Movatterモバイル変換

[0]ホーム

Jump to content

Dodecahedron

Edit links

From Wikipedia, the free encyclopedia

Polyhedron with 12 faces

This article is about the three-dimensional shape and is not to be confused withRoman dodecahedron.

Common dodecahedra
I_h, order 120
Regular	Small stellated	Great	Great stellated

T_h, order 24	T, order 12	O_h, order 48	Johnson (J₈₄)
Pyritohedron	Tetartoid	Rhombic	Triangular

D_4h, order 16		D_3h, order 12
Rhombo-hexagonal	Rhombo-square	Trapezo-rhombic	Rhombo-triangular

Ingeometry, adodecahedron (from Ancient Greek δωδεκάεδρον (dōdekáedron); from δώδεκα (dṓdeka) 'twelve' and ἕδρα (hédra) 'base, seat, face')^[1] orduodecahedron^[2] is anypolyhedron with twelve flat faces. The most familiar dodecahedron is theregular dodecahedron with regular pentagons as faces, which is aPlatonic solid. There are also threeregular star dodecahedra, which are constructed asstellations of the convex form. All of these haveicosahedral symmetry, order 120.

Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular:Thepyritohedron, a common crystal form inpyrite, haspyritohedral symmetry, while thetetartoid hastetrahedral symmetry.

Therhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it hasoctahedral symmetry. Theelongated dodecahedron andtrapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, arespace-filling. There are numerousother dodecahedra.

While the regular dodecahedron shares many features with other Platonic solids, one unique property of it is that one can start at a corner of the surface and draw an infinite number of straight lines across the figure that return to the original point without crossing over any other corner.^[3]

Regular dodecahedron

[edit]

Main article:Regular dodecahedron

Theregular dodecahedron is a convex polyhedron with regular pentagonal faces, three meeting at each vertex. It has 12 faces, 30 edges, and 20 vertices.^[4] It is one of the five regularPlatonic solids, named afterPlato who described them and considered the other four to symbolize theclassical elements; he assigned the regular dodecahedron to the cosmos.^[5] Itsdual is theregular icosahedron.^[6]

Left to right: Regular dodecahedron, small stellated dodecahedron, great dodecahedron, great stellated dodecahedron

The regular dodecahedron has threestellations, all of which are regular star dodecahedra. They form three of the fourKepler–Poinsot polyhedra. They are thesmall stellated dodecahedron, thegreat dodecahedron, and thegreat stellated dodecahedron.^[7] The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to thegreat icosahedron. All of these regular star dodecahedra have regular pentagonal orpentagrammic faces. The convex regular dodecahedron and great stellated dodecahedron are different realisations of the sameabstract regular polyhedron; the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron.

While the three star forms satisfy the requirements for regularity and have twelve faces, and therefore any of them could also be referred to as a "regular dodecahedron", this specific term is exclusively reserved for the first, convex form. A fifth shape which could be classified as a regular dodecahedron, namely thedodecagonal hosohedron, is also not referred to as such, in part because it exists only as aspherical polyhedron and is degenerate in Euclidean space.

Other pentagonal dodecahedra

[edit]

Incrystallography, two important dodecahedra can occur as crystal forms in somesymmetry classes of thecubic crystal system that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron withpyritohedral symmetry, and thetetartoid withtetrahedral symmetry:

Pyritohedron

[edit]

Apyritohedron (or pentagonal dodecahedron) is a dodecahedron withpyritohedral symmetry T_h. Like theregular dodecahedron, it has twelve identicalpentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis. Its 30 edges are divided into two sets, containing 24 and 6 edges of the same length. The only axes ofrotational symmetry are three mutually perpendicular twofold axes and four threefold axes.^[8]

Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of themineral pyrite,^[8] and it may be an inspiration for the discovery of the regularPlatonic solid form.^[9] The true regular dodecahedron can occur as a shape forquasicrystals (such asholmium–magnesium–zinc quasicrystal) withicosahedral symmetry, which includes true fivefold rotation axes.^[10]

Crystal pyrite

[edit]

The namecrystal pyrite comes from one of the two commoncrystal habits shown bypyrite (the other one being thecube). In pyritohedral pyrite, the faces have aMiller index of (210), which means that thedihedral angle is 2·arctan(2) ≈ 126.87° and each pentagonal face has one angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show the measurements for the face of a perfect crystal (which is rarely found in nature).

