“Dodecahedron”的概念、定义、习题解题方法及翻译-数学辞典

单词

Dodecahedron

释义

Dodecahedron

The regular dodecahedron is the Platonic Solid () composed of 20 Vertices, 30 Edges, and 12 Pentagonal Faces. It is given by the symbol , theSchläfli Symbol . It is also Uniform Polyhedron and has WythoffSymbol . The dodecahedron has the Icosahedral Group of symmetries.

A Plane Perpendicular to a axis of a dodecahedron cuts the solid in a regular HexagonalCross-Section (Holden 1991, p. 27). A Plane Perpendicular to a axis of a dodecahedron cuts thesolid in a regular Decagonal Cross-Section (Holden 1991, p. 24).

The Dual Polyhedron of the dodecahedron is the Icosahedron.

When the dodecahedron with edge length is oriented with two opposite faces parallel to the-Plane, the vertices of the top and bottom faces lie at and the other Vertices lie at , where is the Golden Ratio. The explicit coordinates are

(1)

(2)

with

, 1, ..., 4, where

is the Golden Ratio. Explicitly, these coordinates are

			(3)
			(4)
			(5)

			(6)
			(7)
			(8)
			(9)
			(10)

			(11)
			(12)

where

are the top vertices,

are the vertices above the mid-plane,

are thevertices below the mid-plane, and

are the bottom vertices. The Vertices of adodecahedron can be given in a simple form for a dodecahedron of side length

by (0,

), (

, 0,

), (

, 0), and (

For a dodecahedron of unit edge length , the Circumradius and Inradius of aPentagonal Face are

			(13)
			(14)

The Sagitta

is then given by

(15)

Now consider the following figure.

Using the Pythagorean Theorem on the figure then gives

			(16)
			(17)
			(18)

Equation (3) can be written

(19)

Solving (1), (2), and (19) simultaneously gives

			(20)
			(21)
			(22)

The Inradius of the dodecahedron is then given by

(23)


			(24)

and

(25)

Now,


			(26)

and the Circumradius is

(27)

The Interradius is given by


			(28)

(29)

The Area of a single Face is the Area of a Pentagon,

(30)

The Volume of the dodecahedron can be computed by summing the volume of the 12 constituentPentagonal Pyramids,



			(31)

Apollonius

showed that the Volume

and Surface Area

of the dodecahedron and itsDual the Icosahedron are related by

(32)

The Hexagonal Scalenohedron is an irregular dodecahedron.

See also Augmented Dodecahedron, Augmented Truncated Dodecahedron, Dodecagon,Dodecahedron-Icosahedron Compound, Elongated Dodecahedron,Great Dodecahedron, Great Stellated Dodecahedron, Hyperbolic Dodecahedron, Icosahedron,Metabiaugmented Dodecahedron, Metabiaugmented Truncated Dodecahedron, Parabiaugmented Dodecahedron,Parabiaugmented Truncated Dodecahedron, Pyritohedron, Rhombic Dodecahedron, Small StellatedDodecahedron, Triaugmented Dodecahedron, Triaugmented Truncated Dodecahedron, Trigonal Dodecahedron,Trigonometry Values Pi/5, Truncated Dodecahedron

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989.

Davie, T. ``The Dodecahedron.'' http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/dodecahedron.html.

Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。