On Three-Dimensional Space Groups

An entirely new and independent enumeration of the crystallographic space groups is given, based on obtaining the groups as fibrations over the plane crystallographic groups, when this is possible. For the 35 ``irreducible'' groups for which it is not, an independent method is used that has the advantage of elucidating their subgroup relationships. Each space group is given a short ``fibrifold name'' which, much like the orbifold names for two-dimensional groups, while being only specified up to isotopy, contains enough information to allow the construction of the group from the name.Comment: 26 pages, 8 figure

Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory

We show how discrete Morse theory provides a rigorous and unifying foundation for defining skeletons and partitions of grayscale digital images. We model a grayscale image as a cubical complex with a real-valued function defined on its vertices (the voxel values). This function is extended to a discrete gradient vector field using the algorithm presented in Robins, Wood, Sheppard TPAMI 33:1646 (2011). In the current paper we define basins (the building blocks of a partition) and segments of the skeleton using the stable and unstable sets associated with critical cells. The natural connection between Morse theory and homology allows us to prove the topological validity of these constructions; for example, that the skeleton is homotopic to the initial object. We simplify the basins and skeletons via Morse-theoretic cancellation of critical cells in the discrete gradient vector field using a strategy informed by persistent homology. Simple working Python code for our algorithms for efficient vector field traversal is included. Example data are taken from micro-CT images of porous materials, an application area where accurate topological models of pore connectivity are vital for fluid-flow modelling

Three‐periodic nets and tilings: regular and quasiregular nets

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/115935/1/S0108767302018494.pd

Three‐periodic nets and tilings: semiregular nets

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/115982/1/S0108767303017100.pd

Minimal nets and minimal minimal surfaces

The 3-periodic nets of genus 3 ('minimal nets') are reviewed and their symmetries re-examined. Although they are all crystallographic, seven of the 15 only have maximum-symmetry embeddings if some links are allowed to have zero length. The connection bet

Three‐periodic nets and tilings: minimal nets

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/116005/1/S0108767304015442.pd

Isogonal non-crystallographic periodic graphs based on knotted sodalite cages

This work considers non-crystallographic periodic nets obtained from multiple identical copies of an underlying crystallographic net by adding or flipping edges so that the result is connected. Such a structure is called a `ladder' net here because the 1-periodic net shaped like an ordinary (infinite) ladder is a particularly simple example. It is shown how ladder nets with no added edges between layers can be generated from tangled polyhedra. These are simply related to the zeolite nets SOD, LTA and FAU. They are analyzed using new extensions of algorithms in the program Systre that allow unambiguous identification of locally stable ladder nets

Nets with collisions (unstable nets) and crystal chemistry

Nets in which different vertices have identical barycentric coordinates (i.e. have collisions) are called unstable. Some such nets have automorphisms that do not correspond to crystallographic symmetries and are called non-crystallographic. Examples are

A Catalogue of Periodic Fully-Resonant Azulene-Transitive Azulenoid Tilings Analogous to Clar Structures

A reference catalogue of 1274 fully-resonant azulenoid tiling patterns with a symmetry group that operates transitively on the azulenes has been generated, and may be consulted at http://caagt.ugent.be/azul. Such networks are analogous to Clar structures and fully-naphthalenoid structures among benzenoids.</p