The maximally symmetric surfaces in the 3-torus

Abstract

Suppose an orientation preserving action of a finite group G on the closed surface Σg of genus g > 1 extends over the 3-torus T 3 for some embedding Σg ⊂ T 3 . Then |G| ≤ 12(g − 1), and this upper bound 12(g − 1) can be achieved for g = n 2 + 1, 3n 2 + 1, 2n 3 + 1, 4n 3 + 1, 8n 3 + 1, n ∈ Z+. Those surfaces in T 3 realizing the maximum symmetries can be either unknotted or knotted. Similar problems in non-orientable category is also discussed.Connection with minimal surfaces in T 3 is addressed and when the maximum symmetric surfaces above can be realized by minimal surfaces is identified.

Schwarzite nets: a wealth of 3-valent examples sharing similar topologies and symmetries

“…We emphasize that the nets presented here are all deduced ab initio from two-dimensional tilings of hyperbolic space, then asymmetrized to reticulate the D , P and Gyroid (forming ( n , 3) schwarzite nets) then relaxed in three-dimensional space to give s-nets. Recall that our constrained enumeration via reticulations of the D , P and Gyroid TPMS is guaranteed to generate the most symmetric examples of three-periodic ( n , 3) schwarzites, since those surfaces are the most symmetric embeddings of three-periodic surfaces in three-dimensional space [39].…”

Mapping hyperbolic order in curved materials

“…In the hyperbolic case orbifolds and space groups are not in one-to-one correspondence since TP(M)S can have different extrinsic embeddings. 59 The most symmetric pattern on a TPMS has orbifold symbol * 246, 48 Within each class, there is a unique, maximally symmetric, hyperbolic orbifold describing the intrinsic order of any hyperbolic film in three-space; each case can be realised by a patterned TPMS, as shown in the top row of Fig. 3.…”

“…We note that in fact, among all TP(M)S, TPMS are maximally intrinsically symmetric in , and the most intrinsically symmetric are the Primitive, Gyroid and Diamond surfaces. 48 All three are related to each other by the Bonnet transformation 13,49,50 so they are intrinsically identical and can therefore be superimposed on each other (with suitable cuts), with identical curvature and metric variations over the surface. However, their extrinsic embeddings are different: the same hyperbolic surface admits three different embeddings in , forming different arrangements of channels (labyrinths) on either side of the TPMS, surrounded by “collars” on the TPMS.…”

Optimal packings of three-arm star polyphiles: from tricontinuous to quasi-uniformly striped bicontinuous forms

“…A key feature of the bicontinuous Gyroid, D and P patterns formed by amphiphilic systems is their very high intrinsic (two-dimensional, in-surface) symmetry, characterized by their hyperbolic orbifolds [20]. Among all TPMS (which are themselves more symmetric than other hyperbolic surfaces realizable in three-dimensional space, with smallest area orbifolds), the Gyroid, D and P surfaces are the most symmetric [21]. Thanks to standard formulae for orbifolds [22], we can rank this degree of symmetry in terms of increasing area of the orbifold, as in table 1.…”

“…Figure21. Effective stripe widths for periodic striping patterns, calculated as a fraction of the unit cell edges of conventional unit cells for the Gyroid, D and P surfaces (assuming space groups Im 3m, Pn 3m and Ia 3d, respectively), plotted against the stripe-width mismatch, calculated for a single polyphile -water composition, namely 30% hydrophobic volume fraction.…”