The groups

• Domains of a subgroup
• Every wall names a generator
• Five lattices, and no others
• How fast a group grows
• Orbifold notation, the shorter language
• Seven friezes
• Seventeen dollars
• Systematic absences
• Telling two words apart
• The classification proof, one branch at a time
• The descent with no shortcut
• The domains a lost translation makes, which nothing optical can see
• The fundamental domain
• The lattice underneath
• The seventeen
• The symmetry diffraction adds
• The two that fold into a surface
• Twenty-five of the thirty-two can twin, and seven cannot
• Two colours, and a symmetry that swaps them
• Why it is a group and not a list

absences · cayley · colour-classes · compose · domain-index · friedel · fundamental-domain · seventeen · subgroup-chain · twin-plate · two-colour · wallpaper · word-form

• Five lattices, and no others
• How fast a group grows
• How few operations make a pattern
• How many subgroups of index three
• Reduction modulo three
• Seven friezes
• Seventeen dollars
• Seventy-four colourings, forty-six groups
• The arrangements a crystal keeps at absolute zero
• The classification proof, one branch at a time
• The domains a lost translation makes, which nothing optical can see
• The fundamental domain
• The groups a single hand may sit in
• The groups ornament actually uses
• The normaliser is not a function of the group
• The phase problem
• The same pattern, described twice
• The seventeen
• The symmetry of an average
• The zones that behave as if there were a centre
• Three phases that do not move when the origin does
• Two colours, and a symmetry that swaps them
• Two mirrors a coset cannot tell apart
• Two ways down from a group
• Why it is a group and not a list

cayley · colour-classes · compose · detour · disorder · fundamental-domain · generating-set · motif-matters · normaliser · one-hand · phase-problem · seventeen · subgroup-kinds · wallpaper · word-form

• A lattice is not a subgroup
• Centring, and why cm is not pm
• Domains of a subgroup
• Five lattices, and no others
• How many subgroups of index three
• Seven friezes
• Seventeen dollars
• The classification proof, one branch at a time
• The four groups with a centre
• The same pattern, described twice
• The seventeen
• The symmetry diffraction adds
• The zones that behave as if there were a centre
• Two colours, and a symmetry that swaps them
• Two mirrors a coset cannot tell apart
• Why it is a group and not a list
• Why the bigger cell wins

abelian · colour-classes · detour · domain-index · friedel · isomorphic-subgroup · normaliser · seventeen · wallpaper · word-form

• A glide sticks two levels together
• As sharp as the sphere is wide
• Centring, and why cm is not pm
• How many waves a group permits
• Indexing a powder pattern
• One experiment gives the cosine, the other gives the sine
• Reading Hermann–Mauguin
• Seven friezes
• Seventeen dollars
• Seventeen groups, seven vector sets
• Seventeen, without a picture
• Systematic absences
• The classification proof, one branch at a time
• The degeneracy time reversal forces
• The four motions of the plane
• The groups a single hand may sit in
• The map that needs no phases
• The points a group treats differently
• The seventeen
• The symmetry diffraction adds
• The two that fold into a surface
• Two colours, and a symmetry that swaps them
• Which modes a site can carry
• Why it is a group and not a list
• Why the bigger cell wins

absences · colour-classes · folded-flat · friedel · harker · k-bands · k-star · one-hand · operation · order-parameter · patterson-map · phase-circles · seventeen · wallpaper · wyckoff

• A glide sticks two levels together
• A layer is not a wallpaper
• Counting what a group cannot tell apart
• Domains of a subgroup
• Every colour count at once
• How fast a group grows
• How few operations make a pattern
• Seven friezes
• Seventeen dollars
• Seventy-four colourings, forty-six groups
• Systematic absences
• The cell a zone-boundary mode doubles
• The classification proof, one branch at a time
• The domains a lost translation makes, which nothing optical can see
• The phase problem
• The same site under two names
• The seventeen
• Two colours, and a symmetry that swaps them
• Two ways down from a group
• Why it is a group and not a list
• Why the bigger cell wins

absences · cayley · colour-classes · compose · domain-index · generating-set · k-star · layer-groups · normaliser · order-parameter · phase-problem · seventeen · subgroup-kinds · two-colour

