Notes on the Regular Icosahedron and the Regular Dodecahedron

1942 ◽  
Vol 26 (270) ◽  
pp. 153
Author(s):  
W. Hope-Jones ◽  
Lester S. Hill
Keyword(s):  
Regular Dodecahedron ◽  
Regular Icosahedron

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.

Dodecahedral slices and polyhedral pieces

2010 ◽  
Vol 94 (529) ◽  
pp. 5-17
Author(s):  
Doug French ◽  
David Jordan
Keyword(s):  
Icosahedral Symmetry ◽  
Surprising Result ◽  
Regular Pentagon ◽  
Regular Dodecahedron ◽  
Regular Icosahedron ◽  
Image Position

Figure 1 shows how a regular dodecahedron can be dissected into three slices by two planes through the two sets of vertices, each set defining a regular pentagon parallel to the top and bottom faces. A surprising result emerges if we calculate the ratio of the volumes of the three slices. We first prove this result directly and then show it by a dissection argument using simple polyhedral pieces of five types. These pieces can be used to build many polyhedra, including the regular dodecahedron, the regular icosahedron, the great dodecahedron, the small and great stellated dodecahedra and all the Archimedean polyhedra which have icosahedral symmetry.

scholarly journals Toric Geometry of the Regular Convex Polyhedra

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Fiammetta Battaglia ◽  
Elisa Prato
Keyword(s):  
Convex Polyhedra ◽  
Toric Geometry ◽  
Regular Tetrahedron ◽  
Regular Octahedron ◽  
Regular Dodecahedron ◽  
Regular Convex ◽  
Regular Icosahedron

We describe symplectic and complex toric spaces associated with the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational, and the regular icosahedron is neither simple nor rational. We remark that the last two cases cannot be treated via standard toric geometry.

Theoretical Study of Iron Heterogeneous Growth on the Surface of C60 Molecule

2014 ◽  
Vol 1081 ◽  
pp. 115-118
Author(s):  
Qi Cheng Liu ◽  
Yun Fang Li
Keyword(s):  
Crystallization Process ◽  
Theoretical Study ◽  
Covalent Bond ◽  
Md Simulations ◽  
Nucleation And Growth ◽  
Bond Forming ◽  
Regular Icosahedron ◽  
Molecule Dynamics ◽  
Molecule Surface ◽  
Regular Closed

Direct molecule dynamics (MD) simulations have also been performed to study heterogeneous nucleation and growth of iron on C60 molecule. The grown mechanism of this crystallization process was explored. The results indicate that 92 iron atoms attach to C60 molecule surface can form new covalent bond, forming a closed regular icosahedron. More atoms grow in layer to form bigger regular closed clathrate base on the structure of former one. As increase of atoms number, there will appear some crystal faces.

scholarly journals A Non-Regular Icosahedron Geometry Satellite Form, Mirror Invested Polyhedro Heliotrope, For Optical Tracking

2019 ◽  
Vol 2 (2) ◽  
Keyword(s):  
Optical Tracking ◽  
Square Root ◽  
Common Ratio ◽  
Golden Number ◽  
Equilateral Triangles ◽  
Regular Icosahedron

Working on relationships of three circles in common ratio [4/π or square root of the golden number ] and drawing lines of related tangents, squares and triangles, viewed on the paper plan, a figure having the shape of a section [Hexagonal] similar to that of an Icosahedron or Dodecahedron. This gave me the idea of searching for an existing probable Polyhedron built upon this traced shape. In fact this Polyhedron was built[ 4x scale], whose geometry relates to the Icosahedron and the Dodecahedron. It is a non regular Icosahedron having 12 Isosceli triangles and 8 Equilateral triangles. Mirror triangles cut to size, invested the structure for the configuration of a “Polyhedroheliotrope”Satellite Optical Tracking application.

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.

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