# Glossary

Throughout the EPINET site pages we've made use of various mathematical terms from geometry and topology. Here's a brief glossary of the most important terms organized by four subject areas.

## Terms relating to networks

Transitivity
The transitivity of a periodic net (n, m) gives the number of symmetrically–distinct vertices and symmetrically–distinct edges in the net. These are the number of vertices (n) and edges (m) in the asymmetric unit of the space group.

For the hyperbolic tilings and h-nets we give transitivity numbers for the vertices, edges, and faces as a triple (a, b, c).

Note that the vertex- (and edge-) transitivity of an s-net and its related h-nets can be different.

Degree / Valency / Coordination
The number of edges that are incident at a vertex is referred to as the degree, valency, or coordination of the vertex.
Vertex coordination sequence
Elements in the coordination sequence are defined to be the number of distinct vertices at distance k from some initial vertex. The distance between vertices in a network is the smallest number of edges needed to make a path between them.

In EPINET we report coordination sequences to distance 10.

Periodic net
A euclidean network is periodic if it has translational symmetries in some number of independent directions. These directions are the lattice vectors. The number of lattice vectors may be less than the dimension of the space the network is embedded in, i.e. a network in may be 1–periodic, 2–periodic, or 3–periodic.

A 2D hyperbolic network (or tiling) is periodic if it has translational symmetries that correspond to the covering space of a genus–g surface (for g ≥ 2).

Quotient graph
Given a periodic network, a quotient graph is defined by taking a single representative of each translationally–equivalent vertex, and each translationally–equivalent edge. A particularly useful version of this representation of periodic nets is called the vector method.
• S.J. Chung, T. Hahn, and W.E. Klee. Z. Kristallogr. 212:553–558 (1984).
Equilibrium placement / Barycentric embedding
Given the topology of a euclidean periodic net, (defined by its quotient graph, for example), the equilibrium placement assigns coordinates to its vertices. It does this by placing each vertex at the centre of mass (barycentre) of its neighbours (i.e. those vertices joined to it by an edge). Olaf Delgado–Friedrichs has shown that barycentric embeddings exist for all euclidean 3-connected periodic nets, and that such an embedding has the highest possible (affine) symmetry. This symmetry group is computable only for a restricted class of periodic nets: those that have no vertex collisions.
• O. Delgado–Friedrichs and M. O'Keeffe, "Identification of and symmetry computation for crystal nets", Acta Cryst. A 59, pp. 351–360 (2003)

Where possible, EPINET lists the space group parameters for an equilibrium placement with maximal unit cell volume and mean edge-length of 1 (maximal). We also give vertex coordinates for another embedding where the geometry is relaxed according to an energy function that favours equal edge-lengths (relaxed).

Note that the barycentric symmetry is generally not the same as the surface spacegroup found from the inital surface reticulation.

Vertex collision
A periodic net is said to have a vertex collision, if two distinct vertices are assigned the same coordinates in the equilibrium placement.
Systre key
For networks that have an equilibrium placement with no vertex collisions, it is possible to define a canonical form for the quotient graph. This is called the systre key, after the software developed by Olaf Delgado–Friedrichs. Where possible, EPINET uses the systre key to identify surface reticulations that have the same topology.
• O. Delgado–Friedrichs and M. O'Keeffe, "Identification of and symmetry computation for crystal nets", Acta Cryst. A 59, pp. 351–360 (2003)
Sphere packing
A special class of 3D euclidean nets are those whose edges join sphere centres in tangential contact in a packing of spherical balls. Sphere packings therefore afford a source of nets. Homogenous sphere packings have the property that every edge has the same length, and no two vertices are closer than this distance. The enumeration of homogeneous sphere packings in 3D euclidean space from crystallographic principles has been made over the years by Werner Fischer, Elke Koch and Heidrun Sowa, in Marburg. When known, we give the sphere packing symbol from their enumeration in the other names field of an s-net.
RCSR
RCSR, the Reticular Chemical Structure Resource, is a website and database of periodic structures, useful in the design and analysis of crystalline materials.
• O'Keeffe, M.; Peskov, M. A.; Ramsden, S. J.; Yaghi, O. M. Accts. Chem. Res. 2008, 41, 1782-1789.
Wells' vertex symbol
The Wells' vertex symbol for a 3D net lists the length of the shortest cycle that connects each pair of edges incident at a vertex. A z–coordinated vertex must have z*(z–1)/2 cycles. Where there is more than one type of vertex, a comma separates the symbol for each.

