of the simplex family of polytopes.
Symbols[]
Dynkin symbols for the tetrahedron include:
- x3o3o (regular)
- s4o3o (alternated cube)
- s2s4o (tetragonal disphenoid)
- s2s2s (rhombic disphenoid)
- ox3oo&#x (triangular pyramid)
- xo ox&#x (digonal disphenoid)
- oxo&#x (scalene tet)
- oooo&#x (completely irregular)
Structure and Sections[]
Structure[]
As a triangular pyramid, the tetrahedron is composed of a point that grows into a triangle. It has 3 triangles around each vertex.
Sections[]
When seen vertex first, it is a point that expands into a triangle. When seen edge first, it looks like a line segment expanding into a rectangle, eventually reaching a square as midsection, then reversing in a perpendicular direction, until it becomes a line segment perpendicular to the top.
Hypervolumes[]
Subfacets[]
- 4 points (0D)
- 6 line segments (1D)
- 4 triangles (2D)
- 1 tetrahedron (3D)
Radii[]
- Vertex radius:
- Edge radius:
- Face radius:
Angles[]
- Dihedral angle:
Vertex coordinates[]
The simplest vertex coordinates for a regular tetrahedron can be obtained from alternating a cube. The tetrahedron with side
can be given these coordinates:
- (1,1,1)
- (-1,-1,1)
- (-1,1,-1)
- (1,-1,-1)
Alternatively, a tetrahedron can be given as a triangular pyramid with one face parallel to the xy coordinate plane with the coordinates
- (±1,-√3/3,-√6/6)
- (0,2√3/3,-√6/6)
- (0,0,√6/2)
Notations[]
- Tapertopic notation:
Related shapes[]
- Dual: Self-dual
- Vertex figure: Triangle, side length 1
See also[]
- Polytope Wiki. "Tetrahedron".
- Bowers, Jonathan. "Polyhedron Category 1: Regulars".
Regular polyhedra |
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Convex regular polyhedra: tet · cube · oct · doe · ike
Self-intersecting regular polyhedra: gad · sissid · gike · gissid |
... | |||||||||
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Trigonal hosohedron | Tetrahedron | Cube | Dodecahedron | Hexagonal tiling | Order-3 heptagonal tiling | Order-3 octagonal tiling | ... | Order-3 apeirogonal tiling | Order-3 pseudogonal tiling |
... | |||||||||
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Trigonal dihedron | Tetrahedron | Octahedron | Icosahedron | Triangular tiling | Order-7 triangular tiling | Order-8 triangular tiling | ... | Infinite-order triangular tiling | Imaginary-order triangular tiling |
Regular |
Rectified |
Birectified |
Truncated |
Bitruncated |
Cantellated |
Cantitruncated |
---|---|---|---|---|---|---|
Tetrahedron | Octahedron | Tetrahedron | Truncated tetrahedron | Truncated tetrahedron | Cuboctahedron | Truncated octahedron |
Tetrahedron | Square pyramid | Pentagonal pyramid | Hexagonal pyramid |