The Dodecahedron, having twelve regular pentagons (or five-sided figures) as faces.
"Bygone Beliefs" by H. Stanley Redgrove
Associated Words: dodecagon, dodecahedral, dodecahedron, duodecimal, duodecimfid, duodecimo, duodecuple, duodenary.
"Putnam's Word Book" by Louis A. Flemming
Two of these generate the cube and the octahedron; five of these generate the dodecahedron and the icosahedron.
"Occult Chemistry" by Annie Besant and Charles W. Leadbeater
You might as well call it a dodecahedron, or the cube root of minus nothing.
"'That Very Mab'" by May Kendall and Andrew Lang
Round it describe a dodecahedron; the circle including this will be Mars.
"The Martyrs of Science, or, The lives of Galileo, Tycho Brahe, and Kepler" by David Brewster
About it circumscribe a dodecahedron.
"The Astronomy of Milton's 'Paradise Lost'" by Thomas Orchard
The usual forms are the cube, octahedron and pentagonal dodecahedron {210}.
"Encyclopaedia Britannica, 11th Edition, Volume 6, Slice 5" by Various
The "rhombic dodecahedron," one of the geometrical semiregular solids, is an important crystal form.
"Encyclopaedia Britannica, 11th Edition, Volume 8, Slice 5" by Various
Similarly earth is composed {38} of cubes, and the universe is identified with the dodecahedron.
"A Critical History of Greek Philosophy" by W. T. Stace
Cleavage obtains parallel to the dodecahedron, but is imperfect.
"Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 4" by Various
That for the tetrahedron has 12 for its order, for the cube (or octahedron) 24, and for the icosahedron (or dodecahedron) 60.
"Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6" by Various
A drawing of a crystal showing a combination of the cube, octahedron and rhombic dodecahedron is shown in fig.
"Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 7" by Various
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Suppose S is a dodecahedron and P has edge length of either 2 or 3.
Regular Polyhedra of Index Two, I
Only when S is a dodecahedron, do ordered pairs {u, v} of vertices of S exist that cannot be taken under the action of G+ (S ) = G+ (P ) to every other ordered pair of vertices of S the same distance apart, and then the length of {u, v} must be 2 or 3.
Regular Polyhedra of Index Two, I
If S is a tetrahedron (and the edge length is 1) or a dodecahedron (and the edge length is 1 or 4), then q = 3 and there are only two directions in which any face boundary of P , when pro jected on the circumsphere S of S , can continue at any vertex.
Regular Polyhedra of Index Two, I
If S is a tetrahedron or a dodecahedron, q = 3 and so there are only two directions in which any face boundary of P can continue at any vertex.
Regular Polyhedra of Index Two, I
Figure 4: The four families of polyhedra of type {10, 3}10 with icosahedral symmetry; from left to right, derived by Lemma 7.3 from the Petrie-dual of the dodecahedron, the dodecahedron itself, the great stellated dodecahedron, and the Petrie-dual of the great stellated dodecahedron.
Regular Polyhedra of Index Two, I
Here I discuss several data and references freely available on the Web. After analysis, the common features of these artifacts allow to tell that a Roman Dodecahedron was probably a dioptron.
Roman Dodecahedron as dioptron: analysis of freely available data
B is its baseline. Dα,Dα‘ are the diameters of the two opposite holes of pair (α,α’). In [1,2], data of a dodecahedron from Jublains, France, were used: I found these data because the reference was easily available on the Web, under a Google Search in English.
Roman Dodecahedron as dioptron: analysis of freely available data
Since I have not the precise value for each pair, I suppose N as a constant in this table and in all the following tables. For the Jublains dodecahedron, B = (100 ±4) u.
Roman Dodecahedron as dioptron: analysis of freely available data
Avenches, Suisse It is a dodecahedron having a diameter of 58.5 cm, and then distance B of 46.5 mm.
Roman Dodecahedron as dioptron: analysis of freely available data
Tables I-VII. Integer N depends on the used dodecahedron.
Roman Dodecahedron as dioptron: analysis of freely available data
The dodecahedron of Jublians has a decoration composed of three circles. Each pair of opposite holes of the artifact of Jublains has the same decoration, as shown in the following image (Fig.2) arranged from an image of Ref.3.
Roman Dodecahedron as dioptron: analysis of freely available data
On the right, we see the faces of a dodecahedron found at Jublains . In the lower part of the image, an angle of view for the distance measurement is shown.
Roman Dodecahedron as dioptron: analysis of freely available data
It turns out, that Snub Cube, Snub Dodecahedron, P y r(βd−1) and BP y r(αd−1) are the only polytopes in Tables of this paper, having self-intersecting zigzags.
Zigzag structure of complexes
For example, Petersen graph, embedded on the pro jective plane, is a folding of the Dodecahedron by central inversion.
Zigzag structure of complexes
By applying the Wythoff construction to the three 3-valent Platonic solids (Tetrahedron, Cube and Dodecahedron) one obtains all Archimedean 3-polytopes, except Snub Cube and Snub Dodecahedron; their z -structure is indicated in columns 2 and 3 of Table 2.
Zigzag structure of complexes
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