The 8 motions of a regular 4-gon

• the hex ball of radius 0 ×13
• the king ball of radius 0 ×13
• the square ball of radius 0 ×13
• the pieces of Z/2 ∗ Z/2 outside radius 0 that reach the outer shell do not change when it moves out ×10
• a subgroup of 1 divides the group of 8 ×8
• the ball of radius 0 in F₂ has the size the formula gives ×8
• the ball of radius 0 in Z has the size the formula gives ×8
• the ball of radius 0 in Z² has the size the formula gives ×8
• D3 has an element of order 2 ×7
• D3: every solution but e has order exactly 2 ×7
• D3: the solutions of g^2 = e are a multiple of 2 ×7
• the ball of radius 0 in Z/2 ∗ Z/2 has the size the formula gives ×7
• the ball of radius 0 in Z³ has the size the formula gives ×7
• D4: a class's size times its centraliser is the group ×6
• D4: a group that does not commute reaches at most 5/8 ×6
• D4: the centralisers of one class add up to the group ×6
• D4: the centre is at most a quarter of the group ×6
• D4: the commuting pairs are the size times the number of classes ×6
• the solutions of x^1 = e are a multiple of 1 ×6
• a subgroup of order 8 exists ×5
• A4 has an element of order 2 ×5
• A4: every solution but e has order exactly 2 ×5
• A4: the solutions of g^2 = e are a multiple of 2 ×5
• the pieces of F₂ outside radius 0 that reach the outer shell do not change when it moves out ×5
• the pieces of Z × Z/2 outside radius 2 that reach the outer shell do not change when it moves out ×5
• the pieces of Z outside radius 2 that reach the outer shell do not change when it moves out ×5
• the pieces of Z/6 outside radius 0 that reach the outer shell do not change when it moves out ×5
• the pieces of Z² outside radius 2 that reach the outer shell do not change when it moves out ×5
• the pieces of Z³ outside radius 0 that reach the outer shell do not change when it moves out ×5
• every motion sending corner 1 to 0 is one of 2 ×4
• P2: a class's size times its centraliser is the group ×4
• P2: a group that does not commute reaches at most 5/8 ×4
• P2: the centralisers of one class add up to the group ×4
• P2: the centre is at most a quarter of the group ×4
• P2: the commuting pairs are the size times the number of classes ×4
• the Pauli group on 1 has 4^1 + 1 classes ×4
• the Pauli group on 1 has a centre of two ×4
• the Pauli group on 2 has 2·4^2 elements ×4
• S3: the classes are the partitions of 3 ×3
• S4 has the number of elements it should ×3
• S4: a class's size times its centraliser is the group ×3
• S4: a group that does not commute reaches at most 5/8 ×3
• S4: the centralisers of one class add up to the group ×3
• S4: the centre is at most a quarter of the group ×3
• S4: the commuting pairs are the size times the number of classes ×3
• the elements reachable in 1 steps are exactly those at distance 1 or less ×3
• the radius runs to between 3 and 8 ×3
• A4 has the number of elements it should ×2
• A4: a class's size times its centraliser is the group ×2
• A4: a group that does not commute reaches at most 5/8 ×2
• A4: the centralisers of one class add up to the group ×2
• A4: the centre is at most a quarter of the group ×2
• A4: the commuting pairs are the size times the number of classes ×2
• colouring 1100: orbit times stabiliser is the group ×2
• e leaves 16 colourings alone, one per cycle coloured freely ×2
• exactly 8 of the 24 relabellings preserve every distance ×2
• m₁ leaves 8 colourings alone, one per cycle coloured freely ×2
• m₂ leaves 4 colourings alone, one per cycle coloured freely ×2
• m₃ leaves 8 colourings alone, one per cycle coloured freely ×2
• m₄ leaves 4 colourings alone, one per cycle coloured freely ×2
• r leaves 2 colourings alone, one per cycle coloured freely ×2
• r² leaves 4 colourings alone, one per cycle coloured freely ×2
• r³ leaves 2 colourings alone, one per cycle coloured freely ×2
• S4 has an element of order 2 ×2
• S4: every solution but e has order exactly 2 ×2
• S4: the solutions of g^2 = e are a multiple of 2 ×2
• the classes hold all 16 colourings between them and share none ×2
• the drawing of Z/2 ∗ Z/2 shows as many outward pieces as are counted ×2