${\text{Height}}={\frac {\sqrt {5}}{2}}\cdot {\text{Long side}}$

${\text{Width}}={\frac {4}{3}}\cdot {\text{Long side}}$

${\text{Short sides}}={\sqrt {\frac {7}{12}}}\cdot {\text{Long side}}$

Natural pyrite (with face angles on the right)

Cartesian coordinates

[edit]

The eight vertices of a cube have the coordinates (±1, ±1, ±1).

The coordinates of the 12 additional vertices are

(0, ±(1 +h), ±(1 −h²)),
(±(1 +h), ±(1 −h²), 0) and
(±(1 −h²), 0, ±(1 +h)).

h is the height of thewedge-shaped "roof" above the faces of that cube with edge length 2.

An important case ish =⁠1/2⁠ (a quarter of the cube edge length) for perfect natural pyrite (also the pyritohedron in theWeaire–Phelan structure).

Another one ish =⁠1/φ⁠ = 0.618... for theregular dodecahedron. See sectionGeometric freedom for other cases.

Two pyritohedra with swapped nonzero coordinates are in dual positions to each other like the dodecahedra in thecompound of two dodecahedra.

Orthographic projections of the pyritohedron withh = 1/2

Heights 1/2 and 1/φ

The body of the dodecahedron can also be described as an intersection of twelve half-spaces,

\{(x,y,z)\in \mathbb {R} ^{3}\mid \|(x,y,z)\|_{D}\leq 1\}

where

\|(x,y,z)\|_{D}=\max {\Big \{}|\alpha x+\beta z|,\ |\alpha x-\beta z|,\ |\alpha y+\beta x|,\ |\alpha y-\beta x|,\ |\alpha z+\beta y|,\ |\alpha z-\beta y|{\Big \}}

and

$\alpha ={\frac {3-{\sqrt {5}}}{2}},\qquad \beta ={\frac {{\sqrt {5}}-1}{2}}={\frac {1}{\varphi }}.$

Animations

Honeycomb of alternating convex and concave pyritohedra with heights between ±⁠1/φ⁠	Heights between 0 (cube) and 1 (rhombic dodecahedron)

Geometric freedom

[edit]

The pyritohedron has a geometric degree of freedom withlimiting cases of a cubicconvex hull at one limit of collinear edges, and arhombic dodecahedron as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal.

It is possible to go past these limiting cases, creating concave or nonconvex pyritohedra. Theendo-dodecahedron is concave and equilateral; it can tessellate space with the convex regular dodecahedron. Continuing from there in that direction, we pass through a degenerate case where twelve vertices coincide in the centre, and on to the regulargreat stellated dodecahedron where all edges and angles are equal again, and the faces have been distorted into regularpentagrams. On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fish-shaped self-intersecting equilateral pentagonal faces.

Special cases of the pyritohedron
Versions with equal absolute values and opposing signs form a honeycomb together. (Comparethis animation.) The ratio shown is that of edge lengths, namely those in a set of 24 (touching cube vertices) to those in a set of 6 (corresponding to cube faces).
Ratio	1 : 1	0 : 1	1 : 1	2 : 1	1 : 1	0 : 1	1 : 1
h	−⁠√5 + 1/2⁠	−1	⁠−√5 + 1/2⁠	0	⁠√5 − 1/2⁠	1	⁠√5 + 1/2⁠
h	−1.618...	−1	−0.618...	0	0.618...	1	1.618...
Image	Regular star,great stellated dodecahedron, with regularpentagram faces	Degenerate, 12 vertices in the center	The concave equilateral dodecahedron, called anendo-dodecahedron.^{[clarification needed]}	Acube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions.	A regular dodecahedron is an intermediate case with equal edge lengths.	Arhombic dodecahedron is a degenerate case with the 6 crossedges reduced to length zero.	Self-intersecting equilateral dodecahedron

