Generator

quaternion

A generator in the algebra library, called 6 times across 1 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.

quaternion 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

The quaternion multiplication table, from i² = j² = k² = ijk = −1. A four-by-four multiplication table of the quaternion units, with the row giving the left factor, every entry computed from Hamilton's rule, and the pair that differs between the two orders marked.

`show: "rotate"`

A turn about i and one about j, in both orders. A cube in its starting position and the two positions reached by applying the same two quarter turns in the two possible orders, which are not the same position.

`show: "double"`

One full turn returns the object and not the quaternion. A row of squares showing an object's orientation as it turns through two full revolutions, above a plot of the scalar part of the quaternion performing the turn, which reaches minus one after the first revolution and one only after the second.

`show: "algebras"`

What each doubling costs: the reals, the complex numbers, the quaternions, the octonions. A table of the four division algebras with their dimensions, the number of ordered pairs of basis units that fail to commute, and the number of triples that fail to associate, each count made by multiplying them out.

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.

24 of the 64 ordered pairs of units fail to commute ×1
a quaternion and its negative perform the same rotation ×1
conjugating a pure quaternion leaves it pure ×1
doubling the complex numbers gives Hamilton's own multiplication ×1
each composite is a rotation rather than a reflection ×1
i j k = −1, which is the rule the rest follows from ×1
one full turn of the object leaves the quaternion at −1 ×1
the highlighted product is ij, ji, squares, or none ×1
the multiplication is associative on every triple of units ×1
the octonions are not even associative ×1
the quaternion has length one ×1
the quaternions do not commute ×1
the reals commute ×1
the rotation after one full turn is the identity again ×1
the sweep is taken in 8 to 24 steps, a multiple of four ×1
the tower stops at dimension 1, 2, 4 or 8 ×1
the turn is between a tenth and a half of a full circle ×1
the two orders give different quaternions ×1
the two rotations move at least one axis a long way apart ×1
the view is one the family draws ×1
two different axes from i, j, k ×1
two full turns bring it back to 1 ×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

A multiplication that remembers the order

Four characters — i² = j² = k² = ijk = −1 — define a multiplication in which ab and ba are different numbers. Everything follows from them, including the fact that a rotation of a solid body has two names, and that two full turns are needed to get one of them home.

The whole library · What the figures prove