We get a continuous one-parameter new family of embedded minimal surfaces, of which the period problems are two-dimensional. Moreover, one proves that it has Scherk second surface and Hoffman-Wohlgemuth example as limit-members.
For an embedded singly periodic minimal surface M with genus bigger than or equal to 4 and annular ends, some weak symmetry hypotheses imply its congruence with one of the Hoffman-Wohlgemuth examples. We give a very geometrical proof of this fact, al
ong which they come out many valuable clues for the understanding of these surfaces.
Cellular solids are a class of materials that have many interesting engineering applications, including ultralight structural materials. The traditional method for analyzing these solids uses convex uniform polyhedral honeycombs to represent the geom
etry of the material, and this approach has carried over into the design of digital cellular solids. However, the use of such honeycomb-derived lattices makes the problem of decomposing a three-dimensional lattice into a library of two-dimensional parts non-trivial. We introduce a method for generating periodic frameworks from triply periodic minimal surfaces, which result in geometries that are easier to decompose into digital parts. Additionally, we perform finite element modelling of two cellular solids generated from two TPMS, the P- and D-Schwarz, and two cellular solids, the Kelvin and Octet honeycombs. We show that the simulated behavior of these TMPS-derived structures shows the expected modulus of the cellular solid scaling linearly with relative density, which matches the behavior of the highest-coordination honeycomb structure, the octet truss.
We propose systems with structures defined by self-assembled triply periodic minimal surfaces (STPMS) as candidates for photonic bandgap materials. To support our proposal we have calculated the photonic bands for different STPMS and we have found th
at, at least, the double diamond and gyroid structures present full photonic bandgaps. Given the great variety of systems which crystalize in these structures, the diversity of possible materials that form them and the range of lattice constants they present, the construction of photonic bandgap materials with gaps in the visible range may be presently within reach.
We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that $n$th term is mainly based
on terms around a fixed fraction of $n$. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.