We provide an existence proof for two 1-parameter families of embedded triply periodic minimal surfaces of genus three, namely the tG family with tetragonal symmetry that contains the gyroid, and the rGL family with rhombohedral symmetry that contains the gyroid and the Lidinoid, both discovered numerically in the 1990…
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Weaved helices form mechanically stable 3D structures.
problem Creating stable 3D structures from helical elements.
method Exploiting screw symmetry and invariant cylindrical rod packing to form triply periodic arrangements.
result Demonstrated nineteen triply periodic arrangements of interwoven helices.
We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain of 3-space with the same volume. The minimizer is the so-called srs network with quotient the complete graph on four vertices . The network…
Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.
problem Creating space-filling shapes without sharp corners.
method Edge bending algorithm to deform polyhedral tilings into soft tilings.
result Soft tilings derived from minimal surfaces can be continuously transformed into one another.
Minimal Twin Surfacesmath.DG
We report some minimal surfaces that can be seen as copies of a triply periodic minimal surface (TPMS) related by reflections in parallel mirrors. We call them minimal twin surfaces for the resemblance with twin crystal. Brakke's Surface Evolver is employed to construct twinnings of various classical TPMS, including Sc…
Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfacesmath.GT
We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To a…
Paper constructs and proves existence of chiral triply-periodic minimal surfaces.
problem Existence of chiral triply-periodic minimal surfaces of genus 4.
method Construction based on dual qtz--qzd nets, existence proof using Weierstrass method on branched torus.
result Family of chiral triply-periodic minimal surfaces of genus 4 constructed and proved to exist.