Tetartoid

[edit]

Atetartoid (alsotetragonal pentagonal dodecahedron,pentagon-tritetrahedron, andtetrahedric pentagon dodecahedron) is a dodecahedron with chiraltetrahedral symmetry (T). Like theregular dodecahedron, it has twelve identicalpentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes.

Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry.^[11] The mineralcobaltite can have this symmetry form.^[12]

Abstractions sharing the solid'stopology and symmetry can be created from the cube and the tetrahedron. In the cube each face is bisected by a slanted edge. In the tetrahedron each edge is trisected, and each of the new vertices connected to a face center. (InConway polyhedron notation this is a gyro tetrahedron.)

Orthographic projections from 2- and 3-fold axes

Cubic and tetrahedral form

Relationship to the dyakis dodecahedron

A tetartoid can be created by enlarging 12 of the 24 faces of adyakis dodecahedron.(The tetartoid shown here is based on one that is itself created by enlarging 24 of the 48 faces of thedisdyakis dodecahedron.)

Chiral tetartoids based on the dyakis dodecahedron in the middle

Thecrystal model on the right shows a tetartoid created by enlarging the blue faces of the dyakis dodecahedral core. Therefore, the edges between the blue faces are covered by the red skeleton edges.

Cartesian coordinates

[edit]

The following points are vertices of a tetartoid pentagon undertetrahedral symmetry:

(a,b,c); (−a, −b,c); (−⁠n/d₁⁠, −⁠n/d₁⁠,⁠n/d₁⁠); (−c, −a,b); (−⁠n/d₂⁠,⁠n/d₂⁠,⁠n/d₂⁠),

under the following conditions:^[13]

0 ≤a ≤b ≤c,

n =a²c −bc²,

d₁ =a² −ab +b² +ac − 2bc,

d₂ =a² +ab +b² −ac − 2bc,

nd₁d₂ ≠ 0.

Geometric freedom

[edit]

Theregular dodecahedron is a tetartoid with more than the required symmetry. Thetriakis tetrahedron is a degenerate case with 12 zero-length edges. (In terms of the colors used above this means, that the white vertices and green edges are absorbed by the green vertices.)

Tetartoid variations fromregular dodecahedron totriakis tetrahedron

Rhombic dodecahedron

[edit]

Main article:Rhombic dodecahedron

Therhombic dodecahedron is aCatalan solid with twelve rhombic faces and octahedral symmetry. It is dual to thecuboctahedron, anArchimedean solid,^[14] and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space with centrally symmetric faces, azonohedron.^[15]

Therhombic dodecahedron can be seen as a degeneratepyritohedron where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces.

The rhombic dodecahedron has severalstellations, thefirst of which is also aparallelohedral spacefiller.

Another important rhombic dodecahedron, theBilinski dodecahedron, has twelve faces congruent to those of therhombic triacontahedron, i.e. the diagonals are in the ratio of thegolden ratio. It is also azonohedron and was described byBilinski in 1960.^[16] This figure is another spacefiller, and can also occur in non-periodicspacefillings along with the rhombic triacontahedron, the rhombic icosahedron, and rhombic hexahedra.^[17]

Other dodecahedra

[edit]

There are 6,384,634 topologically distinctconvex dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20.^[18] Two polyhedra aretopologically distinct if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.

