From spherical circle coverings to the roundest polyhedra

Tarnai, Tibor; Gáspár, Zsolt; Lengyel, András

doi:10.1080/14786435.2013.800652

Cited by 6 publications

(11 citation statements)

References 11 publications

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“…In a recent paper [10] the authors presented a numerical iterative method for the surface minimization of convex polyhedra circumscribed about the unit sphere with given number of faces and topology (edge graph). The main point of the method is to regard the surface area of a trivalent polyhedron with n faces as the potential energy of a mechanical system to minimize and to create the associated (dual) polyhedron whose faces are considered as triangle elements.…”

Section: Methodsmentioning

confidence: 99%

“…Mutoh [9] also dealt with polyhedra with minimum volume circumscribed about the unit sphere, and by using a computer-aided search, provided a series of conjectured optimal polyhedra with the number of faces ranging between 4 and 30. In an earlier paper [10] the authors pointed out that in some cases the isoperimetric problem for n faces and the problem of the minimum covering of a sphere by n equal circles have the same proven or conjectured solution (the points of tangency of the faces and the centres of the circles are identical). In some cases, the obtained polyhedra are only topologically identical.…”

Section: Introductionmentioning

confidence: 99%

“…The conjectured solution suggested by Goldberg [5] for n = 16 has tetrahedral symmetry, and for n = 32 and 42 has icosahedral symmetry. The conjectured solution for n = 72 provided in [10] has icosahedral symmetry.…”

Section: Introductionmentioning

confidence: 99%

“…The artistic and practical interest in finding the roundest object as shown above is an important motivation to study the isoperimetric problem for polyhedra with symmetry constraints. The primary aim of this paper is to prove the solutions for small values of n (n = 8,10,14) and to present conjectured solutions for n = 18, 20, 22. Recalling that some solutions for small even numbers of polygonal faces have been proven by Fejes Tóth [6], our new results complete the sequence of proven and conjectured solutions for even n up to 22.…”

Section: Introductionmentioning

confidence: 99%

“…We developed an iterative numerical method for determining a locally optimal solution of the isoperimetric problem, where the starting point for the topology of the polyhedron is a conjectured optimal circle covering on a sphere. Using this method and starting with the conjectured best coverings with 50 [11] and 72 [12] equal circles, conjectural solutions to the isoperimetric problem for n = 50 and 72 were obtained [10].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Roundest Polyhedra with Symmetry Constraints

2017

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Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the sphere, i.e., the roundest polyhedra. A new numerical optimization method developed previously by the authors has been applied to optimize polyhedra to best approximate a sphere if tetrahedral, octahedral, or icosahedral symmetry constraints are applied. In addition to evidence provided for various cases of face numbers, potentially optimal polyhedra are also shown for n up to 132.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Roundest Polyhedra with Symmetry Constraints

2017

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New families of cage-like structures based on Goldberg polyhedra with non-isolated pentagons

Javan

Liu

Xie

2023

Journal of Computational Design and Engineering

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A Goldberg polyhedron is a convex polyhedron made of hexagons and pentagons that have icosahedral rotational symmetry. Goldberg polyhedra have appeared frequently in art, architecture, and engineering. Some carbon fullerenes, inorganic cages, viruses, and proteins in nature exhibit the fundamental shapes of Goldberg polyhedra. According to Euler's polyhedron formula, an icosahedral Goldberg polyhedron always has exactly 12 pentagons. In Goldberg polyhedra, all pentagons are surrounded by hexagons only—this is known as the isolated pentagon rule (IPR). The present study systematically developed new families of cage-like structures derived from the initial topology of Goldberg polyhedra but with the 12 pentagons fused in five different arrangements and different densities of hexagonal faces. These families might be of great significance in biology and chemistry, where some non-IPR fullerenes have been created recently with chemical reactivity and properties markedly different from IPR fullerenes. Furthermore, this study has conducted an optimisation for multiple objectives and constraints, such as equal edge length, equal area, planarity, and spherical shape. The optimised configurations are highly desirable for architectural applications, where a structure with a small number of different edge lengths and planar faces may significantly reduce the fabrication cost and enable the construction of surfaces with flat panels.

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Extending Goldberg’s method to parametrize and control the geometry of Goldberg polyhedra

et al. 2022

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Goldberg polyhedra have been widely studied across multiple fields, as their distinctive pattern can lead to many useful applications. Their topology can be determined using Goldberg’s method through generating topologically equivalent structures, named cages. However, the geometry of Goldberg polyhedra remains underexplored. This study extends Goldberg’s framework to a new method that can systematically determine the topology and effectively control the geometry of Goldberg polyhedra based on the initial shapes of cages. In detail, we first parametrize the cage’s geometry under specified topology and polyhedral symmetry; then, we manipulate the predefined independent variables through optimization to achieve the user-defined geometric properties. The benchmark problem of finding equilateral Goldberg polyhedra is solved to demonstrate the effectiveness of the proposed method. Using this method, we have successfully achieved nearly exact spherical Goldberg polyhedra, with all vertices on a sphere and all faces being planar under extremely low numerical errors. Such results serve as strong numerical evidence for the existence of this new type of Goldberg polyhedra. Furthermore, we iteratively perform k -means clustering and optimization to significantly reduce the number of different edge lengths to benefit the cost reduction for architectural and engineering applications.

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From spherical circle coverings to the roundest polyhedra

Cited by 6 publications

References 11 publications

The Roundest Polyhedra with Symmetry Constraints

The Roundest Polyhedra with Symmetry Constraints

New families of cage-like structures based on Goldberg polyhedra with non-isolated pentagons

Extending Goldberg’s method to parametrize and control the geometry of Goldberg polyhedra

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