3664:
315:
463:
600:, which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be
556:
300:
1914:
958:
1935:
2056:
169:
1813:
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446:, the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid, Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra.
1957:
424:
523:, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.55%). The resulting of both spheres' volumes initially began from the problem by ancient Greeks, determining which of two shapes has a larger volume: an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere. The problem was solved by
1595:
29:
2153:, where the vertices of a decagon are connected to those of two pentagons, one pentagon connected to odd vertices of the decagon and the other pentagon connected to the even vertices. Geometrically, this can be visualized as the ten-vertex equatorial belt of the dodecahedron connected to the two 5-vertex polar regions, one on each side.
2042:'s 1954 short story "The Mathematician's Nightmare: The Vision of Professor Squarepunt", the number 5 said: "I am the number of fingers on a hand. I make pentagons and pentagrams. And but for me dodecahedra could not exist; and, as everyone knows, the universe is a dodecahedron. So, but for me, there could be no universe."
1281:
1403:
563:
As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not
1768:. It is a set of polyhedrons containing hexagonal and pentagonal faces. Other than two Platonic solids—tetrahedron and cube—the regular dodecahedron is the initial of Goldberg polyhedron construction, and the next polyhedron is resulted by truncating all of its edges, a process called
546:
may also related to both regular icosahedron and regular dodecahedron. The regular icosahedron can be constructed by intersecting three golden rectangles perpendicularly, arranged in two-by-two orthogonal, and connecting each of the golden rectangle's vertices with a segment line. There are 12
572:"Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "
418:
by using the
Platonic solids setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from the innermost to the outermost: regular octahedron, regular icosahedron, regular dodecahedron, regular tetrahedron, and cube.
627:
of a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column's elements that occur in or at the row's element. The regular dodecahedron can be represented in the following matrix:
1100:
355:, a dialogue of Plato, Plato hypothesized that the classical elements were made of the five uniform regular solids. Plato described the regular dodecahedron, obscurely remarked, "...the god used for arranging the constellations on the whole heaven".
953:
3620:
706:
516:. The regular dodecahedron has ten three-fold axes passing through pairs of opposite vertices, six five-fold axes passing through the opposite faces centers, and fifteen two-fold axes passing through the opposite sides midpoints.
1590:{\displaystyle {\begin{aligned}r_{u}&={\frac {\phi {\sqrt {3}}}{2}}a\approx 1.401a,\\r_{i}&={\frac {\phi ^{2}}{2{\sqrt {3-\phi }}}}a\approx 1.114a,\\r_{m}&={\frac {\phi ^{2}}{2}}a\approx 1.309a.\end{aligned}}}
2104:, meaning that, whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected. The skeleton of a regular dodecahedron can be represented as a graph, and it is called the
539:
discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio but are taken to different powers.
3267:; Jeff Weeks; Alain Riazuelo; Roland Lehoucq; Jean-Phillipe Uzan (2003-10-09). "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background".
772:
379:, adding that there is a fifth solid pattern which, though commonly associated with the regular dodecahedron, is never directly mentioned as such; "this God used in the delineation of the universe."
1408:
1105:
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514:
125:
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Roukema, Boudewijn; Zbigniew Buliński; Agnieszka
Szaniewska; Nicolas E. Gaudin (2008). "A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data".
1739:
3485:
1276:{\displaystyle {\begin{aligned}(\pm 1,\pm 1,\pm 1),&\qquad (0,\pm \phi ,\pm 1/\phi ),\\(\pm 1/\phi ,0,\pm \phi ),&\qquad (\pm \phi ,\pm 1/\phi ,0).\end{aligned}}}
1772:. This process can be continuously repeated, resulting in more new Goldberg's polyhedrons. These polyhedrons are classified as the first class of a Goldberg polyhedron.
2151:
2994:
1697:
1673:
1646:
1622:
1398:
1364:
1337:
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1303:
841:
821:
798:
1941:
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436:
gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.
3159:
Kai Wu; Jonathan
Nitschke (2023). "Systematic construction of progressively larger capsules from a fivefold linking pyrrole-based subcomponent".
2481:
258:, the children's story, toys, and painting arts. It can also be found in nature and supramolecules, as well as the shape of the universe. The
3518:
3034:
2617:
2464:
2399:
2324:
2031:, a positively curved space consisting of a regular dodecahedron whose opposite faces correspond (with a small twist). This was proposed by
216:. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity.
4086:
1896:, the regular dodecahedron appears as a character in the land of Mathematics. Each face of the regular dodecahedron describes the various
2434:
3707:
2456:
1867:
based his entire artistic oeuvre on the regular dodecahedron and the pentagon, presented as a new art movement coined as
Pentagonism.
774:. The golden ratio can be applied to the regular dodecahedron's metric properties, as well as to construct the regular dodecahedron.
3575:
3469:
3430:
3399:
3193:
Hagino, K., Onuma, R., Kawachi, M. and
Horiguchi, T. (2013) "Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-A in
2804:
2582:
2555:
2491:
2372:
2297:
89:
1859:
479:
3225:
726:
2685:
2661:
2633:
4005:
1783:. The first stellation of a regular dodecahedron is constructed by attaching its layer with pentagonal pyramids, forming a
333:
The regular dodecahedron is a polyhedron with twelve pentagonal faces, thirty edges, and twenty vertices. It is one of the
3422:
3391:
2509:
1780:
2089:
1796:
1784:
454:, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons".
263:
719:
is the ratio between two numbers equal to the ratio of their sum to the larger of the two quantities. It is one of two
4502:
4299:
4240:
3250:
597:
4329:
4289:
3895:
3453:
3118:
2693:
2116:
1849:
623:
in which the rows and columns correspond to the elements of a polyhedron as in the vertices, edges, and faces. The
593:
577:
433:
384:
314:
4324:
4319:
2645:
2227:
2220:
2157:
1985:
1924:
488:
99:
2973:, Coxeter's notation for the circumradius, midradius, and inradius, respectively, also noting that Coxeter uses
2936:
2896:
2856:
4512:
3889:
2665:
2101:
410:
228:
1097:
define the twenty vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented:
4430:
4425:
4304:
4210:
4017:
3945:
3865:
3700:
1843:
1795:. The third stellation is by attaching the great dodecahedron with the sharp triangular pyramids, forming a
1750:
478:. One property of the dual polyhedron generally is that the original polyhedron and its dual share the same
220:
44:
432:
Many antiquity philosophers described the regular dodecahedron, including the rest of the
Platonic solids.
4507:
4294:
4235:
4225:
4170:
3951:
2264:
2192:
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4185:
2254:
1892:
1754:
1094:
581:
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Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.
351:
342:
224:
2096:
of a polyhedron; roughly speaking, a framework of a polyhedron. Such a graph has two properties. It is
2085:
2027:
Various models have been proposed for the global geometry of the universe. These proposals include the
948:{\displaystyle A={\frac {15\phi }{\sqrt {3-\phi }}}a^{2},\qquad V={\frac {5\phi ^{3}}{6-2\phi }}a^{3}.}
462:
2100:, meaning the edges of a graph are connected to every vertex without crossing other edges. It is also
1887:
twisty puzzle is shaped like a regular dodecahedron alongside its larger and smaller order analogues.
