Regular dodecahedron

Regular dodecahedron
Regular dodecahedron
Type	Platonic solid,; Truncated trapezohedron,; Goldberg polyhedron
Faces	12 regular pentagons
Edges	30
Vertices	20
Symmetry group	icosahedral symmetry
Dihedral angle (degrees)	116.565°
Properties	convex, regular
Net

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity.

The regular dodecahedron is the family of truncated trapezohedron because it is the result of truncating axial vertices of a pentagonal trapezohedron. It is also a Goldberg polyhedron because it is the initial polyhedron to construct new polyhedrons by the process of chamfering. It has a relation with other Platonic solids, one of them is the regular icosahedron as its dual polyhedron. Other new polyhedrons can be constructed by using regular dodecahedron.

The regular dodecahedron's metric properties and construction are associated with the golden ratio. The regular dodecahedron can be found in many popular cultures: Roman dodecahedron, 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 skeleton of a regular dodecahedron can be represented as the graph called the dodecahedral graph, a Platonic graph. Its property of the Hamiltonian, a path visits all of its vertices exactly once, can be found in a toy called icosian game.

As a Platonic solid

Regular dodecahedron painting by Johannes Kepler

Kepler's Platonic solid model of the Solar System

The regular dodecahedron is a polyhedron with twelve pentagonal faces, thirty edges, and twenty vertices.^[1] It is one of the Platonic solids, a set of polyhedrons in which the faces are regular polygons that are congruent and the same number of faces meet at a vertex.^[2] This set of polyhedrons is named after Plato. In Theaetetus, 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 [it] for arranging the constellations on the whole heaven". Timaeus, as a personage of Plato's dialogue, associates the other four Platonic solids—regular tetrahedron, cube, regular octahedron, and regular icosahedron—with the four classical elements, 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."^[3] Aristotle also postulated that the heavens were made of a fifth element, which he called aithêr (aether in Latin, ether in American English).^[4]

Following its attribution with nature by Plato, Johannes Kepler in his Harmonices Mundi sketched each of the Platonic solids, one of them is a regular dodecahedron.^[5] In his Mysterium Cosmographicum, Kepler also proposed the Solar System 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.^[6]

3D model of a regular dodecahedron

Many antiquity philosophers described the regular dodecahedron, including the rest of the Platonic solids. Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist. Euclid completely mathematically described the Platonic solids in the Elements, 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. Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons".^[7]

Relation to the regular icosahedron

The regular icosahedron inside the regular dodecahedron

The dual polyhedron of a dodecahedron is the regular icosahedron. One property of the dual polyhedron generally is that the original polyhedron and its dual share the same three-dimensional symmetry group. In the case of the regular dodecahedron, it has the same symmetry as the regular icosahedron, the icosahedral symmetry $\mathrm {I} _{\mathrm {h} }$ .^[8] 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.^[9]

When a regular dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.55%).^[10] 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 Hero of Alexandria, Pappus of Alexandria, and Fibonacci, among others.^[11] Apollonius of Perga discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas.^[12] Both volumes have formulas involving the golden ratio but are taken to different powers.^[1]

Golden rectangle 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 regular icosahedron vertices, considered as the center of 12 regular dodecahedron faces.^[13]

Relation to the regular tetrahedron

Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.

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 the opposing pair). As quoted by Coxeter et al. (1938),^[14]

"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 "golden section". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a compound of five octahedra, which comes under our definition of stellated icosahedron. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated triacontahedron.) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a stella octangula, thus forming a compound of ten tetrahedra. Further, we can choose one tetrahedron from each stella octangula, so as to derive a compound of five tetrahedra, 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. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be chiral."

Configuration matrix

The configuration matrix is a matrix in which the rows and columns correspond to the elements of a polyhedron as in the vertices, edges, and faces. The diagonal 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:^[15]^[16] ${\begin{bmatrix}20&3&3\\2&30&2\\5&5&12\end{bmatrix}}$

Relation to the golden ratio

The golden ratio is the ratio between two numbers equal to the ratio of their sum to the larger of the two quantities.^[17] It is one of two roots of a polynomial, expressed as ${\textstyle \phi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618}$ .^[18] The golden ratio can be applied to the regular dodecahedron's metric properties, as well as to construct the regular dodecahedron.

