The quaternion multiplication table, from i² = j² = k² = ijk = −1

• the units 1 and 2 lie on exactly one line ×21
• 24 of the 64 ordered pairs of units fail to commute ×1
• 3 and 