The Roundest Polyhedra with Symmetry Constraints

Lengyel, András; Gáspár, Zsolt; Tarnai, Tibor

doi:10.3390/sym9030041

Cited by 5 publications

(4 citation statements)

References 15 publications

(25 reference statements)

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“…where V is the volume and S is the surface of the polyhedron. By definition the sphere has an IQ = 1, and this is the maximum possible value of the parameter [53]. The values of IQ for the regular dodecahedron and the truncated octahedron are as follows:…”

Section: Fig 9 a Prototype Of The Planar Meta-array With 7 Triangular...mentioning

confidence: 99%

Full-Digital Microphone Meta-Arrays for Consumer Electronics

Pinardi

Toscani

Binelli

et al. 2023

IEEE Trans. Consumer Electron.

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Microphone arrays of various sizes and shapes are widely employed in consumer electronics devices such as smart speakers, speakerphones, smart TVs, smartphones, PCs, and headphones. Therefore, the possibility of creating arrays of arrays (i.e., meta-arrays), by easily connecting several smaller units, is particularly interesting: it allows for different shapes, with an increased number of microphones and improved performance, while keeping costs low. In this paper, full-digital microphone meta-arrays are presented: they are constituted by an arrangement of several triangular units, each featuring four digital Micro Electro-Mechanical Systems (MEMS) capsules connected in daisy-chain through the Automotive Audio Bus (A 2 B). Two prototypes have been built: a planar meta-array made of seven triangles (28 capsules) and a 3-dimensional meta-array with 14 triangles (56 capsules). Numerical simulations were carried out and compared against experimental measurements performed on the prototypes, showing an excellent matching. Finally, the 3-dimensional meta-array was compared with a spherical microphone array widely considered the state-of-the-art equipment in the last decade, demonstrating its potential in telecommunication applications.

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Section: Fig 9 a Prototype Of The Planar Meta-array With 7 Triangular...mentioning

confidence: 99%

Full-Digital Microphone Meta-Arrays for Consumer Electronics

Pinardi

Toscani

Binelli

et al. 2023

IEEE Trans. Consumer Electron.

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“…10 −30 . This is due to the interference of rounding errors [ 22 ], which result from the inexactness in the representation of real numbers and the arithmetic operations done with them. For example, the theoretical value of ln(1) is 0 obviously, yet it equals 2.2204 × 10 −16 in Matlab, due to the limitation of 16 digits of precision [ 20 ].…”

Section: Optimizing the Cage’s Geometrymentioning

confidence: 99%

“…There are various methods to generate cages with different geometric properties. For example, one can derive spherical cages by vertex elimination [ 5 ], cages with planar faces through three-dimensional reciprocal construction [ 5 ], cages with equal-length edges by trigonometry [ 20 ] or optimization [ 21 ], and ‘the roundest polyhedra’ using numerical approaches [ 7 , 22 , 23 ]. Remarkably, cages with planar faces and simultaneously equal-length edges can be obtained by nulling the dihedral angle discrepancies [ 21 ].…”

Section: Introductionmentioning

confidence: 99%

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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“…The vertices of the resulting mesh, projected centrally onto the sphere, create a polyhedron approximating the sphere, in which only the nodes lie on the sphere's surface. This process can be found in the works of J. Clinton [2,3], T. Tarnai [4][5][6], P. Huybers [7][8][9][10][11][12], G. N. Pavlov [13,14], C. Kitrick [15,16], H. Lalvani [17][18][19][20], M. Wenninger [21,22], J. Rębielak [23], H.S.M. Coxeter [24], J. Fuli ński [25], and many others.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Lightweight Structure Mesh Topology of Geodesic Domes

Bysiec,

Jaszczyński,

Maleska

2023

Applied Sciences

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This paper presents two methods of shaping the mesh topology of lightweight structures as spherical domes. The two given methods of dividing the initial face of the polyhedra determine the obtained structures, which differ in the way of connecting the nodal points. These points were obtained by applying the algorithm for calculating spherical coordinates presented in the paper, which were then converted to the Cartesian system using transformation formulas. Two models of dome structures are presented, based on a 4608-hedron according to the first division method, and on a 4704-hedron, using the second proposed method with numerical analysis. Thus, the novelty of this paper is an implementation of the formulas and algorithms from geodesic domes based on the regular dodecahedron to the regular octahedron, which has not been presented so far. The choice of the shape of the structure has impacts on sustainable development, dictated by structural and visual considerations, leading to the design of a light structure with low consumption of construction material (steel), which can undoubtedly be helpful when making the final structure shape. In addition, according to this research, it can be concluded that using the first method to create a geodesic dome mesh is more straightforward, safer, and requires less design experience.

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The Roundest Polyhedra with Symmetry Constraints

Cited by 5 publications

References 15 publications

Full-Digital Microphone Meta-Arrays for Consumer Electronics

Full-Digital Microphone Meta-Arrays for Consumer Electronics

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

Analysis of Lightweight Structure Mesh Topology of Geodesic Domes

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