• A glide sticks two levels together
• A layer is not a wallpaper
• Reading Hermann–Mauguin
• Seven friezes
• Seventeen dollars
• The cell a zone-boundary mode doubles
• The classification proof, one branch at a time
• The four motions of the plane
• The friezes inside the seventeen
• The seventeen
• The zones that behave as if there were a centre
• Why it is a group and not a list

absences · detour · k-star · operation · seventeen · wallpaper

• A coincidence the group did not ask for
• A glide sticks two levels together
• Every wall names a generator
• Orbifold notation, the shorter language
• Seventeen dollars
• Seventeen groups, seven vector sets
• Systematic absences
• The classification proof, one branch at a time
• The four motions of the plane
• The fundamental domain
• The motif must be a comma
• The reflections a superlattice adds
• The same group means the same pattern
• The same symmetry, somewhere else
• The seventeen
• The star of a wavevector
• The two that fold into a surface
• The zones that behave as if there were a centre
• What a powder pattern loses
• Where symmetry stacks the vectors
• Why it is a group and not a list

absences · cayley · detour · extension-count · folded-flat · fundamental-domain · harker · k-bands · k-star · orbit-build · powder · seventeen · wallpaper

• A row written as a product
• Centring, and why cm is not pm
• Domains of a subgroup
• Five lattices, and no others
• How much pattern is enough
• Seventeen dollars
• The cell is a choice, the lattice is not
• The classification proof, one branch at a time
• The orbit is the pattern
• The seventeen
• The symmetry diffraction adds
• The two that fold into a surface
• Two colours, and a symmetry that swaps them
• Why it is a group and not a list
• Why the bigger cell wins

colour-classes · domain-index · folded-flat · friedel · isomorphic-subgroup · seventeen · wallpaper · window-size

• A layer is not a wallpaper
• An order parameter is a representation
• Centring, and why cm is not pm
• Domains of a subgroup
• How fast a group grows
• Seven friezes
• Seventeen dollars
• Seventy-four colourings, forty-six groups
• The classification proof, one branch at a time
• The orbit is the pattern
• The points a group treats differently
• The same site under two names
• The seventeen
• The symmetry of an average
• The zones that behave as if there were a centre
• Two colours, and a symmetry that swaps them
• Why it is a group and not a list
• Why the bigger cell wins

cayley · detour · disorder · domain-index · normaliser · order-parameter · seventeen · two-colour · wallpaper · wyckoff

• A group in four letters
• A merohedral twin moves no spot at all
• A row written as a product
• A twin is a symmetry the lattice has and the crystal does not
• As sharp as the sphere is wide
• Counting what a group cannot tell apart
• Crystallography in a box
• Domains of a subgroup
• Every wall names a generator
• Five lattices, and no others
• How chiral, as a number
• How fast a group grows
• How few operations make a pattern
• How many subgroups of index three
• How much pattern is enough
• Orbifold notation, the shorter language
• Reading Hermann–Mauguin
• Seventeen dollars
• Telling two words apart
• The arrangements a crystal keeps at absolute zero
• The classification proof, one branch at a time
• The domains a lost translation makes, which nothing optical can see
• The eleven a diffraction pattern reports
• The four groups with a centre
• The friezes inside the seventeen
• The fundamental domain
• The groups ornament actually uses
• The map that needs no phases
• The motif must be a comma
• The orbit is the pattern
• The phase problem
• The reflections a superlattice adds
• The relations a polygon dictates
• The same group in a bigger cell
• The same group means the same pattern
• The same pattern, described twice
• The seventeen
• The solver that knows no symmetry
• The star of a wavevector
• The symmetry of an average
• The tile that needs no reflection
• Turn a lattice against itself and almost nothing lines up
• Two structures, one Patterson
• Two ways down from a group
• What a position becomes on the way down
• What a symmetry actually is
• Where the product is
• Which way the order parameter points
• Why it is a group and not a list

