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Essays arrive in groups rather than one at a time. The most recent group is below in full, and every earlier one after it, newest first.

Essays arrive in groups rather than one at a time, and a group usually opens up a subject not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.

5 November 2026

8 essays on lattices, what symmetry decides, what a lattice forbids, order without repetition, into space, operations, the classification and how it is known

Lattices

One way up and p ways down

Every plane lattice has exactly p + 1 sublattices of prime index p. Restrict an integer form to each and the discriminant can do only three things, so the lattices become the vertices of a graph with a definite shape: a ring of lattices at the top, and below it a tree in which every lattice has one way up and p ways down. Whether the ring exists at all is decided by how p factors, which is why the square lattice has two copies of itself at index five and none at index three.

7 figures

Down one optic axis, at three colours, for two crystals. Interference figures of two model monoclinic crystals, inclined above and crossed below, each a plate cut across the optic axis at the sodium line, at 450, 589 and 650 nm. In the inclined crystal the melatope slides along the horizontal mirror and the figure stays symmetric across it. In the crossed crystal the melatope leaves the line away from sodium, and the pattern loses the mirror.

What symmetry decides

The mirror a plate keeps in one colour

Cut a plate of a crystal across one optic axis, and in the light the plate was cut for its interference figure has a perfect mirror, whatever the crystal's class. Change the colour and the mirror survives only if the crystal really has one. Computed ring by ring, five model crystals give the prediction exactly, including the part the eye would miss: the mirror does not come back when the line is moved to follow the melatope.

7 figures

Every form tried, by its symmetries and by how well it packs. 5902 totally positive forms of the ring of fifteenth roots of unity, sorted by the order of their symmetry group into five rows. Forms with 60 and 120 symmetries form scattered bands. Two lattices have 1,440, the tensor products of A2 with A4 and with its dual. One has 14,400, a member of the icosian line; one has 28,800, the icosian ring with its plain trace; and one is E8. The highest Hermite number among forms with only 60 symmetries is above that of the 28,800 lattice.

What a lattice forbids

One ring is E8 and a lattice with no mirror

An operation of order thirty as one block in eight dimensions preserves a whole family of forms, and E₈ was one point of it. Walking the family finds sixty symmetries almost everywhere and two lines above that. One is tensor products. The other is the icosian ring, on which E₈ is famously built, under a sliding choice of form: 14,400 symmetries all along it, 28,800 and not one reflection at one end, and at the other end E₈, where the group jumps by a factor of nearly fifty thousand.

7 figures

Order without repetition

The count of reflected hats is never one out

In a hat tiling one tile in about eight is reflected, and that ratio was only ever quoted here. Four clusters and a table of small integers give it exactly: seven plus three root five over two, the fourth power of the golden ratio. The same table says something the ratio does not. Grow a cluster to two trillion hats and its reflected count is still within half a tile of the exact share, where the same share chosen at random would be out by nearly half a million.

7 figures

530 printed rows, 499 descriptions, by crystal system. For each crystal system: the number of space group types, the number of distinct descriptions up to an origin shift, the number of rows in the printed list, and the difference. All thirty-one extra rows are monoclinic or orthorhombic.

Into space

Five hundred and thirty rows, four hundred and ninety-nine descriptions

The International Tables list 530 settings of the 230 space groups, and that number has long been quoted as a measure of how much naming a group can take. Counted from the operations, a setting being a description up to a shift of origin, the answer is 499. Thirty-one printed rows name a description already on the list, all of them in six types, and the rule that gives a group its second origin finds twenty-three of the twenty-four and misses one.

7 figures

The subgroups of index two, and what joins them. Every affine type of space group by crystal system: its subgroups of index two, 1813 in all; classes under the origin shifts alone, 1392; classes under the whole affine normaliser, 1159; and the fewest classes one cell can give, 1161. Best cell and affine agree in every system except the triclinic.

Operations

No cell joins the seven halves

Every identification the normaliser makes between Wyckoff positions is made by an isometry of some cell, and it was natural to expect the same of subgroups. It nearly is. Across all 219 types and their 1,813 subgroups of index two, one cell does everything the normaliser does, except in the two triclinic groups. P1 has seven ways to be halved, and its normaliser makes them all one. No cell can, because seven is the number the crystallographic restriction forbids.

