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Lattices
- Five lattices, and no others
- The sublattices that are the same shape
- How many lattices share a determinant
- The three that stay cubic
- A bigger cell, a smaller zone
- Centring, and why cm is not pm
- Centring, counted as a sublattice
- Covering and packing want different lattices
- Discrete, or dense, and nothing between
- Every way down, and no way round
- Five parallelohedra, and no others
- Twenty-five cells, and fourteen lattices
- How far one lattice is from another
- How many points a shape holds
- How many ways there are to thin a lattice
- How many vectors of each length
- The cell is a choice, the lattice is not
- The cell nobody chose
- The cell that settles the argument
- The dual lattice, as a construction
- The lattice that minimises a sum
- The lattice underneath
- The lengths do not name the lattice
- The reflections a superlattice adds
- A lattice described on somebody else's axes
- Reduction, and the shortest basis
- The shortest vector, and where it stops being easy
- The space every lattice lives in
- The sublattices that stay square
- The zones above the first
- Two moves reach every basis
- Why the bigger cell wins
- Forty-eight becomes sixteen
- Every plane lattice is its own dual
- One perfect form in space
- The shapes a lattice in space can thin to
- A reduction with one rule
- The halving a lattice will not permit
- The sum that turns a lattice into its dual
- The sum whose answer depends on the shape
- Where the boundary goes when the atoms differ
- Lattices that agree at every prime
- A lattice cannot have all its vectors long
Diffraction
- Systematic absences
- The symmetry diffraction adds
- Near-symmetry, and the tolerance that is not here
- The reflections that are not there
- The map that needs no phases
- Where symmetry stacks the vectors
- Every reflection, several times over
- The average scatters sharply and the rest does not
- The order a diffuse pattern measures
- The solver that knows no symmetry
- How much of it is the other hand
- A translation that is nearly there
- What one turn of the crystal reaches
- Two structures on a torus, and one Patterson
- How many reflections it takes to know there is a centre
- A twin hides in the statistics
- As sharp as the sphere is wide
- How many reflections there are
- Seventeen groups, seven vector sets
- Solving from the vector set
- The absence that fills itself in
- The average that knows the atoms and not where they are
- One experiment gives the cosine, the other gives the sine
- The formula that has the answer already
- The law that hides handedness
- The phase problem
- The reciprocal lattice
- The streaks a faulted stack makes
- The unknowns against the observations
- The zones that behave as if there were a centre
- Three phases that do not move when the origin does
- Two structures, one Patterson
- What a powder pattern loses
- Where the experiment runs out
- Where the pairs come from
- Whether there is a centre is a statistic
- The threshold a symmetry pins down
- When the atoms are not all the same
- A map of the atoms that break the law
- Symmetry does not rescue a Patterson
- The relation that can say no
- The relation that is an equality
Classification
- p3m1 and p31m
- Seven friezes
- How much pattern is enough
- How many waves a group permits
- How much room a hard question needs
- A layer is not a wallpaper
- A tiling of the whole plane, decided on one tile's edge
- Crystallography in a box
- Eleven duals, one tile each
- Eleven tilings, five groups
- Every net folds onto a torus
- Nothing decides whether a set of tiles tiles the plane
- One shape, two kinds of tile
- Orbifold notation, the shorter language
- Past two, the list does not stop
- Reading Hermann–Mauguin
- Seventeen dollars
- Seventeen, without a picture
- Seventy-five ways to be a thread
- Seventy-four colourings, forty-six groups
- Surrounded twice over, and covering nothing
- The Alhambra question
- The classification proof, one branch at a time
- The friezes inside the seventeen
- The groups ornament actually uses
- The motif must be a comma
- The seventeen
- The two that fold into a surface
- Three answers in whole numbers
- Three colours, and why most patterns cannot have them
- Twenty-one vertices, eleven tilings
- Two colours, and a symmetry that swaps them
- What a cleave leaves
- Which shapes tile by themselves
- Why sixteen become seven
- The argument that closes eleven
- What a thread scatters
- What a half-turn does to three colours
- The angle that is not a fraction of a turn
- Every parallelohedron is a shadow of a cube
- The screw a dimension does not have
- The denominator a group actually needs
Operations
- The points a group treats differently
- One part in however many, and why it is never quite that
- What is left when the order is forgotten
