| 释义 |
TetrahedronThe regular tetrahedron, often simply called ``the'' tetrahedron, is the Platonic Solid  with fourVertices, six Edges, and four equivalentEquilateral Triangular faces  . It is also Uniform Polyhedron . It is described by the Schläfli Symbol  and the Wythoff Symbolis  . It is the prototype of the Tetrahedral Group  ,
The tetrahedron is its own Dual Polyhedron. It is the only simple Polyhedron with noDiagonals, and cannot be Stellated. TheVertices 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
Let a tetrahedron be length  on a side. The Vertices are located at ( , 0, 0),( ,  , 0), and (0, 0,  ). From the figure,
 | (5) |
 is then
This gives the Area of the base as
The height is
 | (8) |
 is found from
 | (9) |
 | (10) |
 | (11) |
 is
 | (12) |
 | (13) |
Plugging in for the Vertices gives
 | (15) |
 , and
 | (16) |
 | (17) |
By slicing a tetrahedron as shown above, a Square can be obtained. This cut divides the tetrahedron into twocongruent solids rotated by 90°.
Now consider a general (not necessarily regular) tetrahedron, defined as a convex Polyhedron consisting of four (notnecessarily identical) Triangular faces. Let the tetrahedron be specified by itsVertices at  where  , ..., 4. Then the Volume is given by
 | (18) |
 | (19) |
 , and  , then
 | (20) |
where  means here the Angle between the Planes formed by the Faces  and  , with Vertex along their intersecting Edge. Then
 | (24) |
The analog of Gauss's Circle Problem can be asked for tetrahedra: how many Lattice Pointslie within a tetrahedron centered at the Origin with a given Inradius (Lehmer 1940, Granville 1991, Xu andYau 1992, Guy 1994). See also Augmented Truncated Tetrahedron, Bang's Theorem, Ehrhart Polynomial, HeronianTetrahedron, Hilbert's 3rd Problem, Isosceles Tetrahedron,Sierpinski Tetrahedron, Stella Octangula, Tetrahedron5-Compound, Tetrahedron 10-Compound, Truncated Tetrahedron
ReferencesDavie, T. ``The Tetrahedron.'' http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/tetrahedron.html.Granville, A. ``The Lattice Points of an  -Dimensional Tetrahedron.'' Aequationes Math. 41, 234-241, 1991. Guy, R. K. ``Gauß's Lattice Point Problem.'' §F1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 240-241, 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, 341-353, 1940. Xu, Y. and Yau, S. ``A Sharp Estimate of the Number of Integral Points in a Tetrahedron.'' J. reine angew. Math. 423, 199-219, 1992.
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