Computing with almost periodic functions

Moody, Robert V.; Nesterenko, Maryna; Patera, J.

doi:10.1107/s0108767308025440

Cited by 12 publications

(9 citation statements)

References 20 publications

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“…The icosahedral group H 3 underlies both the fullerenes and the aperiodic three-dimensional crystals ('quasicrystals') (Moody et al, 2008;Moody & Patera, 1993). It is undoubtedly possible to link the fullerenes and nanotubes with suitably set up infinite quasicrystals.…”

Section: Discussionmentioning

confidence: 99%

C₇₀, C₈₀, C₉₀and carbon nanotubes by breaking of the icosahedral symmetry of C₆₀

2013

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The icosahedral symmetry group H3 of order 120 and its dihedral subgroup H2 of order 10 are used for exact geometric construction of polytopes that are known to exist in nature. The branching rule for the H3 orbit of the fullerene C60 to the subgroup H2 yields a union of eight orbits of H2: four of them are regular pentagons and four are regular decagons. By inserting into the branching rule one, two, three or n additional decagonal orbits of H2, one builds the polytopes C70, C80, C90 and nanotubes in general. A minute difference should be taken into account depending on whether an even or odd number of H2 decagons are inserted. Vertices of all the structures are given in exact coordinates relative to a non-orthogonal basis naturally appropriate for the icosahedral group, as well as relative to an orthonormal basis. Twisted fullerenes are defined. Their surface consists of 12 regular pentagons and 20 hexagons that have three and three edges of equal length. There is an uncountable number of different twisted fullerenes, all with precise icosahedral symmetry. Two examples of the twisted C60 are described.

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Section: Discussionmentioning

confidence: 99%

C₇₀, C₈₀, C₉₀and carbon nanotubes by breaking of the icosahedral symmetry of C₆₀

2013

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“…developing powerful computing systems through conventional neurocomputing 18 as well as quantum computing, with the ability of such a system to factorise large integers among other properties, as has been suggested previously. 9 Quasicrystals also have interesting optical properties.…”

Section: Other Interesting Properties Of Quasicrystalsmentioning

confidence: 99%

Fibonacci, quasicrystals and the beauty of flowers

Gardiner

2012

Plant Signaling & Behavior

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T he appearance of Fibonacci sequences and the golden ratio in plant structures is one of the great outstanding puzzles of biology. Here I suggest that quasicrystals, which naturally pack in the golden ratio, may be ubiquitous in biological systems and introduce the golden ratio into plant phyllotaxy. The appearance of golden ratio-based structures as beautiful indicates that the golden ratio may play a role in the development of consciousness and lead to the aesthetic natural selection of flowering plants. Fibonacci and the Golden RatioIn mathematics and the arts, two quantities are in the golden ratio (τ or 1.6180339…) if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is thought to be aesthetically pleasing and thus appears often in art and architecture. The Fibonacci numbers are simply the arithmetic consequence of multiplying each number by τ, and rounding to the nearest integer or, in other words, the next number in the Fibonacci sequence is the sum of the previous two numbers in the sequence thus 1, 1, 2, 3, 5, 8, 13, 21, 34, … The Fibonacci sequence was first discovered by Leonardo da Pisa (also known as Fibonacci) in 1202 and has puzzled scientists for centuries due to its frequent occurrence in nature, including in spiral molluscan shells and plants. The ubiquity of the Fibonacci sequence in nature seems to indicate that it arises through a process that is fundamental to life.

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“…The Coxeter group H 4 does not have an official name in physics, even though it contains all the other non-crystallographic groups within it (H 2 & H 3 & H 4 ). In general, such groups are used to describe aperiodic sets of points (or quasicrystals) and various spherical molecules (Moody & Patera, 1993;Moody et al, 2008;Dechant et al, 2012Dechant et al, , 2013Baake & Grimm, 2013;Zappa, 2015;Salthouse et al, 2015).…”

Section: Introductionmentioning

confidence: 99%

On symmetry breaking of dual polyhedra of non-crystallographic group H ₃

Myronova

2021

Acta Cryst Sect A

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The study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry-breaking mechanism is applied to the family of polytopes {\cal V}_{H_{3}}(\lambda) constructed for each type of dominant point λ. Here a polytope {\cal V}_{H_{3}}(\lambda) is considered as a dual of a {\cal D}_{H_{3}}(\lambda) polytope obtained from the action of the Coxeter group H 3 on a single point \lambda\in{\bb R}^{3}. The H 3 symmetry is reduced to the symmetry of its two-dimensional subgroups H 2, A 1 × A 1 and A 2 that are used to examine the geometric structure of {\cal V}_{H_{3}}(\lambda) polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the `pancake' structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the {\cal V}_{H_{3}}(\lambda) structure into various nanotubes. Moreover, since a {\cal V}_{H_{3}}(\lambda) polytope may contain the orbits obtained by the action of H 3 on the seed points (a, 0, 0), (0, b, 0) and (0, 0, c) within its structure, the stellations of flat-faced {\cal V}_{H_{3}}(\lambda) polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C20 has the dodecahedral structure of {\cal V}_{H_{3}}(a,0,0), the construction of the smallest fullerenes C24, C26, C28, C30 together with the nanotubes C20+6N , C20+10N is presented.

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Computing with almost periodic functions

Cited by 12 publications

References 20 publications

C₇₀, C₈₀, C₉₀and carbon nanotubes by breaking of the icosahedral symmetry of C₆₀

C₇₀, C₈₀, C₉₀and carbon nanotubes by breaking of the icosahedral symmetry of C₆₀

Fibonacci, quasicrystals and the beauty of flowers

On symmetry breaking of dual polyhedra of non-crystallographic group H ₃

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Computing with almost periodic functions

Cited by 12 publications

References 20 publications

C70, C80, C90and carbon nanotubes by breaking of the icosahedral symmetry of C60

C70, C80, C90and carbon nanotubes by breaking of the icosahedral symmetry of C60

Fibonacci, quasicrystals and the beauty of flowers

On symmetry breaking of dual polyhedra of non-crystallographic group H 3

Contact Info

Product

Resources

About

C₇₀, C₈₀, C₉₀and carbon nanotubes by breaking of the icosahedral symmetry of C₆₀

C₇₀, C₈₀, C₉₀and carbon nanotubes by breaking of the icosahedral symmetry of C₆₀

On symmetry breaking of dual polyhedra of non-crystallographic group H ₃