The surface area $A$ and the volume $V$ of a regular dodecahedron of edge length $a$ are:^[19] $A={\frac {15\phi }{\sqrt {3-\phi }}}a^{2},\qquad V={\frac {5\phi ^{3}}{6-2\phi }}a^{3}.$

Cartesian coordinates of a regular dodecahedron in the following:

: the orange vertices lie at $(\pm1, \pm1, \pm1)$ .
: the green vertices lie at $(0, ±ϕ, ±.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/ϕ⁠)$ and form a rectangle on the $yz$ -plane.
: the blue vertices lie at $(\pm ⁠ 1 / ϕ ⁠, 0, \pm ϕ)$ and form a rectangle on the $xz$ -plane.
: the pink vertices lie at $(\pm ϕ, \pm ⁠ 1 / ϕ ⁠, 0)$ and form a rectangle on the $xy$ -plane.

The following Cartesian coordinates define the twenty vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented:^[20] ${\begin{aligned}(\pm 1,\pm 1,\pm 1),&\qquad (0,\pm \phi ,\pm 1/\phi ),\\(\pm 1/\phi ,0,\pm \phi ),&\qquad (\pm \phi ,\pm 1/\phi ,0).\end{aligned}}$

If the edge length of a regular dodecahedron is $a$ , the radius of a circumscribed sphere $r_{c}$ (one that touches the regular dodecahedron at all vertices), the radius of an inscribed sphere $r_{i}$ (tangent to each of the regular dodecahedron's faces), and the midradius $r_{m}$ (one that touches the middle of each edge) are:^[21] ${\begin{aligned}r_{c}&={\frac {\phi {\sqrt {3}}}{2}}a\approx 1.401a,\\r_{i}&={\frac {\phi ^{2}}{2{\sqrt {3-\phi }}}}a\approx 1.114a,\\r_{m}&={\frac {\phi ^{2}}{2}}a\approx 1.309a.\end{aligned}}$ Note that, given a regular dodecahedron of edge length one, $r_{c}$ is the radius of a circumscribing sphere about a cube of edge length $\phi$ , and $r_{i}$ is the apothem of a regular pentagon of edge length $\phi$ .

The dihedral angle of a regular dodecahedron between every two adjacent pentagonal faces is $2\arctan(\phi )$ , approximately 116.565°.

Other related geometric objects

The regular dodecahedron can be interpreted as a truncated trapezohedron. It is the set of polyhedrons that can be constructed by truncating the two axial vertices of a trapezohedron. Here, the regular dodecahedron is constructed by truncating the pentagonal trapezohedron.

The regular dodecahedron can be interpreted as the Goldberg polyhedron. 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 chamfer. 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.^[22]

The stellations of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra. The first stellation of a regular dodecahedron is constructed by attaching its layer with pentagonal pyramids, forming a small stellated dodecahedron. The second stellation is by attaching the small stellated dodecahedron with wedges, forming a great dodecahedron. The third stellation is by attaching the great dodecahedron with the sharp triangular pyramids, forming a great stellated dodecahedron.^[23]

Appearances

In visual arts

Roman dodecahedron

Regular dodecahedra have been used as dice and probably also as divinatory devices. During the Hellenistic era, small hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe.^[24]^[25] Its purpose is not certain.

In 20th-century art, dodecahedra appears in the work of M. C. Escher, such as his lithographs Reptiles (1943) and Gravitation (1952). In Salvador Dalí's painting The Sacrament of the Last Supper (1955), the room is a hollow regular dodecahedron. Gerard Caris based his entire artistic oeuvre on the regular dodecahedron and the pentagon, presented as a new art movement coined as Pentagonism.

In toys and popular culture

In modern role-playing games, the regular dodecahedron is often used as a twelve-sided die, one of the more common polyhedral dice. The Megaminx twisty puzzle is shaped like a regular dodecahedron alongside its larger and smaller order analogues.

In the children's novel The Phantom Tollbooth, the regular dodecahedron appears as a character in the land of Mathematics. Each face of the regular dodecahedron describes the various facial expressions, swiveling to the front as required to match his mood.^{[citation needed]}

In nature and supramolecules

The fossil record of the coccolithophore Braarudosphaera bigelowii goes back a hundred million years

The faces of a Holmium–magnesium–zinc (Ho-Mg-Zn) quasicrystal are true regular pentagons

Crystal structure of Co20L12 dodecahedron reported by Kai Wu, Jonathan Nitschke and co-workers at University of Cambridge in Nat. Synth. 2023, DOI:10.1038/s44160-023-00276-9^{^[26]}

The fossil coccolithophore Braarudosphaera bigelowii (see figure), a unicellular coastal phytoplanktonic alga, has a calcium carbonate shell with a regular dodecahedral structure about 10 micrometers across.^[27]