• the radius is between 2 and 6 ×2
• a class's size times the relabellings fixing one graph is n! ×1
• a class's size times what commutes with one member is the group ×1
• a colouring gives one colour per corner ×1
• a finite group has nothing left outside a large enough ball ×1
• a pair names two different corners ×1
• a point reached in d steps lies inside d·P ×1
• a subgroup whose blocks differ has an element that shows it ×1
• a tree walk is back after two steps with chance 1/4 ×1
• a tuple no turn changes repeats one element ×1
• a turned tuple still has product e ×1
• above the threshold the diagonal time is the minimum, below it more ×1
• all four pieces are non-empty ×1
• all graphs are drawn for at most four points ×1
• and a finite stretch between the cuts ×1
• and divides the index ×1
• and does not fall away ×1
• and it is already below three fifths ×1
• and nearer at the larger radius ×1
• and no element is in two of them ×1
• and none of them is the identity, which would generate nothing ×1
• and that does not change as the ball grows ×1
• and the second subgroup's are not — which is what makes it the control ×1
• and the two maps are genuinely different graphs ×1
• and the whole group is reached ×1
• and there is a multiple of p of them ×1
• at most 20,000 tuples are enumerated ×1
• axis + one diagonal: |B(r)|/r² is near the polygon's area ×1
• axis steps: |B(r)|/r² is near the polygon's area ×1
• between one and four groups this family knows ×1
• between one and six things are tabled ×1
• between one and three generating sets among square, hex, king, knight ×1
• between one and three generators are named ×1
• between one and three groups this family knows ×1
• between three and eight groups this family knows ×1
• between two and four generating sets ×1
• both maps are of the same group ×1
• column e does too ×1
• column m₁ does too ×1
• column m₂ does too ×1
• column m₃ does too ×1
• column m₄ does too ×1
• column r does too ×1
• column r² does too ×1
• column r³ does too ×1
• corner 1: orbit times stabiliser is the group ×1
• corners times the rotations holding one still is every rotation ×1
• diagonal 1–3: orbit times stabiliser is the group ×1
• e preserves every distance between corners ×1
• each step's size divides the last ×1
• edge 1–2: orbit times stabiliser is the group ×1
• edges times the rotations holding one still is every rotation ×1
• every block is the size of the subgroup ×1
• every class has a size dividing the group's ×1
• every composition of two motions is again one of the motions ×1
• every corner of the cube can be carried to every other ×1
• every corner of the dodecahedron can be carried to every other ×1
• every corner of the icosahedron can be carried to every other ×1
• every corner of the octahedron can be carried to every other ×1
• every corner of the tetrahedron can be carried to every other ×1
• every edge of the cube can be carried to every other ×1
• every edge of the dodecahedron can be carried to every other ×1
• every edge of the icosahedron can be carried to every other ×1
• every edge of the octahedron can be carried to every other ×1
• every edge of the tetrahedron can be carried to every other ×1
• every element found is accounted for in the shells ×1
• every element of the tree of triangles is placed ×1
• every element sits in exactly one shell ×1
• every face of the cube can be carried to every other ×1
• every face of the dodecahedron can be carried to every other ×1
• every face of the icosahedron can be carried to every other ×1
• every face of the octahedron can be carried to every other ×1
• every face of the tetrahedron can be carried to every other ×1
• every generator names an element of the group ×1
• every group drawn has a picture in the plane ×1
• every named element is in the group ×1
• every point of the box is reached ×1
• every reduced word up to the stated length is listed ×1
• every ring's size divides the length ×1
• every set has a non-empty edge inside it ×1
• every shorter word is in exactly one piece of the a split ×1
• every shorter word is in exactly one piece of the b split ×1