abelian · antiphase · cayley · colour-classes · compose · coset-table · disorder · domain-index · extension-count · folded-flat · friedel · fundamental-domain · generating-set · isomorphic-subgroup · k-star · motif-matters · normaliser · one-hand · operation · orbit-build · orbit-count · order-parameter · phase-problem · phase-relations · seventeen · superstructure · twin-diffraction · twin-plate · two-colour · wallpaper · window-size · word-form · wyckoff

• An order parameter is a representation
• Chiral in the plane is not chiral in the room
• Counting what a group cannot tell apart
• Crystallography in a box
• Domains of a subgroup
• Every wall names a generator
• How fast a group grows
• How many waves a group permits
• Near-symmetry, and the tolerance that is not here
• Orbifold notation, the shorter language
• p3m1 and p31m
• Reading Hermann–Mauguin
• Seventeen dollars
• Seventy-four colourings, forty-six groups
• The Alhambra question
• The cell a zone-boundary mode doubles
• The cell is a choice, the lattice is not
• The classification proof, one branch at a time
• The domain in reciprocal space
• The fundamental domain
• The groups a single hand may sit in
• The motif must be a comma
• The points a group treats differently
• The same group in a bigger cell
• The same group means the same pattern
• The same site under two names
• The same symmetry, somewhere else
• The seventeen
• The star of a wavevector
• The strain that arrives with the transition
• The symmetry of an average
• The walls a strain permits
• Two colours, and a symmetry that swaps them
• Two mirrors a coset cannot tell apart
• Two ways down from a group
• What a position becomes on the way down
• Where the product is
• Where two levels must meet
• Which levels join which, on the way out of a point
• Which modes a site can carry
• Which way the order parameter points
• Why it is a group and not a list

cayley · colour-classes · compose · disorder · domain-index · extension-count · folded-flat · fundamental-domain · isomorphic-subgroup · k-bands · k-star · motif-matters · normaliser · one-hand · orbit-count · order-parameter · product · seventeen · subgroup-kinds · tolerance · two-colour · wallpaper · wyckoff

• A glide sticks two levels together
• A group in four letters
• An order parameter is a representation
• How many waves a group permits
• How much pattern is enough
• Orbifold notation, the shorter language
• p3m1 and p31m
• Seventeen dollars
• Seventeen groups, seven vector sets
• Systematic absences
• Telling two words apart
• The Alhambra question
• The axis a product lies on
• The cell a zone-boundary mode doubles
• The cell is a choice, the lattice is not
• The classification proof, one branch at a time
• The reciprocal lattice
• The same group means the same pattern
• The same pattern, described twice
• The seventeen
• The two that fold into a surface
• The zones that behave as if there were a centre
• Where the product is
• Why it is a group and not a list

absences · coset-table · detour · folded-flat · fundamental-domain · k-star · motif-matters · normaliser · order-parameter · patterson-map · presentation · product · seventeen · wallpaper · window-size

• A merohedral twin moves no spot at all
• A twin hides in the statistics
• A twin is a symmetry the lattice has and the crystal does not
• Every wall names a generator
• Five copies, and the gap they leave
• Five lattices, and no others
• How fast a group grows
• How few operations make a pattern
• How many waves a group permits
• p3m1 and p31m
• Quartz has exactly three twin laws, and its lattice is why
• Seventeen dollars
• Seventeen groups, seven vector sets
• The Alhambra question
• The classification proof, one branch at a time
• The crossing at the corner
• The groups a single hand may sit in
• The groups ornament actually uses
• The index and the angle a twin misses by
• The law that hides handedness
• The motif must be a comma
• The orbit is the pattern
• The phase problem
• The points a group treats differently
• The same group in a bigger cell
• The seventeen
• The symmetry diffraction adds
• The symmetry of an average
• Three colours, and why most patterns cannot have them
• What a position becomes on the way down
• Why it is a group and not a list

cayley · disorder · friedel · generating-set · isomorphic-subgroup · k-bands · k-star · motif-matters · patterson-map · phase-problem · presentation · seventeen · subgroup-kinds · twin-diffraction · twin-plate · wallpaper · wyckoff

