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.

Areas, Angles, and Polyhedra

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Eighteenth Century ◽  
Unit Sphere ◽  
Regular Polyhedron ◽  
Spherical Triangle ◽  
French Mathematician ◽  
Regular Polyhedra ◽  
Leonhard Euler ◽  
Spherical Triangles

This chapter explains how to find the area of an angle or polyhedron. It begins with a discussion of how to determine the area of a spherical triangle or polygon. The formula for the area of a spherical triangle is named after Albert Girard, a French mathematician who developed a theorem on the areas of spherical triangles, found in his Invention nouvelle. The chapter goes on to consider Euler's polyhedral formula, named after the eighteenth-century mathematician Leonhard Euler, and the geometry of a regular polyhedron. Finally, it describes an approach to finding the proportion of the volume of the unit sphere that the various regular polyhedra occupy.

Three Facially-Regular Polyhedra

1950 ◽  
Vol 2 ◽  
pp. 326-330
Author(s):  
Hugh Apsimon
Keyword(s):  
Regular Polyhedra ◽  
Definition Of

H. S. M. Coxeter has shown the existence of three infinite regular polyhedra, and has proved that there are no infinite regular polyhedra other than these. In his paper he gives the definition of regularity of a polyhedron :A polyhedron is said to be regular if it possesses two particular symmetries: one which cyclically permutes the vertices ox any face c, and one which cyclically permutes the faces that meet at a vertex C, C being a vertex of c.

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.

The complete set of uniform polyhedra

1975 ◽  
Vol 278 (1278) ◽  
pp. 111-135 ◽  
Keyword(s):  
Natural Extension ◽  
Regular Polygons ◽  
Complete Set ◽  
Definition Of ◽  
Uniform Polyhedron

A definitive enumeration of all uniform polyhedra is presented (a uniform polyhedron has all vertices equivalent and all its faces regular polygons). It is shown that the set of uniform polyhedra presented by Coxeter, Longuet-Higgins & Miller (1953) is essentially complete. However, after a natural extension of their definition of a polyhedron to allow the coincidence of two or more edges, one extra polyhedron is found, namely the great disnub dirhombidodecahedron.

JOINT SEPARATION OF GEOMETRIC CLUSTERS AND THE EXTREME IRREGULARITIES OF REGULAR POLYHEDRA

2005 ◽  
Vol 15 (05) ◽  
pp. 491-510
Author(s):  
SUMANTA GUHA
Keyword(s):  
Regular Polyhedron ◽  
Finite Plane ◽  
Finite Sets ◽  
Graph Theoretic ◽  
Regular Polyhedra ◽  
Joint Separation

We propose a natural scheme to measure the joint separation of a cluster of objects in general geometric settings. In particular, here the measure is developed for finite sets of planes in ℝ3 in terms of extreme configurations of vectors on the planes of a given set. We prove geometric and graph-theoretic results about extreme configurations on arbitrary finite plane sets. We then specialize to the planes bounding a regular polyhedron in order to exploit the symmetries. However, even then results are non-trivial and surprising – extreme configurations on regular polyhedra may turn out to be highly irregular.

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.

THE ORIENTATION BEHAVIOR OF HORSE FLIES AND DEER FLIES (TABANIDAE: DIPTERA): V. THE INFLUENCE OF THE NUMBER AND INCLINATION OF REFLECTING SURFACES ON ATTRACTIVENESS TO TABANIDS OF GLOSSY BLACK POLYHEDRA

1966 ◽  
Vol 44 (2) ◽  
pp. 275-279 ◽  
Author(s):  
A. J. Thorsteinson ◽  
G. K. Bracken ◽  
W. Tostowaryk
Keyword(s):  
Regular Polyhedron ◽  
The Other ◽  
Orientation Behavior ◽  
Regular Polyhedra ◽  
Horse Flies ◽  
Reflecting Surfaces

The orientation of tabanids to glossy black target figures of various geometrical forms was investigated. A maximum number of tabanids is attracted by a sphere and by the regular polyhedron that most closely approximates to the sphere in number of reflecting surfaces (icosahedron). The attractiveness of the other regular polyhedra increases with the number of sides, provided they are suspended so that the inclination of the faces causes sunlight to be reflected into the presumed flight lanes of approaching tabanids. These results were confirmed in experiments with pyramidal targets varying in number of sides and inclination of reflecting faces.

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.

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