Some quasicrystals and cages have dodecahedral shape (see figure). Some regular crystals such as garnet and diamond are also said to exhibit "dodecahedral" habit, but this statement actually refers to the rhombic dodecahedron shape.^[28]^[26]

Shape of the universe

Various models have been proposed for the global geometry of the universe. These proposals include the Poincaré dodecahedral space, a positively curved space consisting of a regular dodecahedron whose opposite faces correspond (with a small twist). This was proposed by Jean-Pierre Luminet and colleagues in 2003,^[29]^[30] and an optimal orientation on the sky for the model was estimated in 2008.^[31]

In Bertrand Russell'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."^[32]

Dodecahedral graph

The dodecahedral graph's Hamiltonian property and the mathematical toy Icosian game

According to Steinitz's theorem, the graph can be represented as the skeleton of a polyhedron; roughly speaking, a framework of a polyhedron. Such a graph has two properties. It is planar, meaning the edges of a graph are connected to every vertex without crossing other edges. It is also three-connected graph, meaning that, whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected.^[33]^[34] The skeleton of a regular dodecahedron can be represented as a graph, and it is called the dodecahedral graph, a Platonic graph.^[35]

This graph can also be constructed as the generalized Petersen graph $G(10,2)$ , 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.^[36] 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.

The high degree of symmetry of the polygon is replicated in the properties of this graph, which are distance-transitive, distance-regular, and symmetric. The automorphism group has order a hundred and twenty. The vertices can be colored with 3 colors, as can the edges, and the diameter is five.^[37]

The dodecahedral graph is Hamiltonian, meaning a path visits all of its vertices exactly once. The name of this property is named after William Rowan Hamilton, who invented a mathematical game known as the icosian game. The game's object was to find a Hamiltonian cycle along the edges of a dodecahedron.^[38]