4517:
4274:
4200:
3414:
3347:
3286:
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2236:
2016:
1709:
1310:
620:
528:
483:
441:
94:
3652:
2156:
The high degree of symmetry of the polygon is replicated in the properties of this graph, which are
555:
482:. In the case of the regular dodecahedron, it has the same symmetry as the regular icosahedron, the
4440:
4309:
4284:
4269:
4205:
4153:
3264:
2188:
2093:
2035:
and colleagues in 2003, and an optimal orientation on the sky for the model was estimated in 2008.
2032:
1996:, has a calcium carbonate shell with a regular dodecahedral structure about 10 micrometers across.
1765:
536:
475:
372:
360:
279:
259:
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232:
49:
584:. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated
4455:
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4279:
4174:
4123:
4062:
3939:
3933:
3693:
3660:
K.J.M. MacLean, A Geometric
Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
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1792:
1769:
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524:
368:
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255:
153:
3381:
1966:
dodecahedron reported by Kai Wu, Jonathan
Nitschke and co-workers at University of Cambridge in
1822:
Regular dodecahedra have been used as dice and probably also as divinatory devices. During the
1339:(one that touches the regular dodecahedron at all vertices), the radius of an inscribed sphere
299:
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588:.) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a
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79:
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1948:
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1830:
were made and have been found in various Roman ruins in Europe. Its purpose is not certain.
1788:
1761:. Here, the regular dodecahedron is constructed by truncating the pentagonal trapezohedron.
701:{\displaystyle {\begin{bmatrix}20&3&3\\2&30&2\\5&5&12\end{bmatrix}}}
589:
576:". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a
543:
404:
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160:
62:
2316:
Number, Shape, & Symmetry: An
Introduction to Number Theory, Geometry, and Group Theory
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3222:
2507:
Buker, W. E.; Eggleton, R. B. (1969). "The
Platonic Solids (Solution to problem E2053)".
547:
regular icosahedron vertices, considered as the center of 12 regular dodecahedron faces.
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3172:
2837:
2389:
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3833:
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3006:
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2109:
1879:, the regular dodecahedron is often used as a twelve-sided die, one of the more common
1703:
1288:
826:
806:
783:
616:
596:. Further, we can choose one tetrahedron from each stella octangula, so as to derive a
573:
334:
271:
135:
131:
39:
3659:
3642:
957:
359:, as a personage of Plato's dialogue, associates the other four Platonic solids—
345:
and the same number of faces meet at a vertex. This set of polyhedrons is named after
250:
The regular dodecahedron's metric properties and construction are associated with the
4496:
4400:
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2012:
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1934:
408:
sketched each of the Platonic solids, one of them is a regular dodecahedron. In his
235:
because it is the initial polyhedron to construct new polyhedrons by the process of
3803:
3367:
3314:
2796:
2743:
Peters, J. M. H. (1978). "An Approximate Relation between π and the Golden Ratio".
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186:
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visits all of its vertices exactly once. The name of this property is named after
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3210:
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2223:(4D polytope whose surface consists of a hundred and twenty dodecahedral cells)
2066:
2055:
383:
also postulated that the heavens were made of a fifth element, which he called
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2650:. Vol. 6. University of Toronto Studies (Mathematical Series). p. 4.
2609:
2003:
and cages have dodecahedral shape (see figure). Some regular crystals such as
1787:. The second stellation is by attaching the small stellated dodecahedron with
1776:
447:
193:
149:
3134:
3087:
2707:
Schielack, Vincent P. (1987). "The Fibonacci Sequence and the Golden Ratio".
4460:
4450:
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4379:
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4042:
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3543:
3449:
The Mathematics of Finite Networks: An Introduction to Operator Graph Theory
1371:
532:
380:
3306:
1956:
3461:
2733:
This source contains an elementary derivation of the golden ratio's value.
423:
247:. Other new polyhedrons can be constructed by using regular dodecahedron.
28:
4346:
3742:
3281:
2720:
2216:
1884:
1706:
of a regular dodecahedron between every two adjacent pentagonal faces is
451:
3298:
3109:
Hill, Christopher (1994). "Gallo-Roman Dodecahedra: A Progress Report".
2728:
4470:
4445:
4047:
3680:
3669:
2792:
The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
2764:
2530:
2075:
The dodecahedral graph's Hamiltonian property and the mathematical toy
2008:
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1367:
236:
282:
visits all of its vertices exactly once, can be found in a toy called
2004:
1306:
801:
602:
520:
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2522:
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3394:, vol. 221 (2nd ed.), Springer-Verlag, pp. 235–244,
2343:, Jowett translation ; the Greek word translated as delineation is
239:. It has a relation with other Platonic solids, one of them is the
3342:
2242:
1811:
961:
Cartesian coordinates of a regular dodecahedron in the following:
956:
554:
461:
421:
346:
254:. The regular dodecahedron can be found in many popular cultures:
205:
3621:
Editable printable net of a dodecahedron with interactive 3D view
4138:
3648:
1993:
1880:
1625:
4082:
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440:
completely mathematically described the Platonic solids in the
3681:
The Greek, Indian, and Chinese Elements – Seven Element Theory
3066:
Guggenberger, Michael (2013). "The Gallo-Roman Dodecahedron".
3685:
1597:
Note that, given a regular dodecahedron of edge length one,
3672:: Software used to create some of the images on this page.
2483:
The Penguin Dictionary of Curious and Interesting Geometry
2360:
John Philoponus' Criticism of Aristotle's Theory of Aether
1753:. It is the set of polyhedrons that can be constructed by
767:{\textstyle \phi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618}
565:
3417:(1995). "Chapter 4: Steinitz' Theorem for 3-Polytopes".
1900:, swiveling to the front as required to match his mood.
2292:. Wooden Books. Bloomsbury Publishing USA. p. 55.
466:
The regular icosahedron inside the regular dodecahedron
1779:
of the regular dodecahedron make up three of the four
1370:
to each of the regular dodecahedron's faces), and the
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640:
3074:(4). Springer Science and Business Media LLC: 56–60.
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3425:. Vol. 152. Springer-Verlag. pp. 103–126.
2574:
Calculus Gems: Brief Lives and Memorable Mathematics
2172:
has order a hundred and twenty. The vertices can be
262:
of a regular dodecahedron can be represented as the
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2577:. Mathematical Association of America. p. 50.
1863:(1955), the room is a hollow regular dodecahedron.
1764:The regular dodecahedron can be interpreted as the
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16:Convex polyhedron with 12 regular pentagonal faces
2949:
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2853:Table I(i), pp. 292–293. See the columns labeled
2795:(First trade paperback ed.). New York City:
1749:The regular dodecahedron can be interpreted as a
1624:is the radius of a circumscribing sphere about a
3676:How to make a dodecahedron from a Styrofoam cube
2550:. Courier Dover Publications. pp. 138–140.
