EPINET Project 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 subject area.
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 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. 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 /
Covering Map
-
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.
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.
-
Surface Tiling /
Surface Reticulation
-
A surface tiling is the projection of a 2D hyperbolic
tiling onto a
triply periodic minimal surface.
via a symmetry-preserving
covering map.
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.
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.
References:
- 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.
References:
- 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.
-
Caveat: Currently within EPINET, the tile–transitive tilings
have Delaney–Dress symbols in their canonical form, but this is not
the case for their dual vertex–transitive tilings.
Thus, a tiling which is both vertex–transitive and tile–transitive
will appear in the database with two different Delaney–Dress symbols.
This will be fixed in a later release of the EPINET site.
- Schläfli Symbol
-
The Schläfli symbol encodes the topology of a 2D
tiling, or equivalently, a 2D periodic network. A Schläfli
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 Schläfli symbol lists each vertex symbol 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. For 3D crystalline nets, Mike O'Keeffe has defined an
Extended Schläfli Symbol.
-
Vertex–k–Transitive /
Tile–k–Transitive
-
The numeral k describes the number of
symmetrically–distinct vertices or tiles in a tiling.
Currently, the EPINET pages list vertex– or tile–
1– and 2– transitive tilings.
A vertex–k–transitive tiling is always the
dual
of a tile–k–transitive tiling.
- 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. This abstract symmetry reduces
the number of different Delaney–Dress tiling symbols found when enumerating
all possible glue or split operations that 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 Canadian 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 Relating to Networks
- Valency / Degree / Coordination
-
The number of edges that are incident at a vertex is referred to as
the valency, degree, or coordination
of the vertex.
- Coordination Sequence
-
Elements in the coordination sequence are defined by the
number of
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.
- Periodic Network
-
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).
- Extended Schäfli Symbol
-
This is one way to generalise the
2D Schäfli symbol
to 3D crystalline nets. It is particularly
useful for four–coordinated networks, such as zeolite
frameworks. The symbol requires the notion of “ring”,
and there are various definitions of “strong ring”,
“essential ring”, and so on.
-
The extended Schläfli
symbol of a z–coordinated vertex contains z*(z–1)/2 entries: one for each pair
of edges coincident to that vertex. For four–coordinated
nets, complementary pairs are listed consecutively. We are
exploring other possible notations for 3D frameworks of high vertex
degree.
References:
-
The
IZA Structure Database
-
Crystal Structures: I Patterns and Symmetry by
M. O'Keeffe and B.G. Hyde (Mineralogical Society of America, 1996)
- 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.
References:
-
S.J. Chung, T. Hahn, and W.E. Klee. Z. Kristallogr. 212:553–558 (1984).
-
Barycentric Embedding /
Equilibrium Placement
-
Given the topology of a euclidean
periodic network,
(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 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 networks,
however: those that have no
vertex collisions.
-
Where possible, EPINET lists the barycentric embedding of a net to
define the simplest possible network geometry consistent with the
network topology. Note that the barycentric symmetry is generally
not the same as the
surface spacegroup
found from the inital
surface reticulation.
References:
-
O. Delgado–Friedrichs and M. O'Keeffe,
"Identification of and symmetry computation for crystal nets", Acta
Cryst. A 59, pp. 351–360 (2003)
- 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.
References:
-
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 networks are those that correspond to
the system of contacts in a packing of spherical balls.
Homogenous sphere packings have the property that every edge
has the same length, and no two vertices are closer than this
distance.
- CrystalMaker Files
-
The
CrystalMaker
file links on the EPINET 3D network pages are text files
formatted to be read directly into the CrystalMaker software.
We have provided more details of EPINET's use of CrystalMaker on
the
EPINET and CrystalMaker page.
