For newcomers
Nets, tilings and why topology matters
Why a database of abstract graphs is the useful thing to have when you are designing a real material.
The move that makes reticular chemistry work
Take a metal-organic framework and throw away the chemistry. Replace each metal cluster with a point. Replace each linker with a line joining two points. What remains is a periodic graph — an infinite scaffold of vertices and edges.
That graph is the net, and the surprise is how few of them matter. Tens of
thousands of frameworks have been synthesised; the great majority adopt one of
a few dozen nets. MOF-5 and hundreds of unrelated materials are all pcu, the
primitive cubic net.
This is why a database of abstract graphs earns its place next to a database of
structures. If you know that a six-connected node and a linear linker tend to
give pcu, you can predict what you will get before you synthesise it — and
you can ask the opposite question, which is the useful one: given the net I
want, what would I have to build?
What makes one net different from another
Two nets can have the same coordination numbers everywhere and still be different graphs. Distinguishing them takes finer invariants.
Coordination sequence. Stand on a vertex. Count the vertices one step away,
then two steps, then three. For pcu: 6, 18, 38, 66, 102. Nets that agree for
three shells usually diverge by the fourth or fifth — which is why the search
matches on however many shells you give it, and why giving more shells than you
trust will lose you the right answer rather than confirm it.
Every record's sequences are the source's own values, shown as the source sent them. Beside them a record now also shows the sequence walked from the net's own graph, for every vertex — including the 997 vertex rows whose upstream record sent nothing. Where both exist they agree: 122 318 vertex rows compared, none disagreeing.
Vertex symbol. Describes the shortest rings passing through each pair of edges at a vertex. It catches differences in local ring structure that the sequence smooths over.
Two conventions go by that name, and a record here shows both rather than
picking one. Take the pair of edges — the angle — and look for the shortest
closed path through both. The cycle symbol records the shortest such
cycle per angle with how many there are: this is the convention MOF+ stores,
and it is what the stored vs string is comparable with. The RCSR ring
vertex symbol records the shortest ring per angle instead — a ring being a
cycle that is not the sum of two smaller ones, so a cycle with a shortcut
across it does not count. An angle can have cycles and no ring at all, and RCSR
writes * there. The point symbol throws the counts away and tallies only
the sizes, 4^12.6^3 style.
They differ in practice, not just in principle: sra has cycle symbol
4.4.6.6.6(2).8(6) and ring vertex symbol 4.4.6.6.6.8(2). Quoting one where
the other is meant is the ordinary way to make two identical nets look
different.
Transitivity pqrs. How many kinds of vertex, edge, face and tile the net
has. pcu is 1,1,1,1 — one kind of everything. This is not a curiosity: nets
with low transitivity are the ones frameworks actually adopt, because a highly
symmetric net asks less of the chemistry. When you are choosing a target, low
transitivity is a reason to prefer it.
p and q are counted from the net's own graph and are on 2806 records.
r and s count faces and tiles, which needs a tiling — and this site does
not have one. They are blank on all but one curated record, a search treats them
as wildcards, and the section below says exactly why, because "not computed" and
"none" are different things and only the first is true here.
Tilings, and why r and s are blank
A net can be filled with tiles — the natural cages it encloses — and that
tiling says things the graph alone does not: pore shape, how cages connect,
what would fit inside. The r and s of pqrs count distinct faces and
tiles, and getting them means computing the natural tiling.
This site does not compute one, and the reason is now a measured one rather
than an absence of effort. The tool it would be computed with is here: the
Systre jar this project already runs is the whole Gavrog library, tilings and
all. It was tried, on 2026-09-12, and the result is written up in
docs/reports/tiling-spike-2026-09-12.md in the repository — the short version:
- For a 3-periodic net there is no route from a net to a tiling in that
library at all. Exactly one method in its 650 classes turns a net into the
Delaney symbol a tiling is built from, and its first line throws
Only nets of dimension 2 are supported. That is an unimplemented feature, not a display problem and not a configuration problem. 2730 of the records here are 3-periodic, so for almost the whole collectionrandscannot be computed from the tool in hand. - For a 2-periodic net it works completely: all 200 layer nets here build a
tiling in milliseconds, and its vertex and edge orbit counts agree with
Systre's
pandqon every one of them.
And they still get no r or s, deliberately. A plane tiling has faces and
no tiles — s would be a count of something that does not exist, and a zero
there would be a measurement of nothing. Its faces are not the faces a
3-periodic tiling has either: a pqrs quadruple means vertices, edges, faces and
tiles of a space-filling tiling, and putting a plane tiling's face count into
r would be two different quantities sharing one column — the same mistake the
plane group has its own column to avoid. The honest
shape for a layer net is a triple, and that is a pair of new columns and a
decision nobody has taken, not a number to slip into an existing field.
So r and s are blank on all but one curated record, a search treats them as
wildcards by default, and blank here means not computed, never none. A
net's tiling exists; this site cannot tell you about it.
What a net's symmetry is, and why computing it is not guessing
A record shows a space group, and it is fair to ask where it comes from when neither source supplies one and a topology file has no space-group line.