6 figures

One tile, two lattices. The smallest polyomino on exactly two lattices with no slide between them, of 5 cells. On the left its copies sit on one lattice and on the right on another, and neither tiling can be turned into the other by moving rows.

The classification

One tile, two lattices, and a row that slides

A crystal chooses its lattice, and it is natural to think a tile does too. Among all polyominoes to thirteen cells, more than five hundred that are not rectangles lie on more than one lattice, and some on seven. They fall into at most two families, and a family is a row of copies that can slide against the next row, which is exactly what makes polytypes possible in a real crystal. A tile with a prime number of cells has no such row unless it is a bar, and so it lies on at most two lattices.

6 figures

Completeness against sweep, for four spindles. Completeness against rotation range for one tetragonal crystal, class 4/mmm, on four different spindles. The four-fold spindle rises fastest at first, then stops at 88.9 per cent at forty degrees and never improves. The two-fold spindles reach completion near 87 degrees. The general spindle is highest at forty-five degrees and reaches completion at the same 87.

How it is known

The same measurements, and a hole that stays

How much of the sphere must be swept for a given redundancy? The question has two answers, because it has two parts. The mean number of observations depends only on the sweep, to within two per cent whichever axis the crystal turns on. Completeness depends entirely on the axis, and on one kind of axis no sweep ever finishes: when the Laue group keeps the spindle's line, every copy of a blind reflection is blind too, and fifteen of a tetragonal crystal's 135 reflections are never measured.

6 figures

Before that

Everything published earlier, newest first. Titles only — the cards are on the full listing.

3 November 2026

6 essays on what a lattice forbids, order without repetition, into space, operations, the classification and how it is known

Thirty as one block lives on two lattices — what a lattice forbids
Three tiles keep a number that asks which way they face — order without repetition
A screw keeps a line and a glide keeps a plane — into space
One cell reaches every Wyckoff set — operations
In space a motif can lift two positions in three — the classification
Fit the scale, then put the lean back — how it is known

30 October 2026

4 essays on what a lattice forbids, the classification, lattices and what symmetry decides

The thirties share a lattice — what a lattice forbids
Two forty-sixes that do not match — the classification
The centring survives the limit — lattices
Three places for the two-fold axis — what symmetry decides

25 October 2026

7 essays on what a lattice forbids, order without repetition, the classification, how it is known, operations, lattices and into space

Three hundred and forty ways for twelve pentagons to touch — what a lattice forbids
Three waves measure the cube through the slope — order without repetition
Twenty-five colours, and the mirrors come back — the classification
Two leads need their vector to half a step — how it is known
The offset a quarter turn allows lies on a mirror — operations
The shortest vector is a polynomial with two full fibres — lattices
The last two crossings fall at eighty-one and twelve thousand — into space

21 October 2026

8 essays on order without repetition, the classification, how it is known, operations, lattices, into space and what symmetry decides

Two waves measure the curvature one wave hides — order without repetition
A rhombohedral box saves a factor of two the hexagon never offered — the classification
The lead is wherever it is taken out — how it is known
A three-fold axis puts a grid point on itself — operations
At seventeen the densest power of 1 − ζ moves — lattices
In space the sublattices a group keeps grow at six speeds — into space
A squeeze magnetises a cubic crystal across itself — what symmetry decides
A six-fold crystal hides its frame until rank six — what symmetry decides

19 October 2026

8 essays on what a lattice forbids, the classification, how it is known, lattices, operations, what symmetry decides and into space

The wrong model moves too far for first order — what a lattice forbids
The corner of a hexagon fits in three cells — the classification
The lead taken out solves what no threshold could — how it is known
Thirteen-fold, and nothing is its own dual — lattices
No Petersen graph in the plane — operations
Fifteen constants an energy would have forbidden — what symmetry decides
The copies a group keeps of itself grow at four speeds — into space
Thirty-five classes a squeeze can magnetise — what symmetry decides

14 October 2026

8 essays on what a lattice forbids, order without repetition, the classification, how it is known, lattices, operations, what symmetry decides and into space

A cubic site is blind to a degree-two error — what a lattice forbids
The second order cannot see a curved size effect alone — order without repetition
Five colours, seven colours, and the primes that split — the classification
The wavelengths that phase best do not find the edge — how it is known
The densest eleven-fold lattice is not its own dual — lattices
Thirteen sites obey Friedel's law without a centre — operations
A crystal in 422 needs two resonances — what symmetry decides
Three mirrors never make a crystal either — into space