1400:(one that touches the middle of each edge) are:
1285:If the edge length of a regular dodecahedron is
398:Following its attribution with nature by Plato,
3154:
3152:
570:
3484:: CS1 maint: DOI inactive as of August 2024 (
519:When a regular dodecahedron is inscribed in a
337:, a set of polyhedrons in which the faces are
4094:
3701:
3499:Pisanski, Tomaž; Servatius, Brigitte (2013).
208:in his dialogues, and it was used as part of
8:
2822:"Exact Dihedral Metric for Common Polyhedra"
2015:, but this statement actually refers to the
2547:A Mathematical History of the Golden Number
2313:Herrmann, Diane L.; Sally, Paul J. (2013).
509:{\displaystyle \mathrm {I} _{\mathrm {h} }}
120:{\displaystyle \mathrm {I} _{\mathrm {h} }}
4367:
4101:
4087:
4079:
3708:
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3686:
2966:{\displaystyle {}_{2}\!\mathrm {R} /\ell }
2926:{\displaystyle {}_{1}\!\mathrm {R} /\ell }
2886:{\displaystyle {}_{0}\!\mathrm {R} /\ell }
2394:. Cambridge University Press. p. 57.
2115:This graph can also be constructed as the
219:The regular dodecahedron is the family of
167:
27:
3614:"3D convex uniform polyhedra o3o5x – doe"
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3502:Configuration from a Graphical Viewpoint
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2281:
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2011:are also said to exhibit "dodecahedral"
3021:(2nd ed.). Springer. p. 127.
2850:
2601:A Ludic Journey into Geometric Topology
2275:
3639:– 3-d model that works in your browser
3477:
2486:. Penguin Books. p. 57–58.
580:, which comes under our definition of
18:
2414:
2319:. Taylor & Francis. p. 252.
1837:, dodecahedra appears in the work of
1679:of a regular pentagon of edge length
7:
1989:(see figure), a unicellular coastal
204:, described as cosmic stellation by
3384:(2003), "13.1 Steinitz's theorem",
2457:Mathematical Association of America
2207:along the edges of a dodecahedron.
551:Relation to the regular tetrahedron
458:Relation to the regular icosahedron
3725:Listed by number of faces and type
3633:– Models made with Modular Origami
2951:
2911:
2871:
2203:. The game's object was to find a
500:
494:
428:3D model of a regular dodecahedron
111:
105:
14:
3243:"Is The Universe A Dodecahedron?"
2289:Platonic & Archimedean Solids
1942:Holmium–magnesium–zinc (Ho-Mg-Zn)
1928:goes back a hundred million years
564:the opposing pair). As quoted by
305:Regular dodecahedron painting by
2664:(1973) . "§1.8 Configurations".
2347:, painting in semblance of life.
2065:
2054:
1973:, DOI:10.1038/s44160-023-00276-9
1955:
1933:
1912:
1860:The Sacrament of the Last Supper
480:three-dimensional symmetry group
313:
298:
1745:Other related geometric objects
1734:{\displaystyle 2\arctan(\phi )}
1230:
1145:
892:
3645:The Encyclopedia of Polyhedra
3568:Graph Theory with Applications
3068:The Mathematical Intelligencer
2996:as the edge length (see p. 2).
2604:. Cham: Springer. p. 23.
2140:
2128:
1728:
1722:
1263:
1231:
1220:
1188:
1178:
1146:
1135:
1108:
723:of a polynomial, expressed as
1:
3936:(two infinite groups and 75)
3665:Dodecahedron 3D Visualization
3570:, North Holland, p. 53,
3423:Graduate Texts in Mathematics
3392:Graduate Texts in Mathematics
2544:Herz-Fischler, Roger (2013).
2510:American Mathematical Monthly
968:: the orange vertices lie at
196:faces, three meeting at each
4481:Degenerate polyhedra are in
3954:(two infinite groups and 50)
3670:Stella: Polyhedron Navigator
3211:10.1371/journal.pone.0081749
2357:Wildberg, Christian (1988).
1904:In nature and supramolecules
1797:great stellated dodecahedron
1785:small stellated dodecahedron
1757:the two axial vertices of a
1080:and form a rectangle on the
1042:and form a rectangle on the
1004:and form a rectangle on the
978:: the green vertices lie at
711:Relation to the golden ratio
323:Platonic solid model of the
223:because it is the result of
4300:pentagonal icositetrahedron
4241:truncated icosidodecahedron
3241:Dumé, Belle (Oct 8, 2003).
3027:10.1007/978-0-387-92714-5_9
2692:(2nd ed.). Cambridge:
2571:Simmons, George F. (2007).
2388:Cromwell, Peter R. (1997).
2029:Poincaré dodecahedral space
1871:In toys and popular culture
1054:: the pink vertices lie at
1016:: the blue vertices lie at
598:compound of five tetrahedra
414:, Kepler also proposed the
4534:
4330:pentagonal hexecontahedron
4290:deltoidal icositetrahedron
3454:Cambridge University Press
3360:10.1051/0004-6361:20078777
3330:Astronomy and Astrophysics
3223:Dodecahedral Crystal Habit
3181:10.1038/s44160-023-00276-9
3119:Cambridge University Press
2694:Cambridge University Press
2670:(3rd ed.). New York:
2183:The dodecahedral graph is
2117:generalized Petersen graph
2092:can be represented as the
1841:, such as his lithographs
1741:, approximately 116.565°.
594:compound of ten tetrahedra
578:compound of five octahedra
4479:
4370:
4325:disdyakis triacontahedron
4320:deltoidal hexecontahedron
4006:Kepler–Poinsot polyhedron
3723:
3643:Virtual Reality Polyhedra
3511:10.1007/978-0-8176-8364-1
3446:Rudolph, Michael (2022).
3195:Braarudosphaera bigelowii
3127:10.1017/s0003581500024458
3080:10.1007/s00283-013-9403-7
2690:Regular Complex Polytopes
2647:The Fifty-Nine Icosahedra
2610:10.1007/978-3-031-07442-4
2449:Erickson, Martin (2011).
2228:Braarudosphaera bigelowii
1986:Braarudosphaera bigelowii
1925:Braarudosphaera bigelowii
1919:The fossil record of the
474:of a dodecahedron is the
166:
26:
3505:. Springer. p. 81.
2824:. In Arvo, James (ed.).
2745:The Mathematical Gazette
2436:A History of Mathematics
2231:− A dodecahedron shaped
1890:In the children's novel
1781:Kepler–Poinsot polyhedra
411:Mysterium Cosmographicum
229:pentagonal trapezohedron
4431:gyroelongated bipyramid
4305:rhombic triacontahedron
4211:truncated cuboctahedron
4018:Uniform star polyhedron
3946:quasiregular polyhedron
3464:(inactive 2024-08-21).
3352:2008A&A...482..747L
3111:The Antiquaries Journal
2820:Paeth, Alan W. (1991).