As an example, the symbol for the simple cubic net (sqc1, or pcu) is [4^12.6^3] because there are 15 distinct pairs of edges at each degree-6 vertex and 12 of these have shortest cycles of length 4, while 3 of them are of length 6.

Systre crystallographic geometry file
The Systre crystallographic geometry file is a text file given the .cgd extension used by Olaf Delgado-Friedrichs Systre and Gavrog software packages. Each s-net links to a .cgd file that contains three specifications of the net:
• First is the periodic graph format with an edge defined on each line by v1 v2 i j k, where v1 is the start vertex in the base unit cell, and v2 is the end vertex in the unit cell offset by a translation vector (i,j,k).
• The next two versions give geometric specifications using a crystallographic space group with coordinates for each symmetry-distinct vertex followed by start and end coordinates for each symmetry-distinct edge.
• The maximal net geometry is a systre equilibrium placement (barycentric embedding) maximising the unit cell volume.
• The relaxed net geometry gives systre coordinates favouring equal edge-lengths"

## Terms from tiling theory

Orbifold
An orbifold is a generalised manifold with cone–like singularities and boundaries. It is formed by identifying symmetrically–identical points in a symmetric pattern, such as a periodic tiling. The 2D orbifold symbol is a naming scheme due to John Conway. For more details see the orbifold page.
Orbifold curvature
For discrete 2D hyperbolic symmetry groups, the orbifold curvature is essentially the area of a single asymmetric domain (actually the area is -2π times the orbifold curvature). The order of a subgroup in a given group is equal to the ratio of their orbifold curvatures. The curvature is readily computed from Conway's orbifold symbol. The formula is given on the orbifold page.
Delaney-Dress symbol
The Delaney–Dress symbol is a compact encoding of the symmetries and combinatorial structure of a tiling. There are a number of different representations for this symbol, which are discussed within the combinatorial tiling theory page. It is possible to reconstruct a tiling from its D-symbol, up to the isomorphism class of its symmetry group.
• AWM Dress. Presentations of discrete groups, acting on simply connected manifolds, in terms of parametrized systems of Coxeter matrices - A systematic approach. Advances in Mathematics 63, 196–212 (1987).
• Rüdiger Zeller, Olaf Delgado-Friedrichs, Daniel H. Huson. Tegula – exploring a galaxy of two-dimensional periodic tilings, Computer Aided Geometric Design 90, (2021).
2D Vertex symbol
The 2D vertex symbol encodes the topology of a 2D tiling, or equivalently, a 2D periodic network. A vertex symbol is given for each symmetrically distinct vertex and lists the number of edges of each tile (the "ring size"), for all tiles coincident with that vertex. The ring sizes are given in cyclic fashion around the vertex so that adjacent entries refer to tiles with a common edge. Thus, {6.6.6} refers to the hexagonal tiling of the 2D euclidean plane, {5.5.5} to the edge skeleton of the dodecahedron (which is a tiling of the sphere).

If there is more than one type of vertex, the vertex symbol lists the ring sizes for each vertex in an arbitrary order: for example {8.3.8.3} {8.3.6.3} is one of the hyperbolic tilings compatible with the P, D, and G, minimal surfaces.

