n-dimensional enrichment for Further Mathematicians

2004 ◽  
Vol 89 (516) ◽  
pp. 409-416 ◽  
Author(s):  
Martin Griffiths
Keyword(s):  
Higher Dimensions ◽  
Three Dimensions ◽  
Regular Tetrahedron ◽  
Regular Polygons ◽  
Regular Polytopes ◽  
Platonic Solids ◽  
Mathematical Objects ◽  
Regular Octahedron ◽  
Regular Polyhedra

There are infinitely many regular polygons, but we find, on extending the idea of polygons to three dimensions, that there are only five regular polyhedra, the Platonic solids. What happens then if we try to extend this idea beyond three dimensions? It turns out that, of the five Platonic solids, just the regular tetrahedron, cube and regular octahedron have analogues in all higher dimensions, the so-called regular polytopes. Brief descriptions of these mathematical objects are to be found in [1], for example.

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.

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 ).

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.

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.

scholarly journals The correlation functions of the regular tetrahedron and octahedron

2014 ◽  
Vol 47 (4) ◽  
pp. 1445-1448 ◽  
Author(s):  
Salvino Ciccariello
Keyword(s):  
Correlation Functions ◽  
Rational Functions ◽  
Regular Tetrahedron ◽  
Numerical Determination ◽  
Algebraic Form ◽  
Autocorrelation Functions ◽  
Regular Octahedron ◽  
Size Dispersion ◽  
Arctangent Function

The expressions of the autocorrelation functions (CFs) of the regular tetrahedron and the regular octahedron are reported. They have an algebraic form that involves the arctangent function and rational functions of r and (a + br 2)1/2, a and b being appropriate integers and r a distance. The CF expressions make the numerical determination of the corresponding scattering intensities fast and accurate even in the presence of a size dispersion.

Structure of Liquid–Vapor Interfaces in the Ising Model

1997 ◽  
Vol 08 (03) ◽  
pp. 583-588 ◽  
Author(s):  
L. L. Moseley
Keyword(s):  
Computer Simulation ◽  
Asymptotic Behavior ◽  
Ising Model ◽  
Density Profile ◽  
Error Function ◽  
Higher Dimensions ◽  
Three Dimensions ◽  
Fluid Interface ◽  
Hyperbolic Tangent ◽  
Structure Of Liquid

The asymptotic behavior of the density profile of the fluid-fluid interface is investigated by computer simulation and is found to be better described by the error function than by the hyperbolic tangent in three dimensions. For higher dimensions the hyperbolic tangent is a better approximation.

Pyramids, Prisms, Antiprisms, and Deltahedra

1988 ◽  
Vol 81 (4) ◽  
pp. 261-265
Author(s):  
Donovan R. Lichtenberg
Keyword(s):  
Mathematics Teacher ◽  
Platonic Solids ◽  
Regular Polyhedra ◽  
Archimedean Solids

Each of the nine covers of the Mathematics Teacher for 1985 contained pictures of two polyhedra. The covers for January through May showed the five regular polyhedra, or Platonic solids, along with their truncated versions. The latter are semiregular polyhedra, or Archimedean solids. For the months of September through December the covers displayed the remaining eight Archimedean solids.

Critical intensities of Boolean models with different underlying convex shapes

2002 ◽  
Vol 34 (01) ◽  
pp. 48-57
Author(s):  
Rahul Roy ◽  
Hideki Tanemura
Keyword(s):  
Unit Area ◽  
Higher Dimensions ◽  
Boolean Model ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Boolean Models ◽  
Regular Polytope ◽  
Minimum Value ◽  
Critical Intensity ◽  
Poisson Boolean Model

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

Asymptotic properties of the equilibrium probability of identity in a geographically structured population

1977 ◽  
Vol 9 (2) ◽  
pp. 268-282 ◽  
Author(s):  
Stanley Sawyer
Keyword(s):  
Nearest Neighbor ◽  
Asymptotic Properties ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Order Zero ◽  
Structured Population ◽  
Equilibrium Probability ◽  
Migration Law

Let I(x, u) be the probability that two genes found a vector distance x apart are the same type in an infinite-allele selectively-neutral migration model with mutation rate u. The creatures involved inhabit an infinite of colonies, are diploid and are held at N per colony. Set in one dimension and in higher dimensions, where σ2 is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Then in one dimension, in two dimensions, and in three dimensions uniformly for Here C0 is a constant depending on the migration law, K0(y) is the Bessel function of the second kind of order zero, and are the eigenvalues of σ2. For symmetric nearest-neighbor migrations, in one dimension and log mi in two. For is known in one dimension and C0 does not appear. In two dimensions, These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.

Sign in / Sign up

Export Citation Format

Share Document