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17 pages, 649 KB

Open AccessEditor’s ChoiceArticle

Bi-Symmetric Polyhedral Cages with Three, Four, Five or Six Connected Faces and Small Holes

by Bernard Piette

Symmetry 2025, 17(9), 1537; https://doi.org/10.3390/sym17091537 - 15 Sep 2025

Cited by 1 | Viewed by 1394

Polyhedral cages (p-cages) are Euclidean geometric structures corresponding to polyhedra with holes. They are a good example of the geometry of some artificial protein cages. In this paper we identify p-cages made out of two families of equivalent polygonal faces, where the face of one family is attached to three other faces while the faces of the other family are attached to three, four, five or six other faces. To restrict ourselves to p-cages with small holes, we consider p-cages where each hole comprises at most four faces. The construction starts from planar graphs made out of two families of equivalent nodes. One can then construct the dual of the solid corresponding to that graph and tile its faces with regular or nearly regular polygons. An energy function is then defined to quantify the amount of irregularity of the p-cages which is then minimised using a simulated annealing algorithm. We have analysed nearly 100,000 possible configurations, ruling out the p-cages made out of faces with deformations exceeding 10%. We then present graphically some of the most interesting geometries. Full article

(This article belongs to the Section C: Physics)

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23 pages, 925 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Bi-Symmetric Polyhedral Cages with Nearly Maximally Connected Faces and Small Holes

by Bernard Piette

Symmetry 2025, 17(6), 940; https://doi.org/10.3390/sym17060940 - 12 Jun 2025

Cited by 2 | Viewed by 2062

Abstract

Polyhedral cages (p-cages) provide a good description of the geometry of some families of artificial protein cages. In this paper we identify p-cages made out of two families of equivalent polygonal faces/protein rings, where each face has at least four neighbours and where the holes are contributed by at most four faces. We start the construction from a planar graph made out of two families of equivalent nodes. We construct the dual of the solid corresponding to that graph, and we tile its faces with regular or nearly regular polygons. We define an energy function describing the amount of irregularity of the p-cages, which we then minimise using a simulated annealing algorithm. We analyse over 600,000 possible geometries but restrict ourselves to p-cages made out of faces with deformations not exceeding 10%. We then present graphically some of the most promising geometries for protein nanocages. Full article

(This article belongs to the Special Issue Chemistry: Symmetry/Asymmetry—Feature Papers and Reviews)

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20 pages, 731 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Bi-Symmetric Polyhedral Cages with Maximally Connected Faces and Small Holes

by Bernard Piette and Árpad Lukács

Symmetry 2025, 17(1), 101; https://doi.org/10.3390/sym17010101 - 10 Jan 2025

Cited by 3 | Viewed by 1388

Abstract

Polyhedral cages (p-cages) describe the geometry of some families of artificial protein cages. We identify the p-cages made out of families of equivalent polygonal faces such that the faces of one family have five neighbors and

P_{1}

edges, while those of the [...] Read more.

P_{1}

edges, while those of the other family have six neighbors and

P_{2}

edges. We restrict ourselves to polyhedral cages where the holes are adjacent to four faces at most. We characterize all p-cages with a deformation of the faces, compared to regular polygons, not exceeding 10%. Full article

(This article belongs to the Section B: Mathematics)

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29 pages, 756 KB

Open AccessArticle

Near-Miss Bi-Homogenous Symmetric Polyhedral Cages

by Bernard Piette and Árpad Lukács

Symmetry 2023, 15(9), 1804; https://doi.org/10.3390/sym15091804 - 21 Sep 2023

Cited by 5 | Viewed by 2371

Abstract

Following the discovery of an artificial protein cage with a paradoxical geometry, we extend the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages, for which all the faces are equivalent, and define bi-homogeneous symmetric polyhedral cages made of two different types of faces, where all the faces of a given type are equivalent. We parametrise the possible connectivity configurations for such cages, analytically derive p-cages that are regular, and numerically compute near-symmetric p-cages made of polygons with 6 to 18 edges and with deformation not exceeding 10%. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Nature-Inspired, Bio-Based Materials)

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28 pages, 825 KB

Open AccessEditor’s ChoiceArticle

Near-Miss Symmetric Polyhedral Cages

by Bernard M. A. G. Piette and Árpad Lukács

Symmetry 2023, 15(3), 717; https://doi.org/10.3390/sym15030717 - 13 Mar 2023

Cited by 7 | Viewed by 5190

Abstract

Following the experimental discovery of several nearly symmetric protein cages, we define the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages made out of P-gons. We use group theory to parameterize the possible configurations and we minimize the irregularity of the P-gons [...] Read more.

P = 6

P = 20

with deformation of up to 10%. Full article

(This article belongs to the Section B: Mathematics)

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