5 October 2026

3 essays on what a lattice forbids, lattices and what symmetry decides

A normaliser must permute the axes — what a lattice forbids
A seven-fold lattice lives on a folded torus — lattices
Two classes go dark far from absorption — what symmetry decides

2 October 2026

5 essays on order without repetition, the classification, how it is known, operations and into space

In space, a five-fold axis is a line — order without repetition
A whole star needs a cubic box — the classification
A stack lends each sheet a tension — how it is known
The square net draws eleven groups — operations
Two mirrors never make a crystal — into space

30 September 2026

7 essays on what a lattice forbids, order without repetition, the classification, how it is known, lattices, operations and into space

A scroll cannot be turned end for end — what a lattice forbids
Five-fold centres stay few, and mirrors do not — order without repetition
Every ambiguity in a diffraction pattern, sorted by its cause — the classification
A scale error between two wavelengths has a sign — how it is known
A five-fold lattice has one free number — lattices
A cubic cell has fourteen chances to fail — operations
A crystal with no hand holds rods of both hands — into space

28 September 2026

16 essays on what a lattice forbids, order without repetition, operations, the classification, lattices, into space, what symmetry decides and how it is known

A sheet with two faces rolls into two tubes — what a lattice forbids
The last three contacts go at once — what a lattice forbids
The satellite beside the zero-order reflection — order without repetition
The third survives a rectangle — order without repetition
Three species hide a spot at the golden ratio — operations
Seventy-five is twenty-nine and forty-six — the classification
Ninety-six four-colour designs — the classification
The body-centred hypercube is the checkerboard lattice — lattices
Two operations make rock salt's group, and Pmmm needs six — into space
What the twenty lack is a product of mirrors — what symmetry decides
A homometric twin is the structure moved class by class — how it is known
The third wavelength is there to measure the edge — how it is known
The self-dual width is not flat — lattices
Where the plane had places, space has pitches — operations
Where a gradient term survives at the edge of the zone — into space
Every magnetic class counts its properties by mirrors — what symmetry decides

26 September 2026

10 essays on what a lattice forbids, order without repetition, operations, the classification, lattices, into space, what symmetry decides and how it is known

No lattice carries all three twelves — what a lattice forbids
The second order reads the strain again — order without repetition
The spots say the cell and little else — operations
Four colours, five ways to permute them — the classification
Only two and three can centre a lattice — lattices
Every band group is a ribbon — into space
A field lets every class rotate light — what symmetry decides
A second wavelength completes the map — how it is known
The best texture is a cone — what symmetry decides
A flapping sheet keeps its peaks in the plane — how it is known

25 September 2026

11 essays on what a lattice forbids, order without repetition, operations, the classification, lattices, into space, what symmetry decides and how it is known

Three rotations of order twelve — what a lattice forbids
The centring that turns into a screw — what a lattice forbids
Better counting, and the wrong model — what a lattice forbids
The normaliser only acts modulo two — what a lattice forbids
The strain wave that tips the satellites — order without repetition
Six groups are made of glides — operations
Where a symmetric motif lifts the group — the classification
The two lattices where a four-dimensional descent stops short — lattices
Seven space groups cannot make their own translations — into space
A looser fibre, more piezoelectric — what symmetry decides
The two most anomalous wavelengths are the wrong pair — how it is known

24 September 2026

13 essays on what a lattice forbids, order without repetition, operations, the classification, lattices, into space and how it is known

A rolled sheet turns by a square root — what a lattice forbids
Two hundred and nineteen over seventy-three — what a lattice forbids
What refinement takes back from a wrong model — what a lattice forbids
A chain with no mirror has mirrors everywhere — order without repetition
An occupation wave is its own Fourier series — order without repetition
One invariant for every chain that can be undone — order without repetition
Every superstructure counted once — operations
Seventy-three Smith forms — the classification
Two groups no single operation reveals — the classification
In eight dimensions one lattice wins at every width — lattices
A ribbon can be achiral where its edges are not — into space
The heavy atom the flipping solver cannot see past — how it is known
A warm plane crystal has no Bragg peaks — how it is known