References

^ ^a ^b Sutton, Daud (2002). Platonic & Archimedean Solids. Wooden Books. Bloomsbury Publishing USA. p. 55. ISBN 9780802713865.
^ Herrmann, Diane L.; Sally, Paul J. (2013). Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory. Taylor & Francis. p. 252. ISBN 978-1-4665-5464-1.
^ Plato, Timaeus, Jowett translation [line 1317–8]; the Greek word translated as delineation is diazographein, painting in semblance of life.
^ Wildberg, Christian (1988). John Philoponus' Criticism of Aristotle's Theory of Aether. Walter de Gruyter. pp. 11–12. ISBN 9783110104462.
^ Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 57. ISBN 978-0-521-55432-9.
^ Livio (2003), p. 147.
^ Florian Cajori, A History of Mathematics (1893)
^ Erickson, Martin (2011). Beautiful Mathematics. Mathematical Association of America. p. 62. ISBN 978-1-61444-509-8.
^ Weils, David (1991). The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books. p. 57–58. ISBN 9780140118131.
^ Buker, W. E.; Eggleton, R. B. (1969). "The Platonic Solids (Solution to problem E2053)". American Mathematical Monthly. 76 (2): 192. doi:10.2307/2317282. JSTOR 2317282.
^ Herz-Fischler, Roger (2013). A Mathematical History of the Golden Number. Courier Dover Publications. pp. 138–140. ISBN 9780486152325.
^ Simmons, George F. (2007). Calculus Gems: Brief Lives and Memorable Mathematics. Mathematical Association of America. p. 50. ISBN 9780883855614.
^ Marar, Ton (2022). A Ludic Journey into Geometric Topology. Cham: Springer. p. 23. doi:10.1007/978-3-031-07442-4. ISBN 978-3-031-07442-4.
^ Coxeter, H.S.M.; du Val, Patrick; Flather, H. T.; Petrie, J. F. (1938). The Fifty-Nine Icosahedra. Vol. 6. University of Toronto Studies (Mathematical Series). p. 4.
^ Coxeter, H. S. M. (1973) [1948]. "§1.8 Configurations". Regular Polytopes (3rd ed.). New York: Dover Publications.
^ Coxeter, H. S. M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge: Cambridge University Press. p. 117.
^ Schielack, Vincent P. (1987). "The Fibonacci Sequence and the Golden Ratio". The Mathematics Teacher. 80 (5): 357–358. doi:10.5951/MT.80.5.0357. JSTOR 27965402. This source contains an elementary derivation of the golden ratio's value.
^ Peters, J. M. H. (1978). "An Approximate Relation between π and the Golden Ratio". The Mathematical Gazette. 62 (421): 197–198. doi:10.2307/3616690. JSTOR 3616690. S2CID 125919525.
^ Livio, Mario (2003) [2002]. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). New York City: Broadway Books. pp. 70–71. ISBN 0-7679-0816-3.
^ Paeth, Alan W. (1991). "Exact Dihedral Metric for Common Polyhedra". In Arvo, James (ed.). Graphics Gems II. Academic Press. p. 177. Bibcode:1991grge.book.....A.
^ Coxeter (1973) Table I(i), pp. 292–293. See the columns labeled ${}_{0}\!\mathrm {R} /\ell$ , ${}_{1}\!\mathrm {R} /\ell$ , and ${}_{2}\!\mathrm {R} /\ell$ , Coxeter's notation for the circumradius, midradius, and inradius, respectively, also noting that Coxeter uses $2\ell$ as the edge length (see p. 2).
^ Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie (ed.). Shaping Space (2nd ed.). Springer. p. 127. doi:10.1007/978-0-387-92714-5_9. ISBN 978-0-387-92713-8.
^ Cromwell (1997), p. 265.
^ Guggenberger, Michael (2013). "The Gallo-Roman Dodecahedron". The Mathematical Intelligencer. 35 (4). Springer Science and Business Media LLC: 56–60. doi:10.1007/s00283-013-9403-7. ISSN 0343-6993. S2CID 122337773.
^ Hill, Christopher (1994). "Gallo-Roman Dodecahedra: A Progress Report". The Antiquaries Journal. 74. Cambridge University Press: 289–292. doi:10.1017/s0003581500024458. ISSN 0003-5815. S2CID 161691752.
^ ^a ^b Kai Wu; Jonathan Nitschke (2023). "Systematic construction of progressively larger capsules from a fivefold linking pyrrole-based subcomponent". Nature Synthesis. 2 (8): 789. Bibcode:2023NatSy...2..789W. doi:10.1038/s44160-023-00276-9.
^ Hagino, K., Onuma, R., Kawachi, M. and Horiguchi, T. (2013) "Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-A in Braarudosphaera bigelowii (Prymnesiophyceae)". PLoS One, 8(12): e81749. doi:10.1371/journal.pone.0081749.
^ Dodecahedral Crystal Habit Archived 12 April 2009 at the Wayback Machine
^ Dumé, Belle (Oct 8, 2003). "Is The Universe A Dodecahedron?". PhysicsWorld. Archived from the original on 2012-04-25.
^ Luminet, Jean-Pierre; 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". Nature. 425 (6958): 593–5. arXiv:astro-ph/0310253. Bibcode:2003Natur.425..593L. doi:10.1038/nature01944. PMID 14534579. S2CID 4380713.
^ Roukema, Boudewijn; Zbigniew Buliński; Agnieszka Szaniewska; Nicolas E. Gaudin (2008). "A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data". Astronomy and Astrophysics. 482 (3): 747. arXiv:0801.0006. Bibcode:2008A&A...482..747L. doi:10.1051/0004-6361:20078777. S2CID 1616362.
^ Russell, Bertrand. "Nightmares of Eminent Persons and Other Stories". Internet Archive. Retrieved 10 November 2024.
^ Grünbaum, Branko (2003), "13.1 Steinitz's theorem", Convex Polytopes, Graduate Texts in Mathematics, vol. 221 (2nd ed.), Springer-Verlag, pp. 235–244, ISBN 0-387-40409-0
^ Ziegler, Günter M. (1995). "Chapter 4: Steinitz' Theorem for 3-Polytopes". Lectures on Polytopes. Graduate Texts in Mathematics. Vol. 152. Springer-Verlag. pp. 103–126. ISBN 0-387-94365-X.
^ Rudolph, Michael (2022). The Mathematics of Finite Networks: An Introduction to Operator Graph Theory. Cambridge University Press. p. 25. doi:10.1007/9781316466919 (inactive 1 November 2024). ISBN 9781316466919.{{cite book}}: CS1 maint: DOI inactive as of November 2024 (link)
^ Pisanski, Tomaž; Servatius, Brigitte (2013). Configuration from a Graphical Viewpoint. Springer. p. 81. doi:10.1007/978-0-8176-8364-1. ISBN 978-0-8176-8363-4.
^ Weisstein, Eric W. "Dodecahedral Graph". MathWorld.
^ Bondy, J. A.; Murty, U. S. R. (1976), Graph Theory with Applications, North Holland, p. 53, ISBN 0-444-19451-7

External links

Weisstein, Eric W. "Regular Dodecahedron". MathWorld.
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
Dodecahedron – 3-d model that works in your browser
Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- VRML#Regular dodecahedron
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
The Greek, Indian, and Chinese Elements – Seven Element Theory

[sutton-1] Sutton, Daud (2002). Platonic & Archimedean Solids. Wooden Books. Bloomsbury Publishing USA. p. 55. ISBN 9780802713865.