Dual tiling
For a given 2D tiling, its dual is constructed by replacing each tile by a vertex, and each vertex by a tile. The degree of the new vertex is the number of edges of the old tile, and the new tiles have as many edges as the old vertex degree. An edge between new vertices exists only if the old tiles share a face. An illustrative example is given on the combinatorial tiling theory page.
Automorphism type
Many orbifold symbols have some abstract symmetry, or an automorphism of their form. For example, *2626 denotes a quadrilateral orbifold with two types of vertex and one type of edge. The abstract symmetries reduce the number of different Delaney–Dress tiling symbols found when enumerating all possible glue or split operations that may apply. A generic *pqrs quadrilateral orbifold will have four different edge–glue and four vertex–glue operations, for example, while *2626 has only one edge–glue and two vertex–glue operations.

When we represent the *2626 orbifold as a subgroup of *246, however, the abstract symmetry is broken and we find that some Delaney–Dress symbols may have multiple (non–isometric) embeddings in the *246 grid. See the hyperbolic tiling page for further details.

Petrie polygon
The Petrie polygon was named by the great geometer HSM Coxeter after his school friend John Flinders Petrie (the son of the famous Egyptologist Flinders Petrie). It is a non–planar polygon, whose edges are a subset of the edges of a polyhedron. It is defined by choosing a pair of edges sharing a vertex on the polyhedron, then extending the edge–path to form a closed polygon such that any two consecutive edges, and no more, belong to a face of the polyhedron.

## Terms specific to EPINET

U-Tiling
A U-tiling or universal cover tiling is a two-dimensional hyperbolic tiling within a specific surface-compatible symmetry subgroup and translational unit cell. These tilings are represented by unfolded genus-3 Delaney-Dress symbols augmented with cuts.
O-Tiling
An O-Tiling is a tritorus surface reticulation obtained by projecting a U-tiling onto the tritorus.
E-Tiling
An E-tiling is a tiling of a triply-periodic minimal surface obtained by projecting a U-tiling onto the periodic minimal surface via the covering map. Such tilings are identified by pairing the Delaney-Dress symbol of the U-tiling with a covering map for the surface.
h-net
An h-net or hyperbolic net is a two-dimensional net in distinguished by topology only, represented by the unique maximal-symmetry Delaney-Dress symbol with that topology.
o-net
An o-net is a finite net formed by the edges of an O-tiling embedded in 3D space. Such nets are finite ‘molecular’ nets and equivalent up to ambient isotopy class.
e-net
An e-net is a net formed by vertices and edges of an E-tiling. They are equivalent up to ambient isotopy class.
s-net
An s-net or systre net is a three-dimensional periodic net, defined by topology only, and represented by the systre key.
Surface Tiling
A surface tiling is the projection of a 2D hyperbolic tiling onto a triply periodic minimal surface. via a symmetry-preserving covering map.
Surface Reticulation
A surface reticulation is the network obtained by considering the edge–skeleton of the surface tiling.
Surface spacegroups
The surface spacegroups referred to in the Subgroups index are the 3D euclidean space groups of the surface tilings generated by wrapping an tiling onto the given surface.
Net names
Epinet has its own naming schema, currently implemented for h-nets, e-nets and s-nets. These names contain a three-letter prefix followed by an integer, e.g. hqcN (h-nets), epcN, edcN, egcN (e-nets) and sqcN (s-nets) respectively (here N is an integer). The second letter in the prefix is currently restricted to ‘q’ (generated from reticulations of genus-three cubic TPMS) and ‘p’, ‘d’ or ‘g’ (the relevant TPMS). The third letter ‘c’ refers to the Coxeter class of orbifolds used to generate the net. Further details on the naming schema can be found in S.J. Ramsden, V. Robins and Stephen Hyde, “3D euclidean nets from 2D hyperbolic tilings: Kaleidoscopic examples”, Acta Cryst. A, A65, 81-108 (2009).
Other names
Many of our examples correspond to nets already identified by other means. In those cases, we flag these as ‘known nets’ and give their alternate names. In particular, where possible, we give the three-letter name in the Reticular Chemistry Structural Resource (RCSR) database, the sphere packing symbol from Fischer, Koch, and Sowa's enumerations, and common key words such as 'zeolite', 'sphere packing' or 'lattice'. Keywords, names, and symbols, can all be searched for in the 'other names' s-net filter.
Transpose pair
Most U-tilings are distinguished by a unique Delaney-Dress symbol in the orbifold 'ooo', however a small class of U-Tilings have the same D-symbol, but are related with an automorphism that swaps the Ts and Taus. We call these U-Tilings 'Transpose Pairs', and they are characterised by the fact that the transposition swaps the nets arising from the P and D surfaces. The nets on the G surface, however, are distinct.