23 September 2026

15 essays on order without repetition, operations, into space, the classification, lattices, what symmetry decides and how it is known

A four is not two twos — order without repetition
A winding does not dilute — order without repetition
An experiment sees a Farey sequence — order without repetition
Swapping the species does not halve the count — operations
Two conditions are not one condition twice — operations
Four groups are made of mirrors — operations
Three on a plane, and never four — into space
Eighty is seventeen, seventeen and forty-six — the classification
Four signs say where the charge sits — into space
No lattice in space wins at every width — lattices
Magnetic reflections land where nuclear ones cannot — what symmetry decides
Where a powder goes blind — what symmetry decides
Cutting a sheet can give it a hand — into space
Unequal atoms break the equality, and very unequal ones mend it — how it is known
A fibre of achiral crystals can rotate light — what symmetry decides

19 September 2026

20 essays on what a lattice forbids, order without repetition, into space, operations, the classification and how it is known

A centre at every other ring — what a lattice forbids
Everything except the hexagons — what a lattice forbids
How close the twelve must be — what a lattice forbids
Which groups a crystal could have — what a lattice forbids
Straight lines, and no distances — what a lattice forbids
Finitely many is not few — what a lattice forbids
What a defect costs the count — order without repetition
The count that depends on the edge — order without repetition
Three colours on a chessboard — order without repetition
An ideal across and a prime along — into space
A row written as a product — into space
A lattice is not a subgroup — into space
The richest group has the poorest arithmetic — into space
Closing the plane from two centres — operations
The axis a product lies on — operations
Two patterns laid over one another — operations
The screw a dimension does not have — the classification
The denominator a group actually needs — the classification
The relation that can say no — how it is known
The relation that is an equality — how it is known

16 September 2026

2 essays on what a lattice forbids and order without repetition

A rolled sheet is never one of a pair — what a lattice forbids
Aperiodic is two words in space — order without repetition

15 September 2026

7 essays on what a lattice forbids, operations, into space, order without repetition and how it is known

Most sheets roll into a tube that never repeats — what a lattice forbids
The molecule size that hides a disorder — what a lattice forbids
Going up costs the cell a parameter — operations
A thread's hand is not a choice — into space
Every fraction holds a window — order without repetition
Symmetry does not rescue a Patterson — how it is known
A line carries one screw — into space

14 September 2026

4 essays on operations, how it is known, into space and order without repetition

Three of them, and they are equivalent — operations
A map of the atoms that break the law — how it is known
One matrix, four rules — into space
A window that is not an interval — order without repetition

13 September 2026

6 essays on what a lattice forbids, order without repetition, operations and into space

What forces a lattice — what a lattice forbids
The tube has a screw no lattice allows — what a lattice forbids
How much of the hat is a crystal — order without repetition
The ice rule is a conservation law — order without repetition
Two mirrors a coset cannot tell apart — operations
The primes a cell can grow by — into space

12 September 2026

8 essays on what a lattice forbids, order without repetition, into space and operations

Eleven, eleven and ten — what a lattice forbids
Seven friezes round a cylinder — what a lattice forbids
A gap the sphere does not have — what a lattice forbids
The occupancy does not name the disorder — what a lattice forbids
The hat and the turtle are one tiling — order without repetition
A hand made of pieces that have none — into space
Chiral in the plane is not chiral in the room — into space
The same site under two names — operations

10 September 2026

5 essays on what a lattice forbids, how it is known and operations

The twelve belongs to the vertex — what a lattice forbids
The surfaces a count by genus skips — what a lattice forbids
How many orientations a disorder needs — what a lattice forbids
When the atoms are not all the same — how it is known
The normaliser is not a function of the group — operations

8 September 2026

7 essays on symmetry at work, into space, what symmetry decides and how it is known

The room a thirteenth sphere would need — symmetry at work
The point defect whose charge has no sign — symmetry at work
What six lengths decide and nine do not — symmetry at work
A game that decides what counting only bounds — symmetry at work
The phase a symmetry turns into a number — into space
The strain the atoms do not follow — what symmetry decides
The threshold a symmetry pins down — how it is known

7 September 2026

22 essays on symmetry at work, the classification, lattices, what symmetry decides, into space and operations

