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

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### Literature Review

## 2. Methodology

#### 2.1. Geometry Concepts

^{2}+ y

^{2}+ (z − 0.5)

^{2}= (0.5)

^{2}, and the equation for the plane is z = 1 (upper tangent plane). The center of the stereographic projection is the origin of coordinates (this projection is an inversion using the unit sphere centered at the origin). In order to obtain a point-to-point correspondence between the sphere and the plane, the working Euclidean plane is extended, introducing a point at infinity (inversive plane) [14].

#### 2.2. Transformations Procedure

`→`P’; Q’

`→`Q’). Another third point (A’) as well as its transformed (B’), define the magnitude of the rotation to be performed (A’

`→`B’).

`→`A’, B’

`→`B’). The constant of the inversion is established by setting the correspondence of any point with its transformed. With respect to any circle on the sphere, the polar points (P and Q) of the axis (PQ) perpendicular to the plane of such a circle are transformed into each other (P’

`→`Q’).

#### 2.3. Fundamental Grid and Propagation of Sites

- •
- Figure 7a:
- 1.
- Draw side A’B’ (there are two possible arches, keep the one at the opposite side point O’).
- 2.
- Obtain projections P’ and Q’ from the endpoints of the polar axis perpendicular to the plane of the great circle containing A and B on the sphere. These points are the intersection of the lines perpendicular to the side A’B’ from A’ and B’.
- 3.
- Divide side A’B’ into n equal parts (points N’i) by intersection with the elliptic lines dividing angle A’P’B’ into n equal parts (these lines belong to an elliptic pencil of circles through P’ and Q’).
- •
- Figure 7b:
- 4.
- Draw sides A’C’ and B’C’ and locate the middle point (M’) of the projected spherical triangle by intersection of the medians of both sides.
- •
- Figure 7c:
- 5.
- Define the Möbius transformation that performs the rotations of consecutive division points (N’i) of A’B’. Apply, in the triangle A’M’B’, such rotations to side B’M’ (direction B’
`→`A’) and to side A’M’ (direction A’`→`B’). The intersections of the rotated sides define the points of the fundamental grid (which belong to a hyperbolic pencil of circles [18]). - •
- Figure 7d:
- 6.
- Propagate the fundamental grid to the whole projection of the spherical triangle by means of reflection symmetry, with respect to lines A’M’ and B’M’.
- 7.
- Propagate the sites of the whole projection of the spherical triangle to the adjacent ones, by means of reflection symmetry, with respect to its three sides. Unknown vertices of the projections of the new spherical triangles are obtained by inversion (for example, vertex D’ is the inverse of A’, with respect of side B’C’).

#### 2.4. Generalization of Models by Means of Power Diagrams

- The sites are substituted by circles with variable diameter.
- The Voronoi diagrams and Delaunay triangulations are substituted by power diagrams (planar subdivision by radical axes) and their dual graphs [1].

## 3. Study Cases

## 4. Discussion

## 5. Conclusions

## 6. Further Research

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Geodesic spatial frame approximating a quadric surface: American Society for Metals International Headquarters and Geodesic Dome, at the Materials Park campus in Russell Township, Ohio, United States (Source: https://theredlist.com/wiki-2-19-879-605-285216-view-buckminster-fuller-richard-profile--1.html).

**Figure 2.**Different classes for geodesic structures depending on the orientation of the triangular tessellation with respect to the basic triangle (parameters b and c) as explained in the literature [11].

**Figure 3.**The definition of fundamental grid is cutting using parallel trihedra procedure [17]. Rotations of great circles AM and BM around axis OE (perpendicular to OAB through O), each one with an angular value α (angle that must divide AOB in equal parts) and opposite directions, defines a bi-directional grid in the spherical triangle ABM as a result of the intersection of the two families of arcs. This is called the fundamental grid. From this grid, it is possible to optimally discretize the interior of a spherical patch by applying symmetry operations in space.

**Figure 4.**Stereographic projection of the fundamental grid deployed on the spherical triangle. Great circles and concentric small circles are transformed into elliptic lines and hyperbolic pencils of coaxal circles, respectively, in the inversive plane. The limiting points of those pencils (P´ in the figure) will coincide with the projection of the points where OE intersects the sphere.

**Figure 5.**Homologous transformation of a rotation on the sphere around the polar axis PQ. The Eq. circle is the projection of the equatorial great circle (intersection of the sphere with the plane z = 0.5).

**Figure 6.**Homologous transformation of an inversion on the sphere with respect to a circle c. Correspondence of points: A’→A’; B’→B’; P’→Q’. The Eq. circle is the projection of the equatorial great circle (intersection of the sphere with the plane z = 0.5).

**Figure 7.**Propagation of the fundamental grid in the inversive plane. The Eq. Circ. is the projection of the equatorial great circle (intersection of the sphere with the plane z = 0.5). All elliptic lines intersect the Eq. Circ. in antipodal points.

**Figure 8.**

**Left**: Projection of a power diagram on z = 1 for defining the most generic polyhedron approximating the sphere.

**Right**: The fundamental grid used as a template permits selecting a set of sites to locate the circles of the diagram, so that it leads to solutions that are more symmetrical.

**Figure 9.**Surfaces resulting from the aggregation of Parallel Trihedra [17], according to an icosahedral symmetry: inflated (

**left**), unaltered (

**middle**) and flattened (

**right**). It is also possible to obtain these configurations by the juxtaposition of congruent spherical triangles with non-aligned centers.

**Figure 10.**CR-tangent mesh, application to a paraboloid of revolution (

**left**) and inflated mesh, application to a hyperboloid of revolution (

**right**).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Diaz-Severiano, J.A.; Gomez-Jauregui, V.; Manchado, C.; Otero, C. Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams. *Symmetry* **2018**, *10*, 356.
https://doi.org/10.3390/sym10090356

**AMA Style**

Diaz-Severiano JA, Gomez-Jauregui V, Manchado C, Otero C. Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams. *Symmetry*. 2018; 10(9):356.
https://doi.org/10.3390/sym10090356

**Chicago/Turabian Style**

Diaz-Severiano, Jose A., Valentin Gomez-Jauregui, Cristina Manchado, and Cesar Otero. 2018. "Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams" *Symmetry* 10, no. 9: 356.
https://doi.org/10.3390/sym10090356