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.
 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 hnets 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 snet and its related hnets 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 E³ 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 3connected
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 edgelength of 1 (maximal). We also give vertex coordinates for another embedding where the geometry is relaxed according to an energy function that favours equal edgelengths (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 snet.
 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, 17821789.
 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 degree6 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 DelgadoFriedrichs Systre and Gavrog software packages.
Each snet 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 symmetrydistinct vertex followed by start and end coordinates for each symmetrydistinct 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 edgelengths"
 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.
 DelaneyDress 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 Dsymbol, 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 DelgadoFriedrichs, Daniel H. Huson. Tegula – exploring a galaxy of twodimensional 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.
 UTiling

A Utiling or universal cover tiling is a twodimensional hyperbolic tiling within a specific surfacecompatible symmetry subgroup and translational unit cell. These tilings are represented by unfolded genus3 DelaneyDress symbols augmented with cuts.
 OTiling

An OTiling is a tritorus surface reticulation obtained by projecting a Utiling onto the tritorus.
 ETiling

An Etiling is a tiling of a triplyperiodic minimal surface obtained by projecting a Utiling onto the periodic minimal surface via the covering map. Such tilings are identified by pairing the DelaneyDress symbol of the Utiling with a covering map for the surface.
 hnet

An hnet or hyperbolic net is a twodimensional net in H² distinguished by topology only, represented by the unique maximalsymmetry DelaneyDress symbol with that topology.
 onet

An onet is a finite net formed by the edges of an Otiling embedded in 3D space. Such nets are finite ‘molecular’ nets and equivalent up to ambient isotopy class.
 enet

An enet is a net formed by vertices and edges of an Etiling. They are equivalent up to ambient isotopy class.
 snet

An snet or systre net is a threedimensional 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 symmetrypreserving
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 H²
tiling onto the given surface.
 Net names

Epinet has its own naming schema, currently implemented for hnets, enets and snets. These names contain a threeletter prefix followed by an integer, e.g. hqcN (hnets), epcN, edcN, egcN (enets) and sqcN (snets) respectively (here N is an integer). The second letter in the prefix is currently restricted to ‘q’ (generated from reticulations of genusthree 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, 81108 (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 threeletter 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' snet filter.
 Transpose pair

Most Utilings are distinguished by a unique DelaneyDress symbol in the orbifold 'ooo', however a small class of UTilings have the same Dsymbol, but are related with an automorphism that swaps the Ts and Taus. We call these UTilings '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.
 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 E².
 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, H²
as the covering space for our
triply periodic minimal surfaces.
 Covering map

A covering map is a continuous function from the hyperbolic
plane H² 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.