• 3m1 and 31m are one class
• A group in four letters
• An orbit is what the invariants cannot tell apart
• Domains of a subgroup
• How fast a group grows
• How few operations make a pattern
• Near-symmetry, and the tolerance that is not here
• Orbifold notation, the shorter language
• p3m1 and p31m
• Reading Hermann–Mauguin
• Seventeen dollars
• The classification proof, one branch at a time
• The cubic term that forbids a continuous change
• The fundamental domain
• The polarisation nobody asked for
• The seventeen
• The symmetry diffraction adds
• Three of them, and they are equivalent
• What is left when the order is forgotten
• Where two levels must meet
• Which way the order parameter points
• Why it is a group and not a list

abelian · cayley · coset-table · domain-index · friedel · fundamental-domain · generating-set · k-bands · motif-matters · normaliser · order-parameter · seventeen · tolerance · wallpaper · wyckoff

• 3m1 and 31m are one class
• An order parameter is a representation
• How fast a group grows
• How few operations make a pattern
• Orbifold notation, the shorter language
• p3m1 and p31m
• Seventeen dollars
• The classification proof, one branch at a time
• The polarisation nobody asked for
• The seventeen
• The symmetry diffraction adds
• The zones that behave as if there were a centre
• Three colours, and why most patterns cannot have them
• What a position becomes on the way down
• What is left when the order is forgotten
• Which levels join which, on the way out of a point
• Which modes a site can carry
• Why it is a group and not a list

abelian · cayley · detour · friedel · fundamental-domain · generating-set · k-bands · order-parameter · seventeen · subgroup-kinds · two-colour · wallpaper · wyckoff

• How many waves a group permits
• p3m1 and p31m
• Seventeen dollars
• Telling two words apart
• The classification proof, one branch at a time
• The degeneracy time reversal forces
• The descent with no shortcut
• The domains a lost translation makes, which nothing optical can see
• The groups a single hand may sit in
• The primes a cell can grow by
• The same group in a bigger cell
• The same group means the same pattern
• The seventeen
• The symmetry of an average
• Two colours, and a symmetry that swaps them
• Why it is a group and not a list

colour-classes · disorder · domain-index · extension-count · isomorphic-subgroup · k-star · one-hand · seventeen · two-colour · wallpaper · word-form

• A group in four letters
• An order parameter is a representation
• Counting what a group cannot tell apart
• Domains of a subgroup
• How fast a group grows
• How many waves a group permits
• Orbifold notation, the shorter language
• p3m1 and p31m
• Seventeen dollars
• Telling two words apart
• The Alhambra question
• The classification proof, one branch at a time
• The crossing at the corner
• The cubic term that forbids a continuous change
• The descent with no shortcut
• The domain in reciprocal space
• The fundamental domain
• The motif must be a comma
• The orbit is the pattern
• The points a group treats differently
• The primes a cell can grow by
• The relations a polygon dictates
• The same symmetry, somewhere else
• The seventeen
• The star of a wavevector
• The symmetry of an average
• The walls a strain permits
• What a position becomes on the way down
• Where the product is
• Where two levels must meet
• Which modes a site can carry
• Which way the order parameter points
• Why it is a group and not a list

cayley · coset-table · disorder · domain-index · fundamental-domain · isomorphic-subgroup · k-bands · k-star · motif-matters · orbit-count · order-parameter · product · seventeen · subgroup-chain · wallpaper · word-form · wyckoff