97. The Use of the Pentagram in Constructing the Net for a Regular Dodecahedron

1962 ◽  
Vol 46 (358) ◽  
pp. 307
Author(s):  
E. M. Bishop
Keyword(s):  
Regular Dodecahedron

Random walks on a dodecahedron

1980 ◽  
Vol 17 (02) ◽  
pp. 373-384 ◽  
Author(s):  
G. Letac ◽  
L. Takács
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Transition Probabilities ◽  
Regular Dodecahedron

We consider the general Markov chain on the vertices of a regular dodecahedron D such that P[Xn +1 = j | Xn = i] depends only on the distance between i and j. We consider also a Markov chain on the oriented edges (i, j) of D for which the only non-zero transition probabilities are and fix a vertex A. This paper computes explicitly P[Xn = A | X 0 = A] and P[In = A | I 0 = A]. The methods used are applicable to other solids.

scholarly journals Symplectic Toric Geometry and the Regular Dodecahedron

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Elisa Prato
Keyword(s):  
Toric Geometry ◽  
Toric Manifold ◽  
Platonic Solids ◽  
Simple Polytope ◽  
Singular Space ◽  
Regular Dodecahedron

The regular dodecahedron is the only simple polytope among the platonic solids which is not rational. Therefore, it corresponds neither to a symplectic toric manifold nor to a symplectic toric orbifold. In this paper, we associate to the regular dodecahedron a highly singular space called symplectic toric quasifold.

Regular polyhedral molecules. P20 and its inclusion compounds

1998 ◽  
Vol 76 (9) ◽  
pp. 1274-1279 ◽  
Author(s):  
Lubomír Rulísek ◽  
Zdenek Havlas ◽  
Stanislav Hermánek ◽  
Jaromír Plesek
Keyword(s):  
Ab Initio Calculations ◽  
Ab Initio ◽  
Inclusion Compounds ◽  
Vibrational Analysis ◽  
Kinetic Stability ◽  
Stabilizing Agents ◽  
Geometrical Properties ◽  
Regular Dodecahedron ◽  
Phosphorus Clusters ◽  
High Level

Based upon the geometrical properties of regular polyhedrons, the possibility of the existence of certain polyhedral molecules composed of only one element is investigated. A very promising candidate - the regular dodecahedron - is selected as the convenient polyhedral structural pattern and phosphorus as the appropriate element. A series of high-level ab initio calculations is performed on the dodecahedral P20 molecule, including predictions of its thermodynamic and kinetic stability, natural bond orbital analysis, vibrational analysis, and inclusion of some elements into the molecular skeleton. Due to the potential stabilizing agents that may eventually form stable inclusion compounds and the estimated high kinetic stability, the question of the possible existence of P20 is answered in the positive.Key words: ab initio calculations, inclusion compounds, P20, phosphorus clusters, polyhedra.

Golden Section and Golden Rectangles When Building Icosahedron, Dodecahedron and Archimedean Solids Based On Them

2019 ◽  
Vol 7 (2) ◽  
pp. 47-55 ◽  
Author(s):  
В. Васильева ◽  
V. Vasil'eva
Keyword(s):  
Computer Modeling ◽  
Golden Section ◽  
Platonic Solids ◽  
Divine Proportion ◽  
Section Ratio ◽  
Archimedean Solids ◽  
Regular Dodecahedron ◽  
History Of ◽  
Dual Connection ◽  
Regular Icosahedron

