scholarly journals Vertex-faithful regular polyhedra

2020 ◽  
Vol 343 (10) ◽  
pp. 112013
Author(s):  
Gabe Cunningham ◽  
Mark Mixer
Keyword(s):  
Regular Polyhedra

On regular polyhedra with hidden symmetries

1987 ◽  
Vol 12 (3-4) ◽  
pp. 450-458
Author(s):  
Jörg M. Wills
Keyword(s):  
Hidden Symmetries ◽  
Regular Polyhedra

Regular polyhedra—old and new

1977 ◽  
Vol 16 (1-2) ◽  
pp. 1-20 ◽  
Author(s):  
Branko Grünbaum
Keyword(s):  
Regular Polyhedra

scholarly journals PRODUCING DESIGN OBJECTS FROM REGULAR POLYHEDRA: A PRACTICAL APPROACH

2018 ◽  
pp. 49-55
Author(s):  
Beniamino Polimeni
Keyword(s):  
3D Printing ◽  
Creative Process ◽  
Practical Approach ◽  
Complex Geometries ◽  
Regular Polyhedra ◽  
Modelling Techniques ◽  
Digital Modelling ◽  
Practical Guidance ◽  
Design Objects ◽  
Architecture And Design

In the last few years, digital modelling techniques have played a major role in architecture and design, influencing, at the same time, the creative process and the fabrication of objects. This revolution has produced a new productive generation of architects and designers focused on the expanding possibilities of material and formal production, reinforcing the idea of architecture as an interaction between art and artisanship. This original perspective inspires this paper, which illustrates the contemporary scenario and provides some practical guidance about tools and technologies the designers most often use for creating geometric sculptures with 3D printing. Creative possibilities of topological mesh modelling are used to generate complex geometries from regular polyhedra. This process explores how combining different geometric operations can activate architectural inquiry and generate fascinating shapes with creative flexibility.

scholarly journals DISTRIBUTION OF MAXIMA OF THE ANTISYMMETRIZED WAVE FUNCTION FOR THE NUCLEONS OF A CLOSED-SHELL AND FOR THE NUCLEONS OF ALL CLOSED-SHELLS IN A NUCLEUS.

2020 ◽  
Vol 15 ◽  
pp. 57
Author(s):  
G. S. Anagnostatos
Keyword(s):  
Wave Function ◽  
Mathematical Problem ◽  
Closed Shell ◽  
Wave Functions ◽  
Pictorial Representation ◽  
One Dimensional ◽  
Regular Polyhedra ◽  
Closed Shell Nuclei ◽  
Simple Systems ◽  
Closed Shells

The significant features of exchange symmetry are displayed by simple systems such as two identical, spinless fermions in a one-dimensional well with infinite walls. The conclusion is that the maxima of probability of the antisymmetrized wave function of these two fermions lie at the same positions as if a repulsive force (of unknown nature) was applied between these two fermions. This conclusion is combined with the solution of a mathematical problem dealing with the equilibrium of identical repulsive particles (of one or two kinds) on one or more spheres like neutrons and protons on nuclear shells. Such particles are at equilibrium only for specific numbers of particles and, in addition, if these particles lie on the vertices of regular polyhedra or their derivative polyhedra. Finally, this result leads to a pictorial representation of the structure of all closed shell nuclei. This representation could be used as a laboratory for determining nuclear properties and corresponding wave functions.

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.

Classifying Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Graph Theory ◽  
Rational Number ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Reconstruction Problem ◽  
Regular Polyhedra ◽  
Four Color Theorem ◽  
Stanislaw Ulam ◽  
Irregular Graphs ◽  
Isomorphic Graphs

This chapter considers the richness of mathematics and mathematicians' responses to it, with a particular focus on various types of graphs. It begins with a discussion of theorems from many areas of mathematics that have been judged among the most beautiful, including the Euler Polyhedron Formula; the number of primes is infinite; there are five regular polyhedra; there is no rational number whose square is 2; and the Four Color Theorem. The chapter proceeds by describing regular graphs, irregular graphs, irregular multigraphs and weighted graphs, subgraphs, and isomorphic graphs. It also analyzes the degrees of the vertices of a graph, along with concepts and ideas concerning the structure of graphs. Finally, it revisits a rather mysterious problem in graph theory, introduced by Stanislaw Ulam and Paul J. Kelly, that no one has been able to solve: the Reconstruction Problem.

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