- An orbit is what the invariants cannot tell apart
- Which modes a site can carry
- A group in four letters
- Counting what a group cannot tell apart
- Domains of a subgroup
- Every colour count at once
- Every motion of space is a screw
- Every wall names a generator
- The fundamental domain
- How fast a group grows
- How few operations make a pattern
- How many subgroups of index three
- The descent with no shortcut
- One crystal, and sixteen coordinate lists
- Telling two words apart
- The domain in reciprocal space
- The four motions of the plane
- The orbit is the pattern
- The same pattern, described twice
- Three reflections, and never four
- What a position becomes on the way down
- What a symmetry actually is
- Where the product is
- Why it is a group and not a list
- The quotient each normal subgroup leaves
- The four groups with a centre
- The relations a polygon dictates
- The boundary a growing region forgets
- The normaliser is not a function of the group
- The same site under two names
- The same symmetry, somewhere else
- Two mirrors a coset cannot tell apart
- Two ways down from a group
- Three of them, and they are equivalent
- Going up costs the cell a parameter
- Closing the plane from two centres
- The axis a product lies on
- Two patterns laid over one another
Restriction
- Before the lattice has a say
- Why there is a list at all
- Reduction modulo three
- A fivefold axis in an ordinary crystal
- Five copies, and the gap they leave
- Five solids from one inequality
- The average that makes it finite
- The crystallographic restriction
- The degrees that name the restriction
- The most of an icosahedron a crystal can keep
- The restriction in three dimensions
- The restriction, with no lattice assumed
- The symmetry of an average
- Thirteen ways to hold a lattice
- Twelve pentagons, and no way round them
- What a molecule gives up to sit in a crystal
- Where five-fold becomes legal
- Why five-fold is impossible
- As many heptagons as pentagons
- Four root systems, and the same four rotations
- The twelve belongs to the vertex
- The surfaces a count by genus skips
- How many orientations a disorder needs
- Eleven, eleven and ten
- Seven friezes round a cylinder
- A gap the sphere does not have
- The occupancy does not name the disorder
- What forces a lattice
- The tube has a screw no lattice allows
- The same group means the same pattern
- Most sheets roll into a tube that never repeats
- The molecule size that hides a disorder
- A rolled sheet is never one of a pair
- A centre at every other ring
- How close the twelve must be
- Everything except the hexagons
- Which groups a crystal could have
- Straight lines, and no distances
- Finitely many is not few
Aperiodic
- The satellites that need a second integer
- A spectrum that is a Cantor set
- The tile that needs no reflection
- How often each patch occurs
- Cut and project
- Eight-fold, with the golden ratio taken out
- Every patch comes back
- How many arrangements one rule allows
- How many patches of each size
- Icosahedral symmetry
- Inflation, and where the golden ratio comes from
- Matching rules, and what actually forces aperiodicity
- n plus one, and no fewer
- Neither a peak nor a bump
- One tile, and no period
- Order is not periodicity
- Penrose tilings
- Six integers, and the lattice that holds them
- The arrangements a crystal keeps at absolute zero
- The average is the same wherever it is taken
- The crystal you get by rounding τ off
- The extra dimension that makes it periodic
- The smallest quasicrystal
- The freedom a crystal has not
- The tiling that points every way
- Three gaps, and never four
- Two lattices, one crystal, and no cell at all
- What Shechtman measured
- Which inflation factors exist
- Superspace groups in the plane
- A facet with no energy in it
- The hat and the turtle are one tiling
- How much of the hat is a crystal
- The ice rule is a conservation law
- A window that is not an interval
- Every fraction holds a window
- Aperiodic is two words in space
- What a defect costs the count
- The count that depends on the edge
- Three colours on a chessboard
Space groups
- One group, three symbols
- The star of a wavevector
- An order parameter is a representation
- Which way the order parameter points
- Which levels join which, on the way out of a point
- The degeneracy time reversal forces
- A bigger cell, and sometimes the mirror
- A glide sticks two levels together
- The half of a translation that is not a choice
- Eleven ways to turn while climbing
- The step a flat surface has no room for
- How chiral, as a number
- One symmorphic group per class
- Six ways to name one group
- Ten ways for space to be flat
- The cell a zone-boundary mode doubles
- The crossing at the corner
- Reflect, then slide by half of something
- The groups a single hand may sit in
- The plan contains the group
- The plane that carries two glides
