We explore the relationship between Kekulé structures and maximum face independence sets in fullerenes: plane trivalent graphs with pentagonal and hexagonal faces. For the class of leap-frog fullerenes, we show that a maximum face independence set corresponds to a Kekulé structure with a maximum number of benzene rings and may be constructed by partitioning the pentagonal faces into pairs and 3...

Tetrahedrally close-packed structures with juxtaposed pentagonal antiprisms, such as the F, C14 Laves and the newly found C phases, were studied by means of HREM and SAD. It was found that each bright spot in the structural image corresponds to an antiprism. Differently oriented domains of these phases intergrow frequently with a fairly good match at the interphase boundary. All diffraction pat...

Euler discovered the pentagonal number theorem in 1740 but was not able to prove it until 1750. He sent the proof to Goldbach and published it in a paper that finally appeared in 1760. Moreover, Euler formulated another proof of the pentagonal number in his notebooks theorem around 1750. Euler did not publish this proof or communicate it to his correspondents, probably because of the difficulty...

(19 June 2012) We introduce the concept of a pentagonal geometry as a generalisation of the pentagon and the Desargues configuration, in the same vein that the generalised polygons share the fundamental properties of ordinary polygons. In short, a pentagonal geometry is a regular partial linear space in which for all points x, the points not collinear with the point x, form a line. We compute b...

A 3rd order numerical model is developed for global advection transport computation. The multi-moment finite volume method has been implemented to the hexagonal-pentagonal geodesic grid for spherical geometry. Two kinds of moments, i.e. point value and volume-integrated average, are used as the constraint conditions to derive the time evolution equations to update the computational variables, w...

The dielectric response of pentagonal defects in multilayer graphene nano-cones has been studied by electron energy loss spectroscopy and ab initio simulations. At the cone apex, a strong modification of the dielectric response is observed below the energy of the π plasmon resonance. This is attributed to π → π* interband transitions induced by topology-specific resonant π bonding states as wel...

(2) σ(n) = σ(n−1)+σ(n−2)−σ(n−5)−σ(n−7)+σ(n−12)+σ(n−15)−... where the numbers 1,2,5,7,12,15,... appearing in the successive terms in (1)-(2) are the positive pentagonal numbers {vm} given by (3) vm = m(3m∓ 1)/2, m = 1, 2, ... In identities (1)-(2) we accept that p(m) = 0, σ(m) = 0 when m < 0. The only formal difference is that (1) is true with the understanding that (4) p(0) = 1, while (2) is va...

We explore numerically the formation of Turing patterns in a confined circular domain with small aspect ratio. Our results show that stable fivefold patterns are formed over a well defined range of disk sizes, offering a possible mechanism for inducing the fivefold symmetry observed in early development of regular echinoids. Using this pattern as a seed, more complex biological structures can b...