Chiral Multi-tori

Diudea, Mircea Vasile

doi:10.1007/978-3-319-64123-2_10

Mircea Vasile Diudea⁴

Part of the book series: Carbon Materials: Chemistry and Physics ((CMCP,volume 10))

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Abstract

Chirality is one of the basic characteristics of biological structures; chirality is a symmetry property. Multi-tori are complex structures consisting of more than one torus, embedded in negatively curved surfaces. Design of multi-tori may be achieved by operations on maps. The “monomers” used to design the chiral multi-tori discussed in this chapter are snubs of Platonic solids. A snub polyhedron s(P) is achieved by dualizing the p ₅(P) transform: s(P) = d(p ₅(P)); since p ₅-operation is prochiral, all the consecutive structures will be chiral; high genus structures of rank k = 3 were thus obtained. The genus and rank (or space dimension) of a structure are parameters of complexity.

Topological symmetry of the structures herein discussed was evaluated by ring signature and centrality index and confirmed by symmetry calculation using the adjacency matrix permutations. C₆₀-related chiral tori were also designed and their symmetry evaluated. An atlas section illustrates the discussed chiral multi-tori.

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Department of Chemistry, Faculty of Chemistry and Chemical Engineering, Babes-Bolyai University, Cluj-Napoca, Romania
Mircea Vasile Diudea

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Chapter 10 Atlas: Chiral Multi-tori

10.1.1 Dodecahedron Related Structures


s(D).60	s(D)@s(D)7S.120	op(ca(D)).200


s(D).60	Op(Ca(D)).200	C_280,S


op(ca(D)).200	C_280,S	t(C₂₈₀).840_cyclic cover


C_280,R	P5.1(C_280,R).1240	s(C_280R).840


Op(Ca(D)).200	d(m(C₂₈₀)).400	C_280,S


s(D).60_5	m(C₂₈₀).420	C_280,S


s(D).60_5	op(ca(D)).200	C_280,S


s(D).60_5	Op(Ca(D)).200	d(p ₄(C₂₈₀)820).840


op(ca(D)).200	C_280,S	p ₄(C₂₈₀).820

10.2 C ₆₀ related structures


s(C₆₀).180	s(C₆₀)@s(C₆₀)7R.360 A₅; Classes 6: \|6{60}\|	op(ca(C₆₀)).600


s(C₆₀).180	op(ca(C₆₀)).600	C_840,7R


op(ca(C₆₀)).600	d(m(C₈₄₀)).1200_3	C_840,7R_3


op(ca(C₆₀)).600_5	m(C₈₄₀).1260_3	C_840,7R_3


s(C₆₀).180	op(ca(C₆₀)).600	C_{840,R_5}


op(ca(C₆₀)).600	C_{840,R_5}	t(C₈₄₀).2520_cyclic cover

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Diudea, M.V. (2018). Chiral Multi-tori. In: Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-64123-2_10

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DOI: https://doi.org/10.1007/978-3-319-64123-2_10
Published: 21 October 2017
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-64121-8
Online ISBN: 978-3-319-64123-2
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)

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Chiral Multi-tori

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Chapter 10 Atlas: Chiral Multi-tori

Chapter 10 Atlas: Chiral Multi-tori

10.1.1 Dodecahedron Related Structures

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