The angle two grains differ by — symmetry at work
What a half-turn does to three colours — the classification
A reduction with one rule — lattices
Thirty-two from fourteen matrices — what symmetry decides
The halving a lattice will not permit — lattices
The defect that needs two laps — symmetry at work
A screw that contains its own mirror image — into space
Seventy-three, without a search — what symmetry decides
The sum that turns a lattice into its dual — lattices
The sum whose answer depends on the shape — lattices
The plane a deformation leaves alone — symmetry at work
The plate that only fits when it is twinned — symmetry at work
The angle that is not a fraction of a turn — the classification
Where the boundary goes when the atoms differ — lattices
The boundary a growing region forgets — operations
How many axes there are is a Sylow count — what symmetry decides
The table that decides every action — what symmetry decides
Lattices that agree at every prime — lattices
Every parallelohedron is a shadow of a cube — the classification
The polyhedra that can flex — symmetry at work
A lattice cannot have all its vectors long — lattices
A polyhedron is two properties of a graph — symmetry at work

6 September 2026

8 essays on order without repetition, what a lattice forbids, lattices, what symmetry decides, symmetry at work and operations

A facet with no energy in it — order without repetition
Four root systems, and the same four rotations — what a lattice forbids
One perfect form in space — lattices
Two turns to come back — what symmetry decides
A beat is not a period — symmetry at work
The relations a polygon dictates — operations
Every alias is a supercell — symmetry at work
The shapes a lattice in space can thin to — lattices

5 September 2026

13 essays on symmetry at work, operations, order without repetition, what a lattice forbids, the classification, lattices and what symmetry decides

The symmetry a net was written with — symmetry at work
Every net with two vertices, counted — symmetry at work
The quotient each normal subgroup leaves — operations
The mechanisms a count cannot see — symmetry at work
Which faces are flat — symmetry at work
A stack with no space group — symmetry at work
Superspace groups in the plane — order without repetition
The four groups with a centre — operations
As many heptagons as pentagons — what a lattice forbids
The argument that closes eleven — the classification
What a thread scatters — the classification
Every plane lattice is its own dual — lattices
Twelve of the thirty-two are free — what symmetry decides

3 September 2026

6 essays on lattices, order without repetition, into space, what symmetry decides and symmetry at work

The three that stay cubic — lattices
How often each patch occurs — order without repetition
The degeneracy time reversal forces — into space
What a texture permits — what symmetry decides
The figure of merit a supercell always beats — symmetry at work
The densest packing of a shape that is not a disc — symmetry at work

1 September 2026

5 essays on into space, how it is known and symmetry at work

How chiral, as a number — into space
Which levels join which, on the way out of a point — into space
Two structures on a torus, and one Patterson — how it is known
How many reflections it takes to know there is a centre — how it is known
A hundred and thirteen orbits, and forty-eight shapes — symmetry at work

31 August 2026

14 essays on operations, lattices, order without repetition, the classification, how it is known, what symmetry decides and symmetry at work

What a position becomes on the way down — operations
A bigger cell, a smaller zone — lattices
One tile, and no period — order without repetition
How much room a hard question needs — the classification
The tile that needs no reflection — order without repetition
Neither a peak nor a bump — order without repetition
The solver that knows no symmetry — how it is known
What a group forbids to happen — what symmetry decides
How much of it is the other hand — how it is known
The axes a class pins down — what symmetry decides
A translation that is nearly there — how it is known
What one turn of the crystal reaches — how it is known
The dislocations a boundary allows — symmetry at work
The wall has a group of its own — symmetry at work

30 August 2026

20 essays on operations, what a lattice forbids, lattices, order without repetition, the classification, into space, what symmetry decides, how it is known and symmetry at work

An orbit is what the invariants cannot tell apart — operations
Which modes a site can carry — operations
The degrees that name the restriction — what a lattice forbids
How many lattices share a determinant — lattices
The lattice that minimises a sum — lattices
The average is the same wherever it is taken — order without repetition
How many waves a group permits — the classification
An order parameter is a representation — into space
How many invariants of each degree — what symmetry decides
The groups whose invariants are free — what symmetry decides
Which way the order parameter points — into space
The cell a zone-boundary mode doubles — into space
Three invariants and one relation — what symmetry decides
The average scatters sharply and the rest does not — how it is known
The cubic term that forbids a continuous change — what symmetry decides
The order a diffuse pattern measures — how it is known
The parts a property splits into — what symmetry decides
The strain that arrives with the transition — symmetry at work
The walls a strain permits — symmetry at work
The polarisation nobody asked for — symmetry at work