2709:The Mathematics Teacher
2146:{\displaystyle G(10,2)}
1751:truncated trapezohedron
221:truncated trapezohedron
183:pentagonal dodecahedron
45:Truncated trapezohedron
4426:truncated trapezohedra
4295:disdyakis dodecahedron
4261:(duals of Archimedean)
4236:rhombicosidodecahedron
4226:truncated dodecahedron
3952:semiregular polyhedron
3598:"Regular Dodecahedron"
2990:
2989:{\displaystyle 2\ell }
2967:
2927:
2887:
2265:Truncated dodecahedron
2193:William Rowan Hamilton
2147:
1826:, small hollow bronze
1819:
1735:
1693:
1669:
1642:
1618:
1591:
1394:
1360:
1333:
1299:
1277:
1090:
949:
837:
817:
794:
768:
702:
608:
560:
510:
467:
429:
395:in American English).
274:. Its property of the
200:. It is an example of
121:
4315:pentakis dodecahedron
4231:truncated icosahedron
4186:truncated tetrahedron
3999:non-convex polyhedron
3626:The Uniform Polyhedra
3462:10.1007/9781316466919
3419:Lectures on Polytopes
3228:12 April 2009 at the
3197:(Prymnesiophyceae)".
2991:
2968:
2928:
2888:
2480:Weils, David (1991).
2452:Beautiful Mathematics
2286:Sutton, Daud (2002).
2255:Pentakis dodecahedron
2148:
2102:three-connected graph
2023:Shape of the universe
1962:Crystal structure of
1893:The Phantom Tollbooth
1815:
1736:
1694:
1692:{\displaystyle \phi }
1670:
1668:{\displaystyle r_{i}}
1643:
1641:{\displaystyle \phi }
1619:
1617:{\displaystyle r_{u}}
1592:
1395:
1393:{\displaystyle r_{m}}
1361:
1359:{\displaystyle r_{i}}
1334:
1332:{\displaystyle r_{u}}
1300:
1278:
1095:Cartesian coordinates
960:
950:
838:
818:
795:
769:
703:
582:stellated icosahedron
566:Coxeter et al. (1938)
558:
511:
465:
427:
375:—with the four
122:
4275:rhombic dodecahedron
4201:truncated octahedron
3653:Regular dodecahedron
3539:"Dodecahedral Graph"
3265:Luminet, Jean-Pierre
3011:"Goldberg Polyhedra"
2977:
2937:
2897:
2857:
2721:10.5951/MT.80.5.0357
2122:
2017:rhombic dodecahedron
1710:
1683:
1652:
1632:
1601:
1404:
1377:
1343:
1316:
1311:circumscribed sphere
1289:
1101:
847:
827:
807:
784:
727:
632:
617:configuration matrix
611:Configuration matrix
529:Pappus of Alexandria
489:
484:icosahedral symmetry
227:axial vertices of a
179:regular dodecahedron
100:
95:icosahedral symmetry
22:Regular dodecahedron
4310:triakis icosahedron
4285:tetrakis hexahedron
4270:triakis tetrahedron
4206:rhombicuboctahedron
3612:Klitzing, Richard.
3299:10.1038/nature01944
3291:2003Natur.425..593L
3173:2023NatSy...2..789W
2838:1991grge.book.....A
2598:Marar, Ton (2022).
2158:distance-transitive
2033:Jean-Pierre Luminet
1766:Goldberg polyhedron
537:Apollonius of Perga
476:regular icosahedron
373:regular icosahedron
361:regular tetrahedron
290:As a Platonic solid
241:regular icosahedron
233:Goldberg polyhedron
50:Goldberg polyhedron
4503:Goldberg polyhedra
4280:triakis octahedron
4165:Archimedean solids
3940:regular polyhedron
3934:uniform polyhedron
3896:Hectotriadiohedron
3595:Weisstein, Eric W.
3536:Weisstein, Eric W.
3415:Ziegler, Günter M.
3015:Senechal, Marjorie
2986:
2963:
2923:
2883:
2799:. pp. 70–71.
2672:Dover Publications
2640:; Flather, H. T.;
2367:. pp. 11–12.
2221:regular polychoron
2170:automorphism group
2143:
2106:dodecahedral graph
2086:Steinitz's theorem
2046:Dodecahedral graph
1898:facial expressions
1877:role-playing games
1820:
1817:Roman dodecahedron
1793:great dodecahedron
1731:
1689:
1665:
1638:
1614:
1587:
1585:
1390:
1356:
1329:
1295:
1273:
1271:
1091:
945:
833:
813:
790:
764:
698:
692:
561:
525:Hero of Alexandria
506:
468:
430:
377:classical elements
369:regular octahedron
268:dodecahedral graph
256:Roman dodecahedron
117:
4490:
4489:
4409:
4408:
4246:snub dodecahedron
4221:icosidodecahedron
4076:
4075:
3977:Archimedean solid
3964:convex polyhedron
3872:Icosidodecahedron
3631:Origami Polyhedra
3520:978-0-8176-8363-4
3036:978-0-387-92713-8
2686:Coxeter, H. S. M.
2667:Regular Polytopes
2662:Coxeter, H. S. M.
2619:978-3-031-07442-4
2466:978-1-61444-509-8
2401:978-0-521-55432-9
2365:Walter de Gruyter
2326:978-1-4665-5464-1
2260:Snub dodecahedron
2205:Hamiltonian cycle
2197:mathematical game
2195:, who invented a
1949:regular pentagons
1828:Roman dodecahedra
1566:
1513:
1510:
1445:
1439:
1298:{\displaystyle a}
930:
877:
876:
836:{\displaystyle a}
816:{\displaystyle V}
793:{\displaystyle A}
756:
750:
592:, thus forming a
175:
174:
63:regular pentagons
4525:
4368:
4364:Dihedral uniform
4339:Dihedral regular
4262:
4178:
4127:
4103:
4096:
4089:
4080:
3912:elemental things
3890:Enneacontahedron
3860:Icositetrahedron
3710:
3703:
3696:
3687:
3617:
3608:
3607:
3581:
3580:
3556:
3550:
3549:
3548:
3531:
3525:
3524:
3496:
3490:
3489:
3483:
3475:
3443:
3437:
3436:
3411:
3405:
3404:
3387:Convex Polytopes
3382:Grünbaum, Branko
3378:
3372:
3371:
3345:
3325:
3319:
3318:
3284:
3282:astro-ph/0310253
3261:
3255:
3254:
3249:. Archived from
3238:
3232:
3220:
3214:
3191:
3185:
3184:
3161:Nature Synthesis
3156:
3147:
3146:
3106:
3100:
3099:
3063:
3057:
3047:
3041:
3040:
3003:
2997:
2995:
2993:
2992:
2987:
2972:
2970:
2969:
2964:
2959:
2954:
2948:
2947:
2942:
2932:
2930:
2929:
2924:
2919:
2914:
2908:
2907:
2902:
2892:
2890:
2889:
2884:
2879:
2874:
2868:
2867:
2862:
2848:
2842:
2841:
2826:Graphics Gems II
2817:
2811:
2810:
2783:
2777:
2776:
2751:(421): 197–198.
