Generator

group

A generator in the algebra library, called 12 times across 2 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

group is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

`show: "orbit"`

`show: "fixed"`

`show: "all"`

`show: "action"`

`show: "table"`

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

a subgroup of 1 divides the group of 8 ×4
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
the classes hold all 16 colourings between them and share none ×2
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
e preserves every distance between corners ×1
every class has a size dividing the group's ×1
every composition of two motions is again one of the motions ×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
no two of the motions do the same thing to the corners ×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
some pair of motions gives a different result in each order ×1
the average number left alone is the number of classes ×1
the classes are drawn for at most 64 colourings ×1
the corners are coloured in between 2 and 4 colours ×1
the exhaustive search is drawn for at most 5 corners ×1
the group is the dihedral one or the cyclic one ×1
the polygon has between 3 and 8 corners ×1
the two subgroups nobody has to look for are among the ones found ×1
the view is one the family draws ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Algebra

Colourings nobody can tell apart

Sixteen ways to colour four corners in two colours, and only six of them are genuinely different. The count can be got by pooling the sixteen — or by never forming a single class and instead averaging how many colourings each motion leaves untouched.

Algebra

Eight ways to leave a square alone

A square can be picked up and put back so that nothing looks different. There are exactly eight ways to do it, and the number is not asserted here — it is what a search through all twenty-four relabellings of the corners comes back with.

The whole library · What the figures prove