
The regular tetrahedron, often simply called ``the'' tetrahedron, is the Platonic Solid with four Vertices, six Edges, and four equivalent Equilateral Triangular faces . It is also Uniform Polyhedron . It is described by the Schläfli Symbol and the Wythoff Symbol is . It is the prototype of the Tetrahedral Group ,
The tetrahedron is its own Dual Polyhedron. It is the only simple Polyhedron with no
Diagonals, and cannot be Stellated. The
Vertices of a tetrahedron are given by
,
,
, and
, or by (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0). In the latter case, the face planes are
(1)  
(2)  
(3)  
(4) 
Let a tetrahedron be length on a side. The Vertices are located at (, 0, 0),
(, , 0), and (0, 0, ). From the figure,
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
By slicing a tetrahedron as shown above, a Square can be obtained. This cut divides the tetrahedron into two congruent solids rotated by 90°.
Now consider a general (not necessarily regular) tetrahedron, defined as a convex Polyhedron consisting of four (not
necessarily identical) Triangular faces. Let the tetrahedron be specified by its
Vertices at where , ..., 4. Then the Volume is given by
(18) 
(19) 
(20) 
(21)  
(22)  
(23) 
(24) 
The analog of Gauss's Circle Problem can be asked for tetrahedra: how many Lattice Points lie within a tetrahedron centered at the Origin with a given Inradius (Lehmer 1940, Granville 1991, Xu and Yau 1992, Guy 1994).
See also Augmented Truncated Tetrahedron, Bang's Theorem, Ehrhart Polynomial, Heronian Tetrahedron, Hilbert's 3rd Problem, Isosceles Tetrahedron, Sierpinski Tetrahedron, Stella Octangula, Tetrahedron 5Compound, Tetrahedron 10Compound, Truncated Tetrahedron
References
Davie, T. ``The Tetrahedron.'' http://www.dcs.stand.ac.uk/~ad/mathrecs/polyhedra/tetrahedron.html.
Granville, A. ``The Lattice Points of an Dimensional Tetrahedron.'' Aequationes Math. 41, 234241, 1991.
Guy, R. K. ``Gauß's Lattice Point Problem.'' §F1 in Unsolved Problems in Number Theory, 2nd ed. New York: SpringerVerlag, pp. 240241, 1994.
Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory, rev. ed. Washington, DC: Math. Assoc. Amer., 1991.
Lehmer, D. H. ``The Lattice Points of an Dimensional Tetrahedron.'' Duke Math. J. 7, 341353, 1940.
Xu, Y. and Yau, S. ``A Sharp Estimate of the Number of Integral Points in a Tetrahedron.'' J. reine angew. Math. 423, 199219, 1992.
© 19969 Eric W. Weisstein