28 August 2026

20 essays on operations, lattices, into space, what symmetry decides, order without repetition, the classification, how it is known and symmetry at work

The domain in reciprocal space — operations
The sublattices that are the same shape — lattices
How far one lattice is from another — lattices
The star of a wavevector — into space
What a group does to a function — what symmetry decides
A spectrum that is a Cantor set — order without repetition
How large a degeneracy may be — what symmetry decides
Where two levels must meet — into space
A coincidence the group did not ask for — what symmetry decides
A glide sticks two levels together — into space
Crystallography in a box — the classification
The crossing at the corner — into space
The shell that splits into kinds — what symmetry decides
Every reflection, several times over — how it is known
A twin hides in the statistics — how it is known
The level that does not move — symmetry at work
A mechanism that is a wave — symmetry at work
Every net with one vertex, counted — symmetry at work
The step that never runs out — symmetry at work
A small angle is a row of dislocations — symmetry at work

27 August 2026

15 essays on operations, lattices, what a lattice forbids, the classification, order without repetition, how it is known and symmetry at work

How fast a group grows — operations
Every wall names a generator — operations
How many points a shape holds — lattices
Every way down, and no way round — lattices
The restriction, with no lattice assumed — what a lattice forbids
Every net folds onto a torus — the classification
The crystal you get by rounding τ off — order without repetition
Three answers in whole numbers — the classification
The streaks a faulted stack makes — how it is known
A structure with the distances thrown away — symmetry at work
Counting outwards — symmetry at work
The placement nobody chose — symmetry at work
A net is a choice of what counts as a bond — symmetry at work
The count that promises a mechanism — symmetry at work
A fold that keeps its symmetry — symmetry at work

26 August 2026

15 essays on operations, lattices, what a lattice forbids, the classification, order without repetition, how it is known and symmetry at work

One crystal, and sixteen coordinate lists — operations
The shortest vector, and where it stops being easy — lattices
What a molecule gives up to sit in a crystal — what a lattice forbids
A tiling of the whole plane, decided on one tile's edge — the classification
One shape, two kinds of tile — the classification
Three gaps, and never four — order without repetition
Surrounded twice over, and covering nothing — the classification
Two lattices, one crystal, and no cell at all — order without repetition
Three phases that do not move when the origin does — how it is known
How many dislocations a lattice has — symmetry at work
The formula that has the answer already — how it is known
The average that knows the atoms and not where they are — how it is known
The index and the angle a twin misses by — symmetry at work
A zone is a vanishing dot product — symmetry at work
A cell from a bag of spots — symmetry at work

24 August 2026

15 essays on operations, lattices, what a lattice forbids, the classification, order without repetition, into space, how it is known and symmetry at work

Three reflections, and never four — operations
Every motion of space is a screw — operations
Covering and packing want different lattices — lattices
Discrete, or dense, and nothing between — lattices
The same group means the same pattern — what a lattice forbids
Five copies, and the gap they leave — what a lattice forbids
Seventeen, without a picture — the classification
How many arrangements one rule allows — order without repetition
Nothing decides whether a set of tiles tiles the plane — the classification
The arrangements a crystal keeps at absolute zero — order without repetition
The groups a single hand may sit in — into space
A bigger cell, and sometimes the mirror — into space
The absence that fills itself in — how it is known
The zones that behave as if there were a centre — how it is known
The circuit that does not close — symmetry at work

22 August 2026

15 essays on operations, lattices, the classification, what a lattice forbids, order without repetition, how it is known, into space, what symmetry decides and symmetry at work

Every colour count at once — operations
The space every lattice lives in — lattices
Two moves reach every basis — lattices
The two that fold into a surface — the classification
Twelve pentagons, and no way round them — what a lattice forbids
Which shapes tile by themselves — the classification
Seventy-five ways to be a thread — the classification
The tiling that points every way — order without repetition
Eight-fold, with the golden ratio taken out — order without repetition
Seventeen groups, seven vector sets — how it is known
Ten ways for space to be flat — into space
The seven groups a field can have — what symmetry decides
The unknowns against the observations — how it is known
What a crystal keeps in a field — what symmetry decides
The four plane groups a molecule packs in — symmetry at work