A brief history of the development of the regular polyhedrons theory is given. The work introduces the reader to modelling of the two most complex regular polyhedrons – Platonic solids: icosahedron and dodecahedron, in AutoCAD package. It is suggested to apply the method of the icosahedron and dodecahedron building using rectangles with their sides’ ratio like in the golden section, having taken the icosahedron’s golden rectangles as a basis. This method is well-known-of and is used for icosahedron, but is extremely rarely applied to dodecahedron, as in the available literature it is suggested to build the latter one as a figure dual to icosahedron. The work provides information on the first mentioning of this building method by an Italian mathematician L. Pacioli in his Divine Proportion book. In 1937, a Soviet mathematician D.I. Perepelkin published a paper On One Building Case of the Regular Icosahedron and Regular Dodecahedron, where he noted that this “method is not very well known of” and provided a building based “solely on dividing an intercept in the golden section ratio”. Taking into account the simplicity and good visualization of the building based on golden rectangles, a computer modeling of icosahedron and dodecahedron inscribed in a regular hexahedron is performed in the article. Given that, if we think in terms of the golden section concepts, the bigger side of the rectangle equals a whole intercept – side of the regular hexahedron, and the smaller sides of the icosahedron and dodecahedron rectangles are calculated as parts of the golden section ratio (of the bigger part and the smaller one, respectively). It is demonstrated how, using the scheme of a wireframe image of the dual connection of these polyhedrons as a basis, to calculate the sides of the rectangles in the golden section ratio in order to build an “infinite” cascade of these dual figures, as well as to build the icosahedron and dodecahedron circumscribed about the regular hexahedron. The method based on using the golden-section rectangles is also applied to building semiregular polyhedrons – Archimedean solids: a truncated icosahedron, truncated dodecahedron, icosidodecahedron, rhombicosidodecahedron, and rhombitruncated icosidodecahedron, which are based on icosahedron and dodecahedron.

Isoperimetric Inequalities and the Dodecahedral Conjecture

1997 ◽  
Vol 08 (06) ◽  
pp. 759-780 ◽  
Author(s):  
Károly Bezdek
Keyword(s):  
Unit Sphere ◽  
Sphere Packing ◽  
Isoperimetric Inequalities ◽  
Sphere Packings ◽  
Voronoi Polyhedron ◽  
Regular Dodecahedron ◽  
Nonoverlapping Unit

The dodecahedrad conjecture, posed more than 50 years ago, says that the volume of any Voronoi polyhedron of a unit sphere packing in [Formula: see text] is at least as large as the volume of a regular dodecahedron of inradius 1. In this paper we show how the dodecahedral conjecture can be obtained from the distance conjecture of 14 and 15 nonoverlapping unit spheres and from the isoperimetric conjecture of Voronoi faces of unit sphere packings.

scholarly journals Model for the architecture of caveolae based on a flexible, net-like assembly of Cavin1 and Caveolin discs

2016 ◽  
Vol 113 (50) ◽  
pp. E8069-E8078 ◽  
Author(s):  
Miriam Stoeber ◽  
Pascale Schellenberger ◽  
C. Alistair Siebert ◽  
Cedric Leyrat ◽  
Ari Helenius ◽  
...  
Keyword(s):  
Plasma Membrane ◽  
Complex Formation ◽  
Membrane Binding ◽  
Coiled Coil ◽  
Structural Basis ◽  
Membrane Domains ◽  
Coiled Coil Domain ◽  
Regular Dodecahedron ◽  
Plasma Membrane Domains ◽  
Electron Cryotomography

Caveolae are invaginated plasma membrane domains involved in mechanosensing, signaling, endocytosis, and membrane homeostasis. Oligomers of membrane-embedded caveolins and peripherally attached cavins form the caveolar coat whose structure has remained elusive. Here, purified Cavin1 60S complexes were analyzed structurally in solution and after liposome reconstitution by electron cryotomography. Cavin1 adopted a flexible, net-like protein mesh able to form polyhedral lattices on phosphatidylserine-containing vesicles. Mutating the two coiled-coil domains in Cavin1 revealed that they mediate distinct assembly steps during 60S complex formation. The organization of the cavin coat corresponded to a polyhedral nano-net held together by coiled-coil segments. Positive residues around the C-terminal coiled-coil domain were required for membrane binding. Purified caveolin 8S oligomers assumed disc-shaped arrangements of sizes that are consistent with the discs occupying the faces in the caveolar polyhedra. Polygonal caveolar membrane profiles were revealed in tomograms of native caveolae inside cells. We propose a model with a regular dodecahedron as structural basis for the caveolae architecture.

Random walks on a dodecahedron

1980 ◽  
Vol 17 (2) ◽  
pp. 373-384 ◽  
Author(s):  
G. Letac ◽  
L. Takács
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Transition Probabilities ◽  
Regular Dodecahedron

We consider the general Markov chain on the vertices of a regular dodecahedron D such that P[Xn+1 = j | Xn = i] depends only on the distance between i and j. We consider also a Markov chain on the oriented edges (i, j) of D for which the only non-zero transition probabilities are and fix a vertex A. This paper computes explicitly P[Xn = A | X0 = A] and P[In = A | I0 = A]. The methods used are applicable to other solids.

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