- Forgetting a group in three dimensions
- Turning and climbing at once
- Sixteen candidates, ten groups
- Eleven groups that are their own reflection's rival
- Two origins for one group
- The operations nobody put in
- Where two levels must meet
- A screw that contains its own mirror image
- The phase a symmetry turns into a number
- Chiral in the plane is not chiral in the room
- A hand made of pieces that have none
- The same group in a bigger cell
- The primes a cell can grow by
- One matrix, four rules
- A line carries one screw
- A thread's hand is not a choice
- An ideal across and a prime along
- A row written as a product
- A lattice is not a subgroup
- The richest group has the poorest arithmetic
Point groups
- The operation that reverses time
- Which magnetism a class permits
- The shell that splits into kinds
- How many invariants of each degree
- The groups whose invariants are free
- Three invariants and one relation
- What a group forbids to happen
- What a texture permits
- 3m1 and 31m are one class
- A character does not know its basis
- A coincidence the group did not ask for
- A fingerprint that gave the right answer
- Fifteen may rotate light, and eleven are chiral
- How large a degeneracy may be
- Neumann's principle, as one sum
- One class, two names
- Each permits what the other forbids
- Reading a class off its own axes
- The axes a class pins down
- The cubic term that forbids a continuous change
- The eleven a diffraction pattern reports
- The holohedry is the ceiling
- The parts a property splits into
- The seven groups a field can have
- The ten with a direction of their own
- Thirty-two, and no others
- Thirty-two classes, eighteen groups
- Three optical characters, and the arithmetic that assigns them
- Twenty of the twenty-one
- Twenty-one, thirteen, nine, three
- What a crystal keeps in a field
- What a group does to a function
- What a trace decides
- A filter of great precision and no predictive power
- Permitted is not present
- Twelve of the thirty-two are free
- Two turns to come back
- Thirty-two from fourteen matrices
- Seventy-three, without a search
- How many axes there are is a Sylow count
- The table that decides every action
- The strain the atoms do not follow
Applied
- Indexing a powder pattern
- Every net with one vertex, counted
- The polarisation nobody asked for
- A hundred and thirteen orbits, and forty-eight shapes
- The figure of merit a supercell always beats
- The densest packing of a shape that is not a disc
- A cell from a bag of spots
- A fold that keeps its symmetry
- A form is an orbit, and whether it closes is an integer question
- A mechanism that is a wave
- A small angle is a row of dislocations
- A structure with the distances thrown away
- A twin is a symmetry the lattice has and the crystal does not
- A zone is a vanishing dot product
- The domains a lost translation makes, which nothing optical can see
- Counting outwards
- Two different lattices never coincide, and the question becomes how nearly
- Every coincidence index is odd, and in the plane most of them do not exist
- How many dislocations a lattice has
- How many domains a transition makes is an index
- How many polytypes there are
- The circuit that does not close
- Turn a lattice against itself and almost nothing lines up
- The angles belong to the substance, the shape to the specimen
- The count that promises a mechanism
- The densest lattice in the plane
- The descent of symmetry is a lattice, not a tree
- The dislocations a boundary allows
- The fast faces are the ones that vanish
- The four plane groups a molecule packs in
- The index and the angle a twin misses by
- Why a crystal face carries small whole numbers
- The level that does not move
- The placement nobody chose
- Five classes grow the same cube
- The step that never runs out
- The strain that arrives with the transition
- Quartz has exactly three twin laws, and its lattice is why
- The wall has a group of its own
- The walls a strain permits
- Twenty-five of the thirty-two can twin, and seven cannot
- Two hundred and forty-seven descents, or two hundred and twelve
- Two stackings, one density
- A merohedral twin moves no spot at all
- A net is a choice of what counts as a bond
- Which faces a crystal shows
- The symmetry a net was written with
- Every net with two vertices, counted
- The mechanisms a count cannot see
- Which faces are flat
- A stack with no space group
- A beat is not a period
- Every alias is a supercell
- The angle two grains differ by
- The defect that needs two laps
- The plane a deformation leaves alone
- The plate that only fits when it is twinned
- The polyhedra that can flex
- A polyhedron is two properties of a graph
- The room a thirteenth sphere would need
- The point defect whose charge has no sign
- What six lengths decide and nine do not
- A game that decides what counting only bounds