## Terms relating to periodic minimal surfaces

Minimal surface
A minimal surface is a surface with locally extremal (usually minimal) surface area. A minimal surface's mean curvature is zero at all points on the surface. For more details go to the minimal surfaces page.
Triply periodic minimal surface
Triply periodic minimal surfaces (TPMS) are minimal surfaces with translational symmetry along three independent directions (i.e. they have three lattice vectors). Examples and more background are given on the minimal surfaces page.
Manifold
An N–dimensional manifold is a space that looks locally like N–dimensional euclidean space at every point. The manifolds used within EPINET are all intrinsically two–dimensional, so are built from patches of .
Genus
The concept of genus is defined only for 2D manifolds which can be embedded inside a ball of finite radius and have no boundary edges. The genus of such a surface is the number of torus–like handles it has. The surface of a donut has genus 1, a pretzel has genus 3.

The genus of a triply periodic minimal surface is defined by taking the primitive translational unit cell (of the oriented surface) and identifying opposite faces to obtain a compact surface. The P surface, D surface and G surface, have genus 3.

Flat point
A flat point is a point on a surface with zero Gaussian curvature. Flat points on triply periodic minimal surfaces occur at isolated sites. At all other (non–flat) points, the Gaussian curvature of the surface is strictly negative and thus saddle–shaped.
Gaussian curvature
The Gaussian curvature of a surface is defined at each point p on the surface. First consider the planar curves formed by the intersection of the surface with a plane that contains the normal vector to the surface at p. On a generic surface, these curves can be locally convex (positive curvature), concave (negative curvature) or straight (zero curvature). There is a continuous family of planar curves, one for each tangent to the surface at p.

The point p (such as a flat point on a minimal surface, or any point on a sphere), is an “umbilic” if all planar curves of intersection have the same curvature at p, and in this case the Gaussian curvature is defined to be the square of this curvature. Now assume that p is not an umbilic. Choose the two tangent directions at p of extremal curvature (these directions are always orthogonal). Then the Gaussian curvature at p is defined to be the product of the maximal and minimal planar curve curvatures.

Mean curvature
The mean curvature at a point p on the surface is the arithmetic mean of the two extremal curvatures described in the definition of Gaussian curvature.
Bonnet transformation
The Bonnet transformation is defined via an isometry of the complex plane and creates distinct embeddings of a minimal surface. The transformation conserves all angles, lengths, and curvatures of the surface. The best–known examples of surfaces related by a Bonnnet transformation are the one–parameter family of cateniod–helicoid surfaces (depicted on the minimal surfaces page). The P, G, and D surfaces are triply periodic minimal surfaces related by a Bonnet transformation.
Covering space
The definitions of covering space (and covering map) formalise the (hopefully) familiar process of wrapping the real line onto the unit circle via the cosine or sine function. Within EPINET we use the hyperbolic plane, as the covering space for our triply periodic minimal surfaces.
Covering map
A covering map is a continuous function from the hyperbolic plane onto the surface, and although it is clearly not one–to–one, it does preserve the local structure of the two spaces. Technically the requirement is that given a sufficiently small connected open neighbourhood of a point on the surface, its preimage in the hyperbolic plane consists of many (actually infinitely many) isomorphic copies of the surface patch. More details about the covering maps specific to surfaces used within EPINET are available here.