2740:
2734:
2732:
2704:
2698:
2697:
2682:
2676:
2675:
2658:
2652:
2651:
2630:
2624:
2623:
2595:
2589:
2588:
2568:
2562:
2561:
2541:
2535:
2534:
2504:
2498:
2497:
2477:
2471:
2470:
2446:
2440:
2428:
2422:
2412:
2406:
2405:
2385:
2379:
2378:
2354:
2348:
2337:
2331:
2330:
2310:
2304:
2303:
2283:
2162:distance-regular
2152:
2150:
2149:
2144:
2069:
2058:
2040:Bertrand Russell
1959:
1937:
1916:
1835:20th-century art
1740:
1738:
1737:
1732:
1698:
1696:
1695:
1690:
1674:
1672:
1671:
1666:
1664:
1663:
1647:
1645:
1644:
1639:
1623:
1621:
1620:
1615:
1613:
1612:
1596:
1594:
1593:
1588:
1586:
1567:
1562:
1561:
1552:
1543:
1542:
1514:
1512:
1511:
1500:
1494:
1493:
1484:
1475:
1474:
1446:
1441:
1440:
1435:
1429:
1420:
1419:
1399:
1397:
1396:
1391:
1389:
1388:
1365:
1363:
1362:
1357:
1355:
1354:
1338:
1336:
1335:
1330:
1328:
1327:
1304:
1302:
1301:
1296:
1282:
1280:
1279:
1274:
1272:
1253:
1201:
1174:
1085:
1079:
1077:
1075:
1074:
1069:
1066:
1053:
1047:
1041:
1035:
1033:
1032:
1027:
1024:
1015:
1009:
1003:
1001:
999:
998:
993:
990:
977:
971:
967:
954:
952:
951:
946:
941:
940:
931:
929:
915:
914:
913:
900:
888:
887:
878:
866:
865:
857:
842:
840:
839:
834:
822:
820:
819:
814:
799:
797:
796:
791:
773:
771:
770:
765:
757:
752:
751:
746:
737:
707:
705:
704:
699:
697:
696:
590:stella octangula
544:Golden rectangle
535:, among others.
515:
513:
512:
507:
505:
504:
503:
497:
426:
405:Harmonices Mundi
339:regular polygons
317:
302:
171:
126:
124:
123:
118:
116:
115:
114:
108:
31:
19:
4533:
4532:
4528:
4527:
4526:
4524:
4523:
4522:
4513:Platonic solids
4493:
4492:
4491:
4486:
4475:
4414:Dihedral others
4405:
4384:
4359:
4334:
4263:
4260:
4259:
4250:
4179:
4168:
4167:
4158:
4121:
4119:Platonic solids
4113:
4107:
4077:
4072:
4023:
4012:Star polyhedron
3994:
3959:
3907:
3884:Hexecontahedron
3866:Triacontahedron
3848:
3839:Enneadecahedron
3829:Heptadecahedron
3819:Pentadecahedron
3814:Tetradecahedron
3787:
3726:
3719:
3714:
3611:
3593:
3592:
3589:
3584:
3578:
3564:Murty, U. S. R.
3558:
3557:
3553:
3534:
3533:
3532:
3528:
3521:
3498:
3497:
3493:
3476:
3472:
3445:
3444:
3440:
3433:
3413:
3412:
3408:
3402:
3380:
3379:
3375:
3327:
3326:
3322:
3275:(6958): 593–5.
3263:
3262:
3258:
3240:
3239:
3235:
3230:Wayback Machine
3221:
3217:
3192:
3188:
3158:
3157:
3150:
3108:
3107:
3103:
3065:
3064:
3060:
3050:Cromwell (1997)
3048:
3044:
3037:
3005:
3004:
3000:
2975:
2974:
2940:
2935:
2934:
2900:
2895:
2894:
2860:
2855:
2854:
2849:
2845:
2832:. p. 177.
2819:
2818:
2814:
2807:
2785:
2784:
2780:
2757:10.2307/3616690
2742:
2741:
2737:
2706:
2705:
2701:
2684:
2683:
2679:
2660:
2659:
2655:
2638:du Val, Patrick
2634:Coxeter, H.S.M.
2632:
2631:
2627:
2620:
2597:
2596:
2592:
2585:
2570:
2569:
2565:
2558:
2543:
2542:
2538:
2523:10.2307/2317282
2506:
2505:
2501:
2494:
2479:
2478:
2474:
2467:
2448:
2447:
2443:
2429:
2425:
2413:
2409:
2402:
2387:
2386:
2382:
2375:
2356:
2355:
2351:
2338:
2334:
2327:
2312:
2311:
2307:
2300:
2285:
2284:
2277:
2273:
2233:coccolithophore
2213:
2120:
2119:
2082:
2081:
2080:
2079:
2072:
2071:
2070:
2061:
2060:
2059:
2048:
2025:
1991:phytoplanktonic
1982:coccolithophore
1978:
1977:
1976:
1975:
1974:
1960:
1952:
1951:
1940:The faces of a
1938:
1930:
1929:
1921:coccolithophore
1917:
1906:
1881:polyhedral dice
1873:
1824:Hellenistic era
1810:
1805:
1747:
1708:
1707:
1681:
1680:
1655:
1650:
1649:
1630:
1629:
1628:of edge length
1604:
1599:
1598:
1584:
1583:
1553:
1544:
1534:
1531:
1530:
1495:
1485:
1476:
1466:
1463:
1462:
1430:
1421:
1411:
1402:
1401:
1380:
1375:
1374:
1346:
1341:
1340:
1319:
1314:
1313:
1287:
1286:
1270:
1269:
1226:
1185:
1184:
1141:
1099:
1098:
1089:
1081:
1070:
1067:
1064:
1063:
1061:
1055:
1051:
1043:
1028:
1025:
1022:
1021:
1019:
1017:
1013:
1005:
994:
991:
988:
987:
985:
979:
975:
969:
965:
932:
916:
905:
901:
879:
858:
845:
844:
825:
824:
805:
804:
782:
781:
738:
725:
724:
713:
691:
690:
685:
680:
674:
673:
668:
663:
657:
656:
651:
646:
636:
630:
629:
613:
586:triacontahedron
553:
492:
487:
486:
472:dual polyhedron
460:
422:
400:Johannes Kepler
335:Platonic solids
331:
330:
329:
328:
327:
318:
310:
309:
307:Johannes Kepler
303:
292:
245:dual polyhedron
231:. It is also a
214:Johannes Kepler
202:Platonic solids
103:
98:
97:
48:
43:
17:
12:
11:
5:
4531:
4529:
4521:
4520:
4515:
4510:
4505:
4495:
4494:
4488:
4487:
4480:
4477:
4476:
4474:
4473:
4468:
4463:
4458:
4453:
4448:
4443:
4438:
4433:
4428:
4423:
4417:
4415:
4411:
4410:
4407:
4406:
4404:
4403:
4398:
4392:
4390:
4386:
4385:
4383:
4382:
4377:
4371:
4365:
4361:
4360:
4358:
4357:
4350:
4342:
4340:
4336:
4335:
4333:
4332:
4327:
4322:
4317:
4312:
4307:
4302:
4297:
4292:
4287:
4282:
4277:
4272:
4266:
4264:
4257:Catalan solids
4255:
4252:
4251:
4249:
4248:
4243:
4238:
4233:
4228:
4223:
4218:
4213:
4208:
4203:
4198:
4196:truncated cube
4193:
4188:
4182:
4180:
4163:
4160:
4159:
4157:
4156:
4151:
4146:
4141:
4136:
4130:
4128:
4115:
4114:
4108:
4106:
4105:
4098:
4091:
4083:
4074:
4073:
4071:
4070:
4068:parallelepiped
4065:
4060:
4055:
4050:
4045:
4040:
4034:
4032:
4025:
4024:
4022:
4021:
4015:
4009:
4002:
4000:
3996:
3995:
3993:
3992:
3986:
3980:
3974:
3971:Platonic solid
3967:
3965:
3961:
3960:
3958:
3957:
3956:
3955:
3949:
3943:
3931:
3926:
3921:
3915:
3913:
3909:
3908:
3906:
3905:
3899:
3893:
3887:
3881:
3875:
3869:
3863:
3856:
3854:
3850:
3849:
3847:
3846:
3841:
3836:
3834:Octadecahedron
3831:
3826:
3824:Hexadecahedron
3821:
3816:
3811:
3806:
3801:
3795:
3793:
3789:
3788:
3786:
3785:
3780:
3775:
3770:
3765:
3760:
3755:
3750:
3745:
3740:
3734:
3732:
3728:
3727:
3724:
3721:
3720:
3715:
3713:
3712:
3705:
3698:
3690:
3684:
3683:
3678:
3673:
3667:
3662:
3657:
3656:
3655:
3640:
3634:
3628:
3623:
3618:
3609:
3588:
3587:External links
3585:
3583:
3582:
3576:
3551:
3526:
3519:
3491:
3470:
3456:. p. 25.