[hs-2] Herrmann, Diane L.; Sally, Paul J. (2013). Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory. Taylor & Francis. p. 252. ISBN 978-1-4665-5464-1.

[3] Plato, Timaeus, Jowett translation [line 1317–8]; the Greek word translated as delineation is diazographein, painting in semblance of life.

[wildberg-4] Wildberg, Christian (1988). John Philoponus' Criticism of Aristotle's Theory of Aether. Walter de Gruyter. pp. 11–12. ISBN 9783110104462.

[cromwell-5] Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 57. ISBN 978-0-521-55432-9.

[FOOTNOTELivio2003[httpsbooksgooglecombooksidbUARfgWRH14CpgPA147_147]-6] Livio (2003), p. 147.

[7] Florian Cajori, A History of Mathematics (1893)

[erickson-8] Erickson, Martin (2011). Beautiful Mathematics. Mathematical Association of America. p. 62. ISBN 978-1-61444-509-8.

[weils-9] Weils, David (1991). The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books. p. 57–58. ISBN 9780140118131.

[be-10] Buker, W. E.; Eggleton, R. B. (1969). "The Platonic Solids (Solution to problem E2053)". American Mathematical Monthly. 76 (2): 192. doi:10.2307/2317282. JSTOR 2317282.

[herz-11] Herz-Fischler, Roger (2013). A Mathematical History of the Golden Number. Courier Dover Publications. pp. 138–140. ISBN 9780486152325.

[simmons-12] Simmons, George F. (2007). Calculus Gems: Brief Lives and Memorable Mathematics. Mathematical Association of America. p. 50. ISBN 9780883855614.

[marar-13] Marar, Ton (2022). A Ludic Journey into Geometric Topology. Cham: Springer. p. 23. doi:10.1007/978-3-031-07442-4. ISBN 978-3-031-07442-4.

[coxeter-1938-14] Coxeter, H.S.M.; du Val, Patrick; Flather, H. T.; Petrie, J. F. (1938). The Fifty-Nine Icosahedra. Vol. 6. University of Toronto Studies (Mathematical Series). p. 4.

[coxeter-1973-15] Coxeter, H. S. M. (1973) [1948]. "§1.8 Configurations". Regular Polytopes (3rd ed.). New York: Dover Publications.

[coxeter-1991-16] Coxeter, H. S. M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge: Cambridge University Press. p. 117.

[schielack-17] Schielack, Vincent P. (1987). "The Fibonacci Sequence and the Golden Ratio". The Mathematics Teacher. 80 (5): 357–358. doi:10.5951/MT.80.5.0357. JSTOR 27965402. This source contains an elementary derivation of the golden ratio's value.

[peters-18] Peters, J. M. H. (1978). "An Approximate Relation between π and the Golden Ratio". The Mathematical Gazette. 62 (421): 197–198. doi:10.2307/3616690. JSTOR 3616690. S2CID 125919525.

[livio-19] Livio, Mario (2003) [2002]. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). New York City: Broadway Books. pp. 70–71. ISBN 0-7679-0816-3.

[paeth-20] Paeth, Alan W. (1991). "Exact Dihedral Metric for Common Polyhedra". In Arvo, James (ed.). Graphics Gems II. Academic Press. p. 177. Bibcode:1991grge.book.....A.

[radii-21] Coxeter (1973) Table I(i), pp. 292–293. See the columns labeled ${}_{0}\!\mathrm {R} /\ell$ , ${}_{1}\!\mathrm {R} /\ell$ , and ${}_{2}\!\mathrm {R} /\ell$ , Coxeter's notation for the circumradius, midradius, and inradius, respectively, also noting that Coxeter uses $2\ell$ as the edge length (see p. 2).

[hart-22] Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie (ed.). Shaping Space (2nd ed.). Springer. p. 127. doi:10.1007/978-0-387-92714-5_9. ISBN 978-0-387-92713-8.

[FOOTNOTECromwell1997[httpsbooksgooglecombooksidOJowej1QWpoCpgPA265_265]-23] Cromwell (1997), p. 265.

[guggenberger-24] Guggenberger, Michael (2013). "The Gallo-Roman Dodecahedron". The Mathematical Intelligencer. 35 (4). Springer Science and Business Media LLC: 56–60. doi:10.1007/s00283-013-9403-7. ISSN 0343-6993. S2CID 122337773.

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