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.

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 Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 356 ◽  
Author(s):  
Jose Diaz-Severiano ◽  
Valentin Gomez-Jauregui ◽  
Cristina Manchado ◽  
Cesar Otero
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Structures ◽  
Inversive Plane ◽  
Delaunay Triangulations ◽  
Möbius Transformations ◽  
Regular Polyhedra ◽  
Complex Design ◽  
Elliptic Geometry ◽  
Mobius Transformations

This paper shows a methodology for reducing the complex design process of space structures to an adequate selection of points lying on a plane. This procedure can be directly implemented in a bi-dimensional plane when we substitute (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Möbius transformations on the plane. These graphs can be obtained by sites, specific points obtained by homological transformations in the inversive plane, following the analogous procedure defined previously in the three-dimensional space. From the sites, it is possible to obtain different partitions of the plane, namely, power diagrams, Voronoi diagrams, or Delaunay triangulations. The first would generate geo-tangent structures on the sphere; the second, panel structures; and the third, lattice structures.

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.

Axioms for Absolute Geometry

1968 ◽  
Vol 20 ◽  
pp. 158-181 ◽  
Author(s):  
J. F. Rigby
Keyword(s):  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Absolute Geometry

The axioms of Euclidean geometry may be divided into four groups: the axioms of order, the axioms of congruence, the axiom of continuity, and the Euclidean axiom of parallelism (6). If we omit this last axiom, the remaining axioms give either Euclidean or hyperbolic geometry. Many important theorems can be proved if we assume only the axioms of order and congruence, and the name absolute geometry is given to geometry in which we assume only these axioms. In this paper we investigate what can be proved using congruence axioms that are weaker than those used previously.

scholarly journals Automated Detection of Interesting Properties in Regular Polygons

2020 ◽  
Vol 14 (4) ◽  
pp. 727-755
Author(s):  
Zoltán Kovács
Keyword(s):  
Euclidean Geometry ◽  
Gröbner Bases ◽  
Automated Detection ◽  
Grobner Bases ◽  
Plane Geometry ◽  
Applied Theory ◽  
Regular Polygons ◽  
Natural Numbers ◽  
Minimal Polynomials ◽  
Bijective Function

Abstract We demonstrate a systematic, automated way of discovery of a large number of new geometry theorems on regular polygons. The applied theory includes a formula by Watkins and Zeitlin on minimal polynomials of $$\cos \frac{2\pi }{n}$$ cos 2 π n , and a method by Recio and Vélez to discover a property in a plane geometry construction. This method exploits Wu’s idea on algebraizing the geometric setup and utilizes the theory of Gröbner bases. Also a bijective function is given that maps the investigated cases to the first natural numbers. Finally, several examples are shown that are all previously unknown results in planar Euclidean geometry.

Networks of the Plane in Absolute Geometry

1906 ◽  
Vol 25 (1) ◽  
pp. 392-394
Author(s):  
Duncan M. Y. Sommerville
Keyword(s):  
Regular Polygons ◽  
Absolute Geometry

AbstractThe problem to divide the plane, without overlapping, into a network of regular polygons with the same length of side, has been completely worked out for the three geometries for the case in which the polygons are all of the same kind. The resulting networks are called regular

Tessellating the Sphere with Regular Polygons

2004 ◽  
Vol 97 (3) ◽  
pp. 165-167
Author(s):  
Hortensia Soto-Johnson ◽  
Dawn Bechthold
Keyword(s):  
High School ◽  
High School Students ◽  
Euclidean Geometry ◽  
Spherical Geometry ◽  
Regular Polygons ◽  
School Students ◽  
Disney World ◽  
Spaceship Earth

Spherical geometry can be used to entice students into a deeper understanding of Euclidean geometry. Determining which regular spherical polygons tessellate the sphere is another motivating topic that is accessible to high school students. The most recognizable tessellations of the sphere are found on balls, such as soccer balls, volleyballs, and golf balls. Even Spaceship Earth at Epcot Center in Disney World involves tessellation.

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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