21 August 2026

15 essays on operations, lattices, the classification, what a lattice forbids, order without repetition, how it is known, into space, what symmetry decides and symmetry at work

How many subgroups of index three — operations
How many vectors of each length — lattices
Telling two words apart — operations
The lengths do not name the lattice — lattices
Twenty-one vertices, eleven tilings — the classification
Eleven tilings, five groups — the classification
Eleven duals, one tile each — the classification
Five solids from one inequality — what a lattice forbids
The average that makes it finite — what a lattice forbids
How many patches of each size — order without repetition
Two structures, one Patterson — how it is known
The same group in a bigger cell — into space
Where the pairs come from — how it is known
Thirty-two classes, eighteen groups — what symmetry decides
The densest lattice in the plane — symmetry at work

20 August 2026

15 essays on operations, lattices, the classification, what a lattice forbids, order without repetition, how it is known, into space, what symmetry decides and symmetry at work

A group in four letters — operations
What is left when the order is forgotten — operations
How few operations make a pattern — operations
Five parallelohedra, and no others — lattices
How much pattern is enough — the classification
Why there is a list at all — what a lattice forbids
Reduction modulo three — what a lattice forbids
n plus one, and no fewer — order without repetition
Every patch comes back — order without repetition
How many reflections there are — how it is known
As sharp as the sphere is wide — how it is known
The plan contains the group — into space
Fifteen may rotate light, and eleven are chiral — what symmetry decides
The fast faces are the ones that vanish — symmetry at work
Indexing a powder pattern — symmetry at work

19 August 2026

15 essays on operations, lattices, the classification, what a lattice forbids, order without repetition, how it is known, into space, what symmetry decides and symmetry at work

Where the product is — operations
Counting what a group cannot tell apart — operations
The zones above the first — lattices
The cell that settles the argument — lattices
Seventy-four colourings, forty-six groups — the classification
Past two, the list does not stop — the classification
Thirteen ways to hold a lattice — what a lattice forbids
The most of an icosahedron a crystal can keep — what a lattice forbids
Six integers, and the lattice that holds them — order without repetition
Whether there is a centre is a statistic — how it is known
One experiment gives the cosine, the other gives the sine — how it is known
The plane that carries two glides — into space
Six ways to name one group — into space
Three optical characters, and the arithmetic that assigns them — what symmetry decides
Which faces a crystal shows — symmetry at work

18 August 2026

15 essays on what a lattice forbids, operations, how it is known, into space, order without repetition, lattices, the classification, symmetry at work and what symmetry decides

The symmetry of an average — what a lattice forbids
The descent with no shortcut — operations
The law that hides handedness — how it is known
Two origins for one group — into space
Which inflation factors exist — order without repetition
One symmorphic group per class — into space
Solving from the vector set — how it is known
The freedom a crystal has not — order without repetition
The same pattern, described twice — operations
The cell nobody chose — lattices
Seventeen dollars — the classification
The reflections a superlattice adds — lattices
What a cleave leaves — the classification
How many polytypes there are — symmetry at work
One class, two names — what symmetry decides

16 August 2026

15 essays on what a lattice forbids, how it is known, order without repetition, into space, operations, lattices, the classification, what symmetry decides and symmetry at work

Before the lattice has a say — what a lattice forbids
A fivefold axis in an ordinary crystal — what a lattice forbids
The map that needs no phases — how it is known
The satellites that need a second integer — order without repetition
One group, three symbols — into space
The extra dimension that makes it periodic — order without repetition
Two ways down from a group — operations
Where symmetry stacks the vectors — how it is known
How many ways there are to thin a lattice — lattices
A layer is not a wallpaper — the classification
The sublattices that stay square — lattices
Three colours, and why most patterns cannot have them — the classification
The operation that reverses time — what symmetry decides
Two stackings, one density — symmetry at work
Which magnetism a class permits — what symmetry decides