Topologically, notably distinct dodecahedra (excluding pentagonal and rhombic forms) include:

Decagonal prism, aprism consisting of ten squares and two decagonal bases. Its symmetry group isD_10h symmetry of order 40.^[19]
Pentagonal antiprism: anantiprism consisting of ten equilateral triangles and two pentagonal bases. It isD_5d symmetry of order 20.^[19]
Pentagonal cupola: aJohnson solid with five triangles, five squares, one pentagon, and one decagon. It isC_5v symmetry of order 10.^[20]^[19]
Snub disphenoid: both Johnson solid anddeltahedron, consisting of twelve equilateral triangles. It isD_2d of order 8, the same symmetries as atetragonal disphenoid.^[19]^[21] Its dual iselongated gyrobifastigium, anoctahedron.
Elongated square dipyramid, a Johnson solid. Obtained by augmenting two opposite faces of acube byequilateral square pyramids, the resulting polyhedron has eight triangles and four squares. It isD_4h symmetry of order 16.^[20]^[19]
Metabidiminished icosahedron: a Johnson solid obtained by removing twopentagonal pyramids from aregular icosahedron, resulting in ten triangles and two pentagons. It isC_2v symmetry of order 4.^[20]^[19]
Hexagonal bipyramid: abipyramid with twelve isoscelestriangles, obtained by attaching twohexagonal pyramids base-to-base.^[22] If both affixed pyramids haveregular bases and apices perpendicular to the center of their base, the bipyramid hasD_6h symmetry of order 24.^[23] Like any other bipyramids, the hexagonal bipyramid isface-transitive and the dual ofhexagonal prism.^[24]^[25]
Hexagonal trapezohedron: atrapezohedron with twelvekites. It is face-transitive, the dual ofhexagonal antiprism, and hasD_6d symmetry of order 24.
Triakis tetrahedron: aCatalan solid and theKleetope of aregular tetrahedron. Obtained by affixing four triangular pyramids onto each face of the regular tetrahedron, the resulting polyhedron has twelve isosceles triangles.^[14]^[26] It is face-transitive andRupert property.^[27]^[28] It is the dual oftruncated tetrahedron, shares the same symmetry asT_d symmetry of order 24.^[14]
Hendecagonal pyramid: the twelve isosceles triangles and one regularhendecagon faces in a pyramid. It hasC_11v symmetry of order 11.^[29]
Trapezo-rhombic dodecahedron: a polyhedron with six rhombi, sixtrapezoids. It is the dual oftriangular orthobicupola, both haveD_3h symmetry of order 12. It is known fortessellating by translating a copy of itself.
Rhombo-hexagonal dodecahedron (also known as "elongated dodecahedron"): a polyhedron with eight rhombi and four equilateralhexagons. it hasD_4h symmetry of order 16. It is one of fiveFederov polyhedra or parallelohedra, generated to create itshoneycomb.^[30]

Practical usage

[edit]

Armand Spitz used a dodecahedron as the "globe" equivalent for hisDigital Dome planetarium projector,^[31] based upon a suggestion fromAlbert Einstein.

References

[edit]