• every tuple built has product e ×1
• F₂ multiplies its ball by more than two at each step ×1
• F₂'s count rises at every radius ×1
• faces times the rotations holding one still is every rotation ×1
• graphs on three to five points ×1
• in a line a second cut leaves two outward pieces ×1
• in a tree the second cut adds outward pieces ×1
• king's moves: |B(r)|/r² is near the polygon's area ×1
• knight's moves: |B(r)|/r² is near the polygon's area ×1
• m₁ preserves every distance between corners ×1
• m₂ preserves every distance between corners ×1
• m₃ preserves every distance between corners ×1
• m₄ preserves every distance between corners ×1
• m₅ preserves every distance between corners ×1
• m₆ preserves every distance between corners ×1
• most of a ball in F₂ is its outermost shell ×1
• multiplying on the left carries every edge to an edge ×1
• no missed point is more than three polygon-steps inside the edge ×1
• no point is reached in fewer steps than its norm ×1
• no two of the motions do the same thing to the corners ×1
• no word is in both pieces of the a split ×1
• no word is in both pieces of the b split ×1
• on the tree the distance grows at about half a step per step ×1
• one block for each place in the orbit ×1
• one rotation for each way of putting down a directed edge ×1
• Q8: a class's size times its centraliser is the group ×1
• Q8: a group that does not commute reaches at most 5/8 ×1
• Q8: the centralisers of one class add up to the group ×1
• Q8: the centre is at most a quarter of the group ×1
• Q8: the commuting pairs are the size times the number of classes ×1
• r preserves every distance between corners ×1
• r² preserves every distance between corners ×1
• r³ preserves every distance between corners ×1
• r⁴ preserves every distance between corners ×1
• r⁵ preserves every distance between corners ×1
• row e contains every element exactly once ×1
• row m₁ contains every element exactly once ×1
• row m₂ contains every element exactly once ×1
• row m₃ contains every element exactly once ×1
• row m₄ contains every element exactly once ×1
• row r contains every element exactly once ×1
• row r² contains every element exactly once ×1
• row r³ contains every element exactly once ×1
• so the subgroup's size times the number of blocks is the group's size ×1
• some group in the table does not reach the identity ×1
• some pair of motions gives a different result in each order ×1
• the average number left alone is the number of classes ×1
• the ball is drawn for the groups with a two-dimensional picture ×1
• the ball keeps growing ×1
• the bars are drawn for at most 24 elements ×1
• the block a product lands in does not depend on which elements were picked ×1
• the blocks between them cover the group exactly once ×1
• the bound is drawn for a group that does not commute ×1
• the box runs to between 6 and 16 ×1
• the chance falls and stays above one half ×1
• the classes are drawn for at most 64 colourings ×1
• the classes hold every element once ×1
• the classes hold every graph exactly once ×1
• the commutators generate a subgroup of the group they came from ×1
• the conjugacy classes account for every element exactly once ×1
• the corners are coloured in between 2 and 4 colours ×1
• the derived subgroup is carried into itself by every conjugation ×1
• the drawing has extent in both directions ×1
• the drawing of F₂ shows as many outward pieces as are counted ×1
• the drawing of Z × Z/2 shows as many outward pieces as are counted ×1
• the drawing of Z shows as many outward pieces as are counted ×1
• the drawing of Z² shows as many outward pieces as are counted ×1
• the drawing reaches at least two steps past the cut ×1
• the dual's corners point at the faces in one of its orientations ×1
• the edge of a ball in F₂ stays a large share of it ×1
• the edge of a ball in Z² is a falling share of it ×1
• the elements are closed under composition ×1
• the estimate falls as the distance grows ×1
• the even symmetries of five letters have no normal subgroup in between ×1
• the exhaustive search is drawn for at most 5 corners ×1
• the first has as many blocks as its index ×1