15 August 2026

15 essays on symmetry at work

Why a crystal face carries small whole numbers — symmetry at work
The angles belong to the substance, the shape to the specimen — symmetry at work
A form is an orbit, and whether it closes is an integer question — symmetry at work
Five classes grow the same cube — symmetry at work
A twin is a symmetry the lattice has and the crystal does not — symmetry at work
Twenty-five of the thirty-two can twin, and seven cannot — symmetry at work
Quartz has exactly three twin laws, and its lattice is why — symmetry at work
A merohedral twin moves no spot at all — symmetry at work
How many domains a transition makes is an index — symmetry at work
The descent of symmetry is a lattice, not a tree — symmetry at work
The domains a lost translation makes, which nothing optical can see — symmetry at work
Two hundred and forty-seven descents, or two hundred and twelve — symmetry at work
Turn a lattice against itself and almost nothing lines up — symmetry at work
Every coincidence index is odd, and in the plane most of them do not exist — symmetry at work
Two different lattices never coincide, and the question becomes how nearly — symmetry at work

13 August 2026

15 essays on what symmetry decides

Thirty-two, and no others — what symmetry decides
A fingerprint that gave the right answer — what symmetry decides
What a trace decides — what symmetry decides
Reading a class off its own axes — what symmetry decides
3m1 and 31m are one class — what symmetry decides
The holohedry is the ceiling — what symmetry decides
The eleven a diffraction pattern reports — what symmetry decides
Neumann's principle, as one sum — what symmetry decides
A character does not know its basis — what symmetry decides
Twenty-one, thirteen, nine, three — what symmetry decides
Permitted is not present — what symmetry decides
The ten with a direction of their own — what symmetry decides
Twenty of the twenty-one — what symmetry decides
Each permits what the other forbids — what symmetry decides
A filter of great precision and no predictive power — what symmetry decides

11 August 2026

15 essays on into space, operations, how it is known and lattices

The step a flat surface has no room for — into space
Forgetting a group in three dimensions — into space
The half of a translation that is not a choice — into space
Sixteen candidates, ten groups — into space
Eleven groups that are their own reflection's rival — into space
Turning and climbing at once — into space
Eleven ways to turn while climbing — into space
One part in however many, and why it is never quite that — operations
The reflections that are not there — how it is known
Reflect, then slide by half of something — into space
Twenty-five cells, and fourteen lattices — lattices
Where the experiment runs out — how it is known
The operations nobody put in — into space
Forty-eight becomes sixteen — lattices
A lattice described on somebody else's axes — lattices

10 August 2026

12 essays on how it is known, lattices, order without repetition, operations and the classification

The symmetry diffraction adds — how it is known
Centring, counted as a sublattice — lattices
Near-symmetry, and the tolerance that is not here — how it is known
The smallest quasicrystal — order without repetition
The points a group treats differently — operations
Domains of a subgroup — operations
Icosahedral symmetry — order without repetition
Why the bigger cell wins — lattices
Why sixteen become seven — the classification
The friezes inside the seventeen — the classification
Two colours, and a symmetry that swaps them — the classification
The Alhambra question — the classification

4–8 August 2026

34 essays on order without repetition, what a lattice forbids, lattices, how it is known, the classification and operations

Penrose tilings — order without repetition
The crystallographic restriction — what a lattice forbids
The lattice underneath — lattices
The reciprocal lattice — how it is known
The seventeen — the classification
What a symmetry actually is — operations
Five lattices, and no others — lattices
Inflation, and where the golden ratio comes from — order without repetition
Reading Hermann–Mauguin — the classification
Systematic absences — how it is known
The four motions of the plane — operations
Why five-fold is impossible — what a lattice forbids
Order is not periodicity — order without repetition
p3m1 and p31m — the classification
The cell is a choice, the lattice is not — lattices
The phase problem — how it is known
The restriction in three dimensions — what a lattice forbids
Why it is a group and not a list — operations
Centring, and why cm is not pm — lattices
Matching rules, and what actually forces aperiodicity — order without repetition
The motif must be a comma — the classification
The orbit is the pattern — operations
What a powder pattern loses — how it is known
Where five-fold becomes legal — what a lattice forbids
The same symmetry, somewhere else — operations
Cut and project — order without repetition
Seven friezes — the classification
Reduction, and the shortest basis — lattices
The fundamental domain — operations
The classification proof, one branch at a time — the classification
The dual lattice, as a construction — lattices
What Shechtman measured — order without repetition
Orbifold notation, the shorter language — the classification
The groups ornament actually uses — the classification

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