Billiard Trajectories in Regular Polygons and Geodesics on Regular Polyhedra

2021 ◽  
Author(s):  
Dmitry Fuchs
Keyword(s):  
Regular Polygons ◽  
Regular Polyhedra

Extremum Properties of the Regular Polyhedra

1950 ◽  
Vol 2 ◽  
pp. 22-31 ◽  
Author(s):  
Lâszlό Fejes Tόth
Keyword(s):  
Regular Polygons ◽  
Surface Areas ◽  
Extremum Problems ◽  
Regular Polyhedra ◽  
Historical Remarks

1. Historical remarks. In this paper we extend some well-known extremum properties of the regular polygons to the regular polyhedra. We start by mentioning some known results in this direction.First, let us briefly consider the problem which has received the greatest attention among all the extremum problems for polyhedra. It is the determination of the polyhedron of greatest volume F of a class of polyhedra of equal surface areas F, i.e., the isepiphan problem.

Historically Speaking— A Geometry Capsule Concerning the Five Platonic Solids

1969 ◽  
Vol 62 (1) ◽  
pp. 42-44
Author(s):  
Howard Eves
Keyword(s):  
Regular Polygons ◽  
Platonic Solids ◽  
Regular Polyhedra ◽  
Number Of Faces

A polyhedron is said to be “regular” if its faces are congruent regular polygons and its polyhedral angles are all congruent. While there are regular polygons of all orders, it is surprising that there are only five different regular polyhedra. These regular polyhedra have been named according to the number of faces each possesses. Thus there is the tetrahedron with four triangular faces, the hexahedron (cube) with six square faces, the octahedron with eight triangular faces, the dodecahedron with twelve pentagonal faces, and the icosahedron with twenty triangular faces. See the accompanying figure.

scholarly journals XXIX.—Semi-regular Networks of the Plane in Absolute Geometry

1906 ◽  
Vol 41 (3) ◽  
pp. 725-747 ◽  
Author(s):  
Duncan M. Y. Sommerville
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Geometry ◽  
Regular Polygons ◽  
Ordinary Space ◽  
Absolute Geometry ◽  
Regular Polyhedra ◽  
Elliptic Geometry ◽  
Regular Networks

§ 1. The networks considered in the following paper are those networks of the plane whose meshes are regular polygons with the same length of side.When the polygons are all of the same kind the network is called regular, otherwise it is semi-regular.The regular networks have been investigated for the three geometries from various standpoints, the chief of which may be noted.1. The three geometries can be treated separately. For Euclidean geometry we have then to find what regular polygons will exactly fill up the space round a point. For elliptic geometry we have to find the regular divisions of the sphere, or, what is the same thing, the regular polyhedra in ordinary space. The regular networks which do not belong to either of these classes are then those of the hyperbolic plane.

5. Tilings and polyhedra

2016 ◽  
pp. 72-85
Author(s):  
Robin Wilson
Keyword(s):  
Three Dimensional ◽  
Regular Polygons ◽  
Regular Polyhedra ◽  
Different Types

A tiling of the plane (or tessellation) is a covering of the whole plane with tiles so that no tiles overlap and there are no gaps. Polyhedra are three-dimensional solids that are bounded by plane faces. How many are there, and can we construct and classify them? ‘Tilings and polyhedra’ describes the different types of tilings and polyhedra that are possible, beginning with regular tilings made up of regular polygons, semi-regular tilings, and irregular tilings. There are only five types of regular polyhedra—the tetrahedron, cube (or hexahedron), octahedron, dodecahedron, and icosahedron—but there are numerous semi-regular polyhedra, including prisms and antiprisms.

Hanging Around with Platonic Solids

2019 ◽  
Vol 112 (5) ◽  
pp. 328-329
Author(s):  
Günhan Caglayan
Keyword(s):  
Regular Polygons ◽  
Platonic Solids ◽  
Regular Polyhedra

The Platonic solids, also known as the five regular polyhedra, are the five solids whose faces are congruent regular polygons of the same type. Polyhedra is plural for polyhedron, derived from the Greek poly + hedros, meaning “multi-faces.” The five Platonic solids include the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. Photographs 1a-d show several regular polyhedra

scholarly journals Stereographic Visualization of 5-Dimensional Regular Polytopes

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 391
Author(s):  
Xingchang Wang ◽  
Tao Yu ◽  
Kwokwai Chung ◽  
Krzysztof Gdawiec ◽  
Peichang Ouyang
Keyword(s):  
Stereographic Projection ◽  
General Strategy ◽  
Two Dimensional ◽  
Regular Polygons ◽  
Regular Polytopes ◽  
Edge Information ◽  
Topological Information ◽  
Regular Polyhedra ◽  
Perfect Symmetry ◽  
Topological Data

Regular polytopes (RPs) are an extension of 2D (two-dimensional) regular polygons and 3D regular polyhedra in n-dimensional ( n ≥ 4 ) space. The high abstraction and perfect symmetry are their most prominent features. The traditional projections only show vertex and edge information. Although such projections can preserve the highest degree of symmetry of the RPs, they can not transmit their metric or topological information. Based on the generalized stereographic projection, this paper establishes visualization methods for 5D RPs, which can preserve symmetries and convey general metric and topological data. It is a general strategy that can be extended to visualize n-dimensional RPs ( n > 5 ).

scholarly journals VIII. (I.) On a tangential property of regular hypoeycloids and epicycloids. (II.) On theorems relating to the regular polyhedra which are analogous to those of Dr. Matthew Stewart on the regular polygons

1883 ◽  
Vol 34 (220-223) ◽  
pp. 105-112
Keyword(s):  
Regular Polygons ◽  
Regular Polyhedra ◽  
Tangential Property

1. The following theorems will be established :— (A.) The product of the perpendiculars drawn from all the cusps in each of these roulettes on any tangent is a function of the perpen­dicular only, which is drawn on the same tangent from the centre of the fixed circle, on which the roulette is generated.

Convex Polyhedra with Regular Faces

1966 ◽  
Vol 18 ◽  
pp. 169-200 ◽  
Author(s):  
Norman W. Johnson
Keyword(s):  
The Other ◽  
Convex Polyhedra ◽  
Regular Polygons ◽  
Platonic Solids ◽  
Geometric Figures ◽  
Regular Polyhedra ◽  
The Subject ◽  
Symmetry Operations

An interesting set of geometric figures is composed of the convex polyhedra in Euclidean 3-space whose faces are regular polygons (not necessarily all of the same kind). A polyhedron with regular faces is uniform if it has symmetry operations taking a given vertex into each of the other vertices in turn (5, p. 402). If in addition all the faces are alike, the polyhedron is regular.That there are just five convex regular polyhedra—the so-called Platonic solids—was proved by Euclid in the thirteenth book of the Elements (10, pp. 467-509). Archimedes is supposed to have described thirteen other uniform, “semi-regular” polyhedra, but his work on the subject has been lost.

Two Vertex-Regular Polyhedra

1951 ◽  
Vol 3 ◽  
pp. 269-271
Author(s):  
Hugh Apsimon
Keyword(s):  
Regular Polyhedron ◽  
Regular Polygons ◽  
Regular Polyhedra ◽  
Definition Of

The definition of a regular polyhedron may be enunciated as follows: (α) A polyhedron is said to be regular if its faces are equal regular polygons, and its vertex figures are equal regular polygons In a recent note1 I gave three examples of uniform non-regular polyhedra, which I called facially-regular, using the definition: (β) A polyhedron is said to be facially-regular if it is uniform and all its faces are equal.

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