# 四面体分析

2019/08/16 12:20

The regular tetrahedron, often simply called "the" tetrahedron, is the Platonic solid  with four polyhedron vertices, six polyhedron edges, and four equivalent equilateral triangular faces, . It is also uniform polyhedron  and Wenninger model . It is described by the Schläfli symbol  and the Wythoff symbol is . It is an isohedron, and a special case of the general tetrahedron and the isosceles tetrahedron.

The regular tetrahedron is implemented in the Wolfram Language as Tetrahedron[], and precomputed properties are available as PolyhedronData["Tetrahedron"].

The tetrahedron has 7 axes of symmetry:  (axes connecting vertices with the centers of the opposite faces) and  (the axes connecting the midpoints of opposite sides).

There are no other convex polyhedra other than the tetrahedron having four faces.

The tetrahedron has two distinct nets (Buekenhout and Parker 1998). Questions of polyhedron coloring of the tetrahedron can be addressed using the Pólya enumeration theorem.

The surface area of the tetrahedron is simply four times the area of a single equilateral triangle face

 (1)

so

 (2)

The height of the regular tetrahedron is

 (3)

 (4) (5)

where  as it must.

Since a tetrahedron is a pyramid with a triangular base, , giving

 (6)

The dihedral angle is

 (7)

The solid angle  subtended from a vertex by the opposite face of a regular tetrahedron is given by

 (8) (9)

The midradius of the tetrahedron is

 (10) (11)

Plugging in for the polyhedron vertices gives

 (12)

The dual polyhedron of an tetrahedron with unit edge lengths is another oppositely oriented tetrahedron with unit edge lengths.

The figure above shows an origami tetrahedron constructed from a single sheet of paper (Kasahara and Takahama 1987, pp. 56-57).

It is the prototype of the tetrahedral group . The connectivity of the vertices is given by the tetrahedral graph, equivalent to the circulant graph  and the complete graph .

The tetrahedron is its own dual polyhedron, and therefore the centers of the faces of a tetrahedron form another tetrahedron (Steinhaus 1999, p. 201). The tetrahedron is the only simple polyhedron with no polyhedron diagonals, and it cannot be stellated. If a regular tetrahedron is cut by six planes, each passing through an edge and bisecting the opposite edge, it is sliced into 24 pieces (Gardner 1984, pp. 190 and 192; Langman 1951).

Alexander Graham Bell was a proponent of use of the tetrahedron in framework structures, including kites (Bell 1903; Lesage 1956, Gardner 1984, pp. 184-185). The opposite edges of a regular tetrahedron are perpendicular, and so can form a universal coupling if hinged appropriately. Eight regular tetrahedra can be placed in a ring which rotates freely, and the number can be reduced to six for squashed irregular tetrahedra (Wells 1975, 1991).

Let a tetrahedron be length  on a side, and let its base lie in the plane  with one vertex lying along the positive -axis. The polyhedron vertices of this tetrahedron are then located at (, 0, 0), (, 0), and (0, 0, ), where

 (13)

is then

 (14)

This gives the area of the base as

 (15)

The height is

 (16)

 (17)
 (18)

Solving gives

 (19) (20) (21)

 (22) (23)

which is also

 (24) (25)

The angle between the bottom plane and center is then given by

 (26) (27) (28) (29)

Given a tetrahedron of edge length  situated with vertical apex and with the origin of coordinate system at the geometric centroid of the vertices, the four polyhedron vertices are located at , with, as shown above

 (30) (31) (32) (33)

The vertices of a tetrahedron of side length  can also be given by a particularly simple form when the vertices are taken as corners of a cube (Gardner 1984, pp. 192-194). One such tetrahedron for a cube of side length 1 gives the tetrahedron of side length having vertices (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0), and satisfies the inequalities

 (34) (35) (36) (37)

The following table gives polyhedra which can be constructed by augmentation of a tetrahedron by pyramids of given heights .

 result triakis tetrahedron 2 cube 3 9-faced star deltahedron

Connecting opposite pairs of edges with equally spaced lines gives a configuration like that shown above which divides the tetrahedron into eight regions: four open and four closed (Steinhaus 1999, p. 246).

Michigan artist David Barr designed his "Four Corners Project" in 1976. It is an Earth-sized regular tetrahedron that spans the planet, with just the tips of its four corners protruding. These visible portions are four-inch tetrahedra, which protrude from the globe at Easter Island, Greenland, New Guinea, and the Kalahari Desert. Barr traveled to these locations and was able to permanently install the four aligned marble tetrahedra between 1981 and 1985 (G. Hart, pers. comm.; Arlinghaus and Nystuen 1986).

By slicing a tetrahedron as shown above, a square can be obtained. This cut divides the tetrahedron into two congruent solids rotated by . The projection of a tetrahedron can be an equilateral triangle or a square (Steinhaus 1999, pp. 191-192).

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