Dodecahedron
A dodecahedron is any
polyhedron with twelve faces, but usually a regular dodecahedron is meant: a
Platonic solid composed of twelve regular
pentagonal faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its
dual polyhedron is the
icosahedron.
Encyclopedia
A
dodecahedron is any
polyhedron with twelve faces, but usually a
regular dodecahedron is meant: a
Platonic solid composed of twelve regular
pentagonal faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its
dual polyhedron is the
icosahedron.
Area and volume
The area
A and the volume
V of a regular dodecahedron of edge length
a are:
Cartesian coordinates
The following
Cartesian coordinates define the vertices of a dodecahedron centered at the origin:
where φ = /2 is the
golden ratio . The side length is 2/φ = -1 + √5.
The
dihedral angle of a dodecahedron is 2arctan or approximately 116.565 degrees.
Geometric relations
The
regular dodecahedron is the third in an infinite set of
truncated trapezohedra which can be constructed by truncating the two axial vertices of a
pentagonal trapezohedron.
Five
cubes can be made from these, with their edges as diagonals of the dodecahedron's faces, and together these make up the regular
polyhedral compound of five cubes. Since two tetrahedra can fit on alternate cube vertices, five and ten tetrahedra can also fit in a dodecahedron.
The
stellations of the dodecahedron make up three of the four
Kepler-Poinsot solids.
Icosahedron vs dodecahedron
Despite appearances, when a dodecahedron is inscribed in a
sphere, it occupies more of the sphere's volume than an
icosahedron inscribed in the same sphere .
A regular dodecahedron with edges length 1 has more than three and a half times the volume of an
icosahedron with the same length edges .
Other dodecahedra
The term dodecahedron is also used for other
polyhedra with twelve faces, most notably the
rhombic dodecahedron which is dual to the
cuboctahedron and occurs in nature as a crystal form. The
Platonic solid dodecahedron can be called a
pentagonal dodecahedron or a
regular dodecahedron to distinguish it.
Other dodecahedra include:
- Uniform polyhedra:
- Pentagonal antiprism - 10 equilateral triangles, 2 pentagons
- Decagonal prism - 10 squares, 2 decagons
- Johnson solids :
- Pentagonal cupola - 5 triangles, 5 squares, 1 pentagon, 1 decagon
- Snub disphenoid - 12 triangles
- Elongated square dipyramid - 8 triangles and 4 squares
- Metabidiminished icosahedron
...
- 10 triangles and 2 pentagons
...
al
pyramid - 11 isosceles triangles and 1
hendecagon- Trapezo-rhombic dodecahedron - 6 rhombi, 6 trapezoids - dual of Triangular orthobicupola
- Rhombo-hexagonal dodecahedron or Elongated Dodecahedron - 8 rhombi and 4 equilateral hexagons.
Uses
- If each edge of a dodecahedron is a one-ohm resistor
|
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...
, the resistance between adjacent vertices is 19/30 ohm, and that between opposite vertices is 7/6 ohm.
- The regular dodecahedron is often used in role-playing games as a twelve-sided die , one of the more common polyhedral dice.
- One of the main characters in the movie and book of The Phantom Tollbooth Is a talking dodecahedron with arms and legs and has 12 different "faces".
Regular dodecahedra in the arts and sciences
- A dodecahedron sits on the table in M. C. Escher's lithograph print "Reptiles" , and a stellated dodecahedron is used in his "Gravitation".
- In Salvador Dalí's painting of the Last Supper , the room is a hollow dodecahedron.
- The 20 vertices and 30 edges of a dodecahedron form the map for an early computer game, Hunt the Wumpus.
- The Dodecahedron was the mysterious power source for an underground city in the Doctor Who episode Meglos .
- "Dodecahedron" is the title of a song by Aphex Twin.
- In 2003, an apparent periodicity in the cosmic microwave background led to the suggestion, by of the Observatoire de Paris and colleagues, that the shape of the Universe is a finite dodecahedron, attached to itself by each pair of opposite faces to form a Poincaré sphere. During the following year, astronomers searched for more evidence to support this hypothesis but found none.
See also
[i]
...
: a
regular polychoron whose surface consists of 120 dodecahedral cells.
External links
- Math Is Fun
- Many links
- Create, print and fold a 12 sided calendar as either a dodecahedron or a rhombic dodecahedron
- The Encyclopedia of Polyhedra
- VRML models
- regular
- quasiregular
- vertex-uniform
- vertex-uniform
- face uniform
- face-uniform
- face-uniform
- regular faces