^Chambers's Twentieth Century Dictionary of the English Language. W. & R. Chambers. 1908. p. 274.
^1913 Webster's Revised Unabridged Dictionary
^Athreya, Jayadev S.; Aulicino, David; Hooper, W. Patrick (2020). "Platonic Solids and High Genus Covers of Lattice Surfaces".Experimental Mathematics.31 (3):847–877.arXiv:1811.04131.doi:10.1080/10586458.2020.1712564.S2CID 119318080.
^Sutton, Daud (2002).Platonic & Archimedean Solids. Wooden Books. Bloomsbury Publishing USA. p. 55.ISBN 9780802713865.
^Cromwell, Peter R. (1997).Polyhedra. Cambridge University Press. p. 53.ISBN 978-0-521-55432-9.
^Erickson, Martin (2011).Beautiful Mathematics.Mathematical Association of America. p. 62.ISBN 978-1-61444-509-8.
^Cromwell (1997), p. 265–266.
^^a ^bHurlbut, Cornelius S.; Sharp, W. Edwin (1998).Dana's Minerals and How to Study Them (4th ed.). John Wiley & Sons. p. 26.ISBN 978-0-471-15677-2.
^Stephenson, Paul (1993). "Plato's Fourth Solid and the "Pyritohedron"".The Mathematical Gazette.77 (479):220–226.doi:10.2307/3619718.JSTOR 3619718.
^Canfield, Paul C.; Fisher, Ian R. (2001). "High-temperature solution growth of intermetallic single crystals and quasicrystals".Journal of Crystal Growth.225 (2–4). Elsevier BV:155–161.Bibcode:2001JCrGr.225..155C.doi:10.1016/s0022-0248(01)00827-2.ISSN 0022-0248.
^Dutch, Steve.The 48 Special Crystal Forms Archived 2013-09-18 at theWayback Machine. Natural and Applied Sciences,University of Wisconsin-Green Bay, U.S.
^Crystal Habit. Galleries.com. Retrieved on 2016-12-02.
^The Tetartoid. Demonstrations.wolfram.com. Retrieved on 2016-12-02.
^^a ^b ^cWilliams, Robert (1979).The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 72.ISBN 978-0-486-23729-9.
^Eppstein, David (1996)."Zonohedra and zonotopes".Mathematica in Education and Research.5 (4):15–21.
^Hafner, I. and Zitko, T.Introduction to golden rhombic polyhedra. Faculty of Electrical Engineering,University of Ljubljana, Slovenia.
^Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2000)."Tilings, coverings, clusters and quasicrystals".Current Science.78:64–72.
^Counting polyhedra. Numericana.com (2001-12-31). Retrieved on 2016-12-02.
^^a ^b ^c ^d ^e ^fJohnson, Norman W. (1966)."Convex polyhedra with regular faces".Canadian Journal of Mathematics.18:169–200.doi:10.4153/cjm-1966-021-8.MR 0185507.S2CID 122006114.Zbl 0132.14603.
^^a ^b ^cBerman, Martin (1971). "Regular-faced convex polyhedra".Journal of the Franklin Institute.291 (5):329–352.doi:10.1016/0016-0032(71)90071-8.MR 0290245.
^Cundy, H. Martyn (1952). "Deltahedra".The Mathematical Gazette.36 (318):263–266.doi:10.2307/3608204.JSTOR 3608204.MR 0051525.S2CID 250435684.
^Aarts, J. M. (2008).Plane and Solid Geometry. Springer. p. 303.doi:10.1007/978-0-387-78241-6.ISBN 978-0-387-78241-6.
^Flusser, Jan; Suk, Tomas; Zitofa, Barbara (2017).2D and 3D Image Analysis by Moments. John & Sons Wiley. p. 126.ISBN 978-1-119-03935-8.
^Chang, Ch.; Patzer, A. B. C.; Sülzle, D.; Hauer, H."Onion-Like Inorganic Fullerenes from a Polyhedral Perspective". In Sattler, Klaus D. (ed.).21st Century Nanoscience: A Handbook. Taylor & Francis. p. 15-4.
^Sibley, Thomas Q. (2015).Thinking Geometrically: A Survey of Geometries. Mathematical Association of America. p. 53.ISBN 978-1-939512-08-6.
^Brigaglia, Aldo; Palladino, Nicla; Vaccaro, Maria Alessandra (2018). "Historical notes on star geometry in mathematics, art and nature". In Emmer, Michele; Abate, Marco (eds.).Imagine Math 6: Between Culture and Mathematics. Springer International Publishing. pp. 197–211.doi:10.1007/978-3-319-93949-0_17.hdl:10447/325250.ISBN 978-3-319-93948-3.
^Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazon (2010). "Catalan Solids Derived From 3D-Root Systems and Quaternions".Journal of Mathematical Physics.51 (4).arXiv:0908.3272.doi:10.1063/1.3356985.
^Fredriksson, Albin (2024). "Optimizing for the Rupert property".The American Mathematical Monthly.131 (3):255–261.arXiv:2210.00601.doi:10.1080/00029890.2023.2285200.
^Alexandroff, Paul (2012).An Introduction to the Theory of Groups. Dover Publications. p. 48.ISBN 978-0-486-48813-4.
^Alexandrov, A. D. (2005). "8.1 Parallelohedra".Convex Polyhedra. Springer. pp. 349–359.
^Ley, Willy (February 1965)."Forerunners of the Planetarium". For Your Information.Galaxy Science Fiction. pp. 87–98.

External links

[edit]

Wikimedia Commons has media related toPolyhedra with 12 faces.

Plato's Fourth Solid and the "Pyritohedron", by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp. 220–226[1]
Stellation of Pyritohedron VRML models and animations of Pyritohedron and itsstellations
Klitzing, Richard."3D convex uniform polyhedra o3o5x – doe".
Editable printable net of a dodecahedron with interactive 3D view
The Uniform Polyhedra
Origami Polyhedra – Models made with Modular Origami
Virtual Reality Polyhedra The Encyclopedia of Polyhedra
K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
Dodecahedron 3D Visualization
Stella: Polyhedron Navigator: Software used to create some of the images on this page.
How to make a dodecahedron from a Styrofoam cube

v t e Polyhedra
Listed by number of faces and type
1–10 faces	Monohedron Dihedron Trihedron Tetrahedron Pentahedron Hexahedron Heptahedron Octahedron Enneahedron Decahedron
11–20 faces	Hendecahedron Dodecahedron Tridecahedron Tetradecahedron Pentadecahedron Hexadecahedron Heptadecahedron Octadecahedron Enneadecahedron Icosahedron
>20 faces	Icositetrahedron (24) Triacontahedron (30) Icosidodecahedron (32) Hexoctahedron (48) Hexecontahedron (60) Enneacontahedron (90) Hectotriadiohedron (132) Apeirohedron (∞)
elemental things	face edge vertex uniform polyhedron (two infinite groups and 75) regular polyhedron (9) quasiregular polyhedron (16) semiregular polyhedron (two infinite groups and 50)
convex polyhedron	Platonic solid (5) Archimedean solid (13) Catalan solid (13) Johnson solid (92)
non-convex polyhedron	Kepler–Poinsot polyhedron (4) Star polyhedron (infinite) Uniform star polyhedron (57)
prismatoid‌s	prism antiprism frustum cupola wedge pyramid parallelepiped

Convexpolyhedra

Platonic solids(regular)

Archimedean solids
(semiregular oruniform)

Catalan solids
(duals of Archimedean)

Dihedral regular

dihedron
hosohedron

Dihedral uniform

prisms antiprisms
duals:	bipyramids trapezohedra

Dihedral others

Degenerate polyhedra are initalics.

v t e Fundamental convexregular anduniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) /D_n	E₆ /E₇ /E₈ /F₄ /G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron •Cube	Demicube		Dodecahedron •Icosahedron
Uniform polychoron	Pentachoron	16-cell •Tesseract	Demitesseract	24-cell	120-cell •600-cell
Uniform 5-polytope	5-simplex	5-orthoplex •5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex •6-cube	6-demicube	1₂₂ •2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex •7-cube	7-demicube	1₃₂ •2₃₁ •3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex •8-cube	8-demicube	1₄₂ •2₄₁ •4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex •9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex •10-cube	10-demicube
Uniformn-polytope	n-simplex	n-orthoplex •n-cube	n-demicube	1_k2 •2_k1 •k₂₁	n-pentagonal polytope
Topics:Polytope families •Regular polytope •List of regular polytopes and compounds •Polytope operations

Authority control databases	GND

Retrieved from "https://en.wikipedia.org/w/index.php?title=Dodecahedron&oldid=1318444301"

Categories:

Hidden categories:

[8]ページ先頭