It comes from the net. An abstract periodic graph has no coordinates, but among all the ways of drawing it in space there is one of maximum symmetry — the barycentric placement, where every vertex sits at the average of its neighbours. That embedding's symmetry group is a property of the graph alone, and that is the space group quoted here. It is derived from the connectivity, not read off a deposited cell, and this matters: a deposited cell reflects how somebody happened to draw the net, while the net's own group does not change when you redraw it.
This is what Systre computes, and it is the same
computation behind the canonical Systre key — a string that is identical for
two graphs exactly when they are the same net. It is how the site can say that a
MOF+ deposit and an RCSR archive entry filed under one name really are one net:
2292 of them are, and one, skl-a, is not.
Two limits are worth knowing. A 2-periodic net — a layer — has a plane group,
numbered 1–17, not one of the 230 space groups, so those 200 nets show their
group in the computed-topology block rather than in the space-group column. And
for the eleven enantiomorphic pairs such as P6222/P6422 the graph fixes the
group only up to the pair: a graph carries no handedness, the drawing does. A
record in that position says so beside the symbol.
Density and genus: two more numbers on a record
TD10 is the topological density: start at a vertex, add up how many vertices
lie within ten steps (the vertex itself included), and average that over the
vertices of one repeat unit — equivalently, a mean over the kinds of vertex
weighted by how many vertices each kind holds, which is not the same number as
an unweighted average whenever the kinds are of different sizes. Denser nets
score higher — pcu 1561 — and it separates nets that share a coordination
sequence prefix. The exact value is served; Systre prints it rounded half up, and
where a Systre run exists that integer is served beside it.
Genus counts the independent cycles of one repeat unit: 1 + E − V on the
smallest cell that tiles the net. pcu 3, dia 3, srs 3, nbo 4, fcu 6,
sod 7. Three is the minimum for a connected 3-periodic net, which is another
way of saying that a framework needs at least three independent directions of
travel. It is defined for a connected 3-periodic net and left blank otherwise —
on a layer, for instance — rather than reported as zero.
Catenation
Two identical nets can interpenetrate without bonding to each other. The result
is a real, distinct material with half the pore volume, and it is a distinct
record here, usually named with a -c suffix.
Whether you want catenated nets in your results depends on what you are asking. Designing for porosity: exclude them. Explaining why your synthesis produced half the surface area you expected: they are the answer.
A name is not a measurement, and the site no longer treats it as one. Interpenetration is counted from the graph: build the quotient graph, work out the lattice its cycles generate, and see how many copies of the net that lattice unfolds into. One copy means not catenated; two means two nets threaded through each other.
Run over the whole collection, that measurement agrees with the -c names on
49 records and disagrees on 8 — qtz-c, qtz-c3, qtz-c4, qtz-c6,
dia-c3s, eta-c, eta-c4 and twt-c each hold a single copy of the parent
net, whatever the record was meant to describe. Nothing was found in the other
direction: no net without the marker turned out to be interpenetrated. Five
records are left undetermined on purpose — four stacks of layers and one
bundle of chains — because whether layers are polycatenated is a question about
how they thread through space, which a graph does not record. Undetermined is
shown as undetermined, never as "no".
54 records are flagged catenated today. Where the name and the measurement disagree, both are on the record: the measurement in the value, the old name heuristic kept beside it.
Augmented nets, and what an -a suffix means
The other suffix you will meet often is -a. An augmented net is what you
get by zooming in on every vertex: replace the vertex by its coordination
figure — the polygon or polyhedron whose corners are the directions its edges
leave in, a triangle for a three-connected vertex, a square or a tetrahedron for
a four-connected one — and replace each original edge by a single link joining
two figures. skl-a is skl treated that way. 456 of the 3077 records here
are named with that suffix, and 452 of them sit beside a record under the
parent symbol.
This is not decoration. A real framework has extended nodes — a paddlewheel, a
carboxylate cluster — and the augmented net is the one whose geometry those
nodes actually match, which is why the reticular literature quotes X-a where
the underlying topology is X. Two properties fall straight out of the
definition and are worth knowing, because they are how you check that a graph
really is one: every vertex of an augmented net carries exactly one link
edge, and contracting each figure back to a point must return the parent net.
That check is not run on every record here, but it has been run in earnest once.
One record, skl-a, holds two graphs under one name, and contracting each of
them is what decided which net the symbol denotes: the MOF+ deposit contracts
back to skl exactly, self-loops included, and the RCSR archive net of that name
does not — 8 of its 56 vertices carry no link edge at all. The evidence and the
ruling are in docs/reports/net-analysis-2026-09-12.md §12 in the repository.
How the nets are ordered
Nets are browsed in RCSR symbol order, which is a column on the record rather than something worked out when the page is drawn. There is a second, unrelated number on every record — the order the nets were ingested in, simplest first — and the two get confused often enough to be worth naming: the two orderings a net carries says which is which, where each comes from, and which fields no source supplies.
Where this leads
The database is organised so that the design question runs forwards. Pick a net by its topology, find building blocks whose connectivity matches its vertices, and assemble — the reverse topological approach.