3438:
3431:
3406:
3400:
3373:
3320:
3256:
3253:on 2012-04-25.
3233:
3215:
3205:(12): e81749.
3186:
3148:
3101:
3058:
3042:
3035:
2998:
2985:
2982:
2962:
2958:
2953:
2946:
2922:
2918:
2913:
2906:
2882:
2878:
2873:
2866:
2851:Coxeter (1973)
2843:
2830:Academic Press
2812:
2805:
2797:Broadway Books
2778:
2735:
2715:(5): 357–358.
2699:
2696:. p. 117.
2677:
2653:
2625:
2618:
2590:
2583:
2563:
2556:
2536:
2499:
2492:
2472:
2465:
2459:. p. 62.
2441:
2431:Florian Cajori
2423:
2407:
2400:
2380:
2373:
2349:
2332:
2325:
2305:
2298:
2274:
2272:
2269:
2268:
2267:
2262:
2257:
2252:
2246:
2230:
2224:
2212:
2209:
2142:
2139:
2136:
2133:
2130:
2127:
2110:Platonic graph
2074:
2073:
2064:
2063:
2062:
2053:
2052:
2051:
2050:
2049:
2047:
2044:
2024:
2021:
1961:
1954:
1953:
1939:
1932:
1931:
1918:
1911:
1910:
1909:
1908:
1907:
1905:
1902:
1872:
1869:
1809:
1808:In visual arts
1806:
1804:
1801:
1746:
1743:
1730:
1727:
1724:
1721:
1718:
1715:
1704:dihedral angle
1688:
1662:
1658:
1637:
1611:
1607:
1582:
1579:
1576:
1573:
1570:
1565:
1560:
1556:
1550:
1547:
1545:
1541:
1537:
1533:
1532:
1529:
1526:
1523:
1520:
1517:
1509:
1506:
1503:
1498:
1492:
1488:
1482:
1479:
1477:
1473:
1469:
1465:
1464:
1461:
1458:
1455:
1452:
1449:
1444:
1438:
1433:
1427:
1424:
1422:
1418:
1414:
1410:
1409:
1387:
1383:
1353:
1349:
1326:
1322:
1294:
1268:
1265:
1262:
1259:
1256:
1252:
1248:
1245:
1242:
1239:
1236:
1233:
1229:
1227:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1200:
1196:
1193:
1190:
1187:
1186:
1183:
1180:
1177:
1173:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1144:
1142:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1106:
1093:The following
1088:
1087:
1049:
1011:
973:
962:
944:
939:
935:
928:
925:
922:
919:
912:
908:
904:
898:
895:
891:
886:
882:
875:
872:
869:
864:
861:
855:
852:
832:
812:
789:
763:
760:
755:
749:
744:
741:
735:
732:
712:
709:
695:
689:
686:
684:
681:
679:
676:
675:
672:
669:
667:
664:
662:
659:
658:
655:
652:
650:
647:
645:
642:
641:
639:
612:
609:
574:golden section
552:
549:
502:
496:
459:
456:
319:
312:
311:
304:
297:
296:
295:
294:
293:
291:
288:
272:Platonic graph
173:
172:
164:
163:
157:
156:
147:
143:
142:
139:
132:Dihedral angle
128:
127:
113:
107:
92:
90:Symmetry group
86:
85:
82:
76:
75:
72:
66:
65:
59:
53:
52:
40:Platonic solid
37:
33:
32:
24:
23:
15:
13:
10:
9:
6:
4:
3:
2:
4530:
4519:
4516:
4514:
4511:
4509:
4508:Planar graphs
4506:
4504:
4501:
4500:
4498:
4484:
4478:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4447:
4444:
4442:
4439:
4437:
4434:
4432:
4429:
4427:
4424:
4422:
4419:
4418:
4416:
4412:
4402:
4399:
4397:
4394:
4393:
4391:
4387:
4381:
4378:
4376:
4373:
4372:
4369:
4366:
4362:
4356:
4355:
4351:
4349:
4348:
4344:
4343:
4341:
4337:
4331:
4328:
4326:
4323:
4321:
4318:
4316:
4313:
4311:
4308:
4306:
4303:
4301:
4298:
4296:
4293:
4291:
4288:
4286:
4283:
4281:
4278:
4276:
4273:
4271:
4268:
4267:
4265:
4258:
4253:
4247:
4244:
4242:
4239:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4191:cuboctahedron
4189:
4187:
4184:
4183:
4181:
4176:
4172:
4166:
4161:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4131:
4129:
4125:
4120:
4116:
4112:
4104:
4099:
4097:
4092:
4090:
4085:
4084:
4081:
4069:
4066:
4064:
4061:
4059:
4056:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4036:
4035:
4033:
4030:
4026:
4019:
4016:
4013:
4010:
4007:
4004:
4003:
4001:
3997:
3990:
3989:Johnson solid
3987:
3984:
3983:Catalan solid
3981:
3978:
3975:
3972:
3969:
3968:
3966:
3962:
3953:
3950:
3947:
3944:
3941:
3938:
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3920:
3917:
3916:
3914:
3910:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3878:Hexoctahedron
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3857:
3855:
3851:
3845:
3842:
3840:
3837:
3835:
3832:
3830:
3827:
3825:
3822:
3820:
3817:
3815:
3812:
3810:
3809:Tridecahedron
3807:
3805:
3802:
3800:
3799:Hendecahedron
3797:
3796:
3794:
3790:
3784:
3781:
3779:
3776:
3774:
3771:
3769:
3766:
3764:
3761:
3759:
3756:
3754:
3751:
3749:
3746:
3744:
3741:
3739:
3736:
3735:
3733:
3729:
3722:
3718:
3711:
3706:
3704:
3699:
3697:
3692:
3691:
3688:
3682:
3679:
3677:
3674:
3671:
3668:
3666:
3663:
3661:
3658:
3654:
3650:
3647:
3646:
3644:
3641:
3638:
3635:
3632:
3629:
3627:
3624:
3622:
3619:
3615:
3610:
3605:
3604:
3599:
3596:
3591:
3590:
3586:
3579:
3577:0-444-19451-7
3573:
3569:
3565:
3561:
3555:
3552:
3546:
3545:
3540:
3537:
3530:
3527:
3522:
3516:
3512:
3508:
3504:
3503:
3495:
3492:
3487:
3481:
3473:
3471:9781316466919