• the first subgroup is named by element labels ×1
• the first subgroup's left and right blocks are the same blocks ×1
• the four pieces and the identity account for every word ×1
• the generating set is one this family knows ×1
• the generators are distinct ×1
• the grid is drawn for at most twelve elements ×1
• the group is a dihedral group D3 to D8 or a named permutation group ×1
• the group is one of Z, Z2, Z3, F2, D, T, ZZ2, C6 ×1
• the group is one this family knows ×1
• the group is the dihedral one or the cyclic one ×1
• the groups are among Z, Z2, Z3, F2, D, T, ZZ2, C6 ×1
• the groups drawn do not all grow alike ×1
• the identity is a class of its own ×1
• the knight misses points inside its polygon ×1
• the labelled trees number n to the power n − 2, as Cayley's formula says ×1
• the lattice counts agree with direct enumeration ×1
• the lattices' roots approach 1 ×1
• the m₁ and m₂ generate the whole group ×1
• the measured rate matches 2√(k − 1)/k ×1
• the motions sending the thing to one place are one coset of the stabiliser ×1
• the named elements are closed under composition ×1
• the outer sphere is (k − 2)/(k − 1) of a large ball ×1
• the outermost shell of Z is a falling share of the ball ×1
• the outermost shell of Z² is a falling share of the ball ×1
• the permutations are of between two and five letters ×1
• the polygon has between 3 and 8 corners ×1
• the r and m₁ generate the whole group ×1
• the r generate the whole group ×1
• the radius removed is between 0 and 3 ×1
• the removed radius runs to between 2 and 5 ×1
• the rings hold every tuple once ×1
• the second is too ×1
• the series never grows ×1
• the sets run to size parameter 4 to 12 ×1
• the share of the polygon reached does not fall as r grows ×1
• the singletons are exactly the elements whose p-th power is e ×1
• the solid is one of the five ×1
• the spread grows like a power near one third ×1
• the subgroup is named by between one and eight element labels ×1
• the Sylow subgroups are exactly the conjugates of one of them ×1
• the three-dimensional count agrees with enumeration ×1
• the translate's centre is inside the drawing ×1
• the tree's root approaches √3/2 ×1
• the trivial subgroup and the whole group are both among them ×1
• the tuples are drawn when there are at most 64 of them ×1
• the tuples with product e number |G| to the power n − 1 ×1
• the two subgroups nobody has to look for are among the ones found ×1
• the view is one the family draws ×1
• the word e lands on e ×1
• the word length is never more than three above the norm ×1
• the word m₁ lands on m₁ ×1
• the word m₁·m₂ lands on r³ ×1
• the word m₁·m₂·m₁ lands on m₄ ×1
• the word m₁·m₂·m₁·m₂ lands on r² ×1
• the word m₁·r lands on m₄ ×1
• the word m₁·r lands on m₅ ×1
• the word m₁·r lands on m₆ ×1
• the word m₁·r·m₁ lands on r⁴ ×1
• the word m₁·r·m₁ lands on r⁵ ×1
• the word m₁·r·r lands on m₄ ×1
• the word m₁·r·r lands on m₅ ×1
• the word m₂ lands on m₂ ×1
• the word m₂·m₁ lands on r ×1
• the word m₂·m₁·m₂ lands on m₃ ×1
• the word r lands on r ×1
• the word r·m₁ lands on m₂ ×1
• the word r·r lands on r² ×1
• the word r·r·m₁ lands on m₃ ×1
• the word r·r·r lands on r³ ×1
• the word r·r·r·m₁ lands on m₄ ×1
• the word r·r·r·r lands on r⁴ ×1
• the word r·r·r·r·r lands on r⁵ ×1
• the words run to between 3 and 8 letters ×1
• the words strictly inside the bound are counted ×1
• their number leaves remainder 1 on division by p ×1
• they differ in something a reader can see — the number of steps across, or the number of arrows ×1
• three or four increasing radii up to 24 ×1
• two generating sets are compared ×1
• two motions with the same target differ by a motion holding the thing still ×1
• two of Z, F2, T and D are compared ×1
• under a prime turn every ring has 1 or p tuples ×1
• Z × Z/2 settles on 0, 1 or 2 ×1
• Z does not ×1
• Z settles on 0, 1 or 2 ×1
• Z/2 ∗ Z/2 does not ×1
• Z/2 ∗ Z/2 settles on 0, 1 or 2 ×1
• Z/2 ∗ Z/3's count rises at every radius ×1
• Z/6 settles on 0, 1 or 2 ×1
• Z² does not ×1
• Z² settles on 0, 1 or 2 ×1
• Z³ does not ×1
• Z³ settles on 0, 1 or 2 ×1

The whole library · What the figures prove