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Quaternions — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    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.

    part 1 · algebra
  2. A rotation of four-space, and the two angles it turns through. The four by four matrix of the map sending x to p x conjugate q, beside two dials showing the angle it turns through in each of its two invariant planes.

    A rotation of four-space takes two of them

    One unit quaternion, conjugating, turns three-space about an axis. Two of them, multiplying from the left and the right, turn four-space — and a rotation of four-space has no axis at all, but two independent angles and two planes it spins in.

    part 2 · algebra
  3. The four-square identity, and how few squares a number needs. A product of two whole quaternions with both sides of the four-square identity evaluated, above a strip colouring every number by the fewest squares that add to it.

    The identity that multiplies sums of squares

    A sum of two squares times a sum of two squares is a sum of two squares, and the same is true of four and of eight. It is true of three of nothing: 3 and 5 are each a sum of three squares, 15 is not, and one small pair of numbers rules the case out forever.

    part 3 · algebra
  4. The twenty-four unit quaternions, at the corners of a four-dimensional solid. The twenty-four units of the Hurwitz quaternions drawn as the vertices of a 24-cell projected into three-space, with the ninety-six edges joining units at distance one.

    The integers among the quaternions

    The obvious integer quaternions are the ones with whole coordinates, and they are the wrong ones. Sixteen more, with every coordinate a half, have to be let in — and with them comes a division algorithm, twenty-four units instead of eight, and a regular solid that exists only in four dimensions.

    part 4 · algebra
  5. The octonion multiplication table, drawn as seven lines. The Fano plane with its seven points labelled by the imaginary octonion units and its seven lines carrying an arrow each, giving the products of every pair.

    What is lost at eight

    Double the quaternions and division still works, but ab times c and a times bc are no longer the same number. What survives is a weaker law that turns out to be enough for a great deal — and the whole multiplication table fits into a picture of seven lines.

    part 5 · algebra
  6. Two hundred and forty roots, built and counted. An eight by eight grid whose upper cells stand for the pairs of coordinates a root can use, beside bars counting how many roots stand at each angle from a given root: 1, 56, 126, 56 and 1.

    Two hundred and forty directions

    The quaternions have twenty-four units and they are the vertices of the most symmetric object in four dimensions. Eight dimensions has two hundred and forty of them, and the quaternions turn out to be how they are built — twice over, with a hundred and ninety-two left to explain.

    part 6 · algebra

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