3467:
3463:
3459:
3455:
3451:
3450:
3442:
3439:
3434:
3432:0-387-94365-X
3428:
3424:
3420:
3416:
3410:
3407:
3403:
3401:0-387-40409-0
3397:
3393:
3389:
3388:
3383:
3377:
3374:
3369:
3365:
3361:
3357:
3353:
3349:
3344:
3339:
3335:
3331:
3324:
3321:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3288:
3283:
3278:
3274:
3270:
3266:
3260:
3257:
3252:
3248:
3244:
3237:
3234:
3231:
3227:
3224:
3219:
3216:
3212:
3208:
3204:
3200:
3196:
3190:
3187:
3182:
3178:
3174:
3170:
3166:
3162:
3155:
3153:
3149:
3144:
3140:
3136:
3132:
3128:
3124:
3120:
3116:
3112:
3105:
3102:
3097:
3093:
3089:
3085:
3081:
3077:
3073:
3069:
3062:
3059:
3055:
3051:
3046:
3043:
3038:
3032:
3028:
3024:
3020:
3019:Shaping Space
3016:
3012:
3008:
3002:
2999:
2983:
2980:
2960:
2956:
2944:
2920:
2916:
2904:
2880:
2876:
2864:
2852:
2847:
2844:
2839:
2835:
2831:
2827:
2823:
2816:
2813:
2808:
2806:0-7679-0816-3
2802:
2798:
2794:
2793:
2788:
2782:
2779:
2774:
2770:
2766:
2762:
2758:
2754:
2750:
2746:
2739:
2736:
2730:
2726:
2722:
2718:
2714:
2710:
2703:
2700:
2695:
2691:
2687:
2681:
2678:
2673:
2669:
2668:
2663:
2657:
2654:
2649:
2648:
2643:
2642:Petrie, J. F.
2639:
2635:
2629:
2626:
2621:
2615:
2611:
2607:
2603:
2602:
2594:
2591:
2586:
2584:9780883855614
2580:
2576:
2575:
2567:
2564:
2559:
2557:9780486152325
2553:
2549:
2548:
2540:
2537:
2532:
2528:
2524:
2520:
2516:
2512:
2511:
2503:
2500:
2495:
2493:9780140118131
2489:
2485:
2484:
2476:
2473:
2468:
2462:
2458:
2454:
2453:
2445:
2442:
2438:
2437:
2432:
2427:
2424:
2420:
2416:
2411:
2408:
2403:
2397:
2393:
2392:
2384:
2381:
2376:
2374:9783110104462
2370:
2366:
2362:
2361:
2353:
2350:
2346:
2345:diazographein
2342:
2336:
2333:
2328:
2322:
2318:
2317:
2309:
2306:
2301:
2299:9780802713865
2295:
2291:
2290:
2282:
2280:
2276:
2270:
2266:
2263:
2261:
2258:
2256:
2253:
2250:
2249:Dodecahedrane
2247:
2244:
2241:
2240:phytoplankton
2238:
2234:
2229:
2226:
2225:
2222:
2218:
2215:
2214:
2210:
2208:
2206:
2202:
2199:known as the
2198:
2194:
2190:
2186:
2181:
2179:
2175:
2171:
2167:
2163:
2159:
2154:
2137:
2134:
2131:
2125:
2118:
2113:
2111:
2107:
2103:
2099:
2095:
2091:
2087:
2084:According to
2078:
2068:
2057:
2045:
2043:
2041:
2036:
2034:
2030:
2022:
2020:
2018:
2014:
2010:
2006:
2002:
2001:quasicrystals
1997:
1995:
1992:
1988:
1987:
1983:
1972:
1969:
1965:
1958:
1950:
1946:
1943:
1936:
1927:
1926:
1922:
1915:
1903:
1901:
1899:
1895:
1894:
1888:
1886:
1882:
1878:
1870:
1868:
1866:
1862:
1861:
1856:
1855:Salvador Dalí
1852:
1851:
1846:
1845:
1840:
1836:
1831:
1829:
1825:
1818:
1814:
1807:
1802:
1800:
1798:
1794:
1790:
1786:
1782:
1778:
1773:
1771:
1767:
1762:
1760:
1759:trapezohedron
1756:
1752:
1744:
1742:
1725:
1719:
1716:
1713:
1705:
1700:
1686:
1678:
1660:
1656:
1635:
1627:
1609:
1605:
1580:
1577:
1574:
1571:
1568:
1563:
1558:
1554:
1548:
1546:
1539:
1535:
1527:
1524:
1521:
1518:
1515:
1507:
1504:
1501:
1496:
1490:
1486:
1480:
1478:
1471:
1467:
1459:
1456:
1453:
1450:
1447:
1442:
1436:
1431:
1425:
1423:
1416:
1412:
1385:
1381:
1373:
1369:
1351:
1347:
1324:
1320:
1312:
1308:
1292:
1283:
1266:
1260:
1257:
1254:
1250:
1246:
1243:
1240:
1237:
1234:
1228:
1223:
1217:
1214:
1211:
1208:
1205:
1202:
1198:
1194:
1191:
1181:
1175:
1171:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1143:
1138:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1096:
1084:
1073:
1059:
1050:
1046:
1039:
1031:
1012:
1008:
997:
983:
974:
964:
963:
959:
955:
942:
937:
933:
926:
923:
920:
917:
910:
906:
902:
896:
893:
889:
884:
880:
873:
870:
867:
862:
859:
853:
850:
830:
810:
803:
787:
780:
775:
761:
758:
753:
747:
742:
739:
733:
730:
722:
718:
710:
708:
693:
687:
682:
677:
670:
665:
660:
653:
648:
643:
637:
626:
622:
618:
610:
607:
605:
604:
599:
595:
591:
587:
583:
579:
575:
569:
567:
557:
550:
548:
545:
541:
538:
534:
530:
526:
522:
517:
485:
481:
477:
473:
464:
457:
455:
453:
449:
445:
444:
439:
435:
425:
420:
417:
413:
412:
407:
406:
401:
396:
394:
390:
386:
382:
378:
374:
370:
366:
362:
358:
354:
353:
348:
344:
340:
336:
326:
322:
316:
308:
301:
289:
287:
285:
281:
277:
273:
269:
265:
261:
257:
253:
248:
246:
242:
238:
234:
230:
226:
222:
217:
215:
211:
207:
203:
199:
195:
192:
188:
184:
180:
170:
165:
162:
158:
155:
151:
148:
144:
141:116.565°
140:
137:
133:
129:
96:
93:
91:
87:
83:
81:
77:
73:
71:
67:
64:
60:
58:
54:
51:
46:
41:
38:
34:
30:
25:
20:
4482:
4401:trapezohedra
4352:
4345:
4149:dodecahedron
4148:
3902:Apeirohedron
3853:>20 faces
3804:Dodecahedron
3637:Dodecahedron
3601:
3567:
3560:Bondy, J. A.
3554:
3542:
3529:
3501:
3494:
3448:
3441:
3418:
3409:
3386:
3376:
3333:
3329:
3323:
3272:
3268:
3259:
3251:the original
3247:PhysicsWorld
3246:
3236:
3218:
3202:
3198:
3194:
3189:
3164:
3160:
3114:
3110:
3104:
3071:
3067:
3061:
3045:
3018:
3007:Hart, George
3001:
2846:
2825:
2815:
2791:
2787:Livio, Mario
2781:
2748:
2744:
2738:
2712:
2708:
2702:
2689:
2680:
2666:
2656:
2646:
2628:
2600:
2593:
2573:
2566:
2546:
2539:
2514:
2508:
2502:
2482:
2475:
2451:
2444:
2435:
2426:
2415:Livio (2003)
2410:
2390:
2383:
2359:
2352:
2344:
2340:
2335:
2315:
2308:
2288:
2201:icosian game
2187:, meaning a
2182:
2155:
2114:
2105:
2083:
2077:Icosian game
2037:
2026:
1998:
1984:
1979:
1970:
1967:
1963:
1945:quasicrystal
1923:
1891:
1889:
1874:
1865:Gerard Caris
1858:
1857:'s painting
1848:
1842:
1839:M. C. Escher
1832:
1821:
1791:, forming a
1774:
1763:
1748:
1701:
1284:
1092:
1082:
1071:
1057:
1044:
1037:
1029:
1006:
995:
981:
970:(±1, ±1, ±1)
779:surface area
776:
717:golden ratio
714:
614:
601:
571:
562:
542:
518:
469:
450:states that
442:
431:
416:Solar System
409:
403:
397:
392:
388:
350:
332:
325:Solar System
284:icosian game
267:
252:golden ratio
249:
218:
212:proposed by
210:Solar System
189:composed of
187:dodecahedron
182:
178:
176:
4518:12 (number)
4171:semiregular
4154:icosahedron
4134:tetrahedron
3844:Icosahedron
3792:11–20 faces
3778:Enneahedron
3768:Heptahedron
3758:Pentahedron
3753:Tetrahedron
3121:: 289–292.
2237:unicellular
2185:Hamiltonian
1980:The fossil
1968:Nat. Synth.
1853:(1952). In
1850:Gravitation
1847:(1943) and
1803:Appearances
1777:stellations
276:Hamiltonian
266:called the
4497:Categories
4466:prismatoid
4396:bipyramids
4380:antiprisms
4354:hosohedron
4144:octahedron
4029:prismatoid
4014:(infinite)
3783:Decahedron
3773:Octahedron
3763:Hexahedron
3738:Monohedron
3731:1–10 faces
3336:(3): 747.
3167:(8): 789.
3052:, p.
2517:(2): 192.
2417:, p.
2271:References
2251:(molecule)
1875:In modern
1755:truncating
448:Iamblichus
434:Theaetetus
391:in Latin,
352:Theaetetus
237:chamfering
225:truncating
194:pentagonal
146:Properties
4461:birotunda
4451:bifrustum
4216:snub cube
4111:polyhedra
4043:antiprism
3748:Trihedron
3717:Polyhedra
3603:MathWorld
3544:MathWorld
3480:cite book
3343:0801.0006
3143:161691752
3135:0003-5815
3096:122337773
3088:0343-6993
2984:ℓ
2961:ℓ
2921:ℓ
2881:ℓ
2789:(2003) .
2773:125919525
2391:Polyhedra
2180:is five.
2166:symmetric
1947:are true
1726:ϕ
1720:
1687:ϕ
1636:ϕ
1572:≈
1555:ϕ
1519:≈
1508:ϕ
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1487:ϕ
1451:≈
1432:ϕ
1372:midradius
1255:ϕ
1244:±
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927:ϕ
921:−
907:ϕ
874:ϕ
871:−
863:ϕ
759:≈
731:ϕ
533:Fibonacci
381:Aristotle
343:congruent
341:that are
4441:bicupola
4421:pyramids
4347:dihedron
3743:Dihedron
3566:(1976),
3307:14534579
3226:Archived
3199:PLoS One
3009:(2012).
2729:27965402
2688:(1991).
2644:(1938).
2217:120-cell
2211:See also
2178:diameter
2094:skeleton
1885:Megaminx
1844:Reptiles
800:and the
625:diagonal
452:Hippasus
443:Elements
321:Kepler's
260:skeleton
80:Vertices
4483:italics
4471:scutoid
4456:rotunda
4446:frustum
4175:uniform
4124:regular
4109:Convex
4063:pyramid
4048:frustum
3368:1616362
3348:Bibcode
3315:4380713
3287:Bibcode
3169:Bibcode
3017:(ed.).
2834:Bibcode
2765:3616690
2531:2317282
2341:Timaeus
2339:Plato,
2174:colored
2019:shape.
2009:diamond
1964:Co20L12
1770:chamfer
1677:apothem
1675:is the
1368:tangent
1086:-plane.
1076:
1062:
1048:-plane.
1034:
1020:
1010:-plane.
1000:
986:
402:in his
357:Timaeus
243:as its
191:regular
154:regular
136:degrees
4436:cupola
4389:duals:
4375:prisms
4053:cupola
3929:vertex
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3269:Nature
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2098:planar
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2005:garnet
1883:. The
1789:wedges
1717:arctan
1648:, and
1307:radius
1305:, the
1052:
1036:, 0, ±
1014:
976:
966:
802:volume
621:matrix
603:chiral
531:, and
521:sphere
438:Euclid
389:aether
385:aithêr
371:, and
198:vertex
150:convex
4058:wedge
4038:prism
3898:(132)
3364:S2CID
3338:arXiv
3311:S2CID
3277:arXiv
3139:S2CID
3092:S2CID
3013:. In
2769:S2CID
2761:JSTOR
2725:JSTOR
2527:JSTOR
2243:algae
2090:graph
2013:habit
1999:Some
1575:1.309
1522:1.114
1454:1.401
1309:of a
980:(0, ±
843:are:
762:1.618
721:roots
619:is a
393:ether
349:. In
347:Plato
264:graph
206:Plato
185:is a
70:Edges
57:Faces
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3991:(92)
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3649:VRML
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3303:PMID
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2189:path
2108:, a
2007:and
1994:alga
1971:2023
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777:The
715:The
615:The
470:The
365:cube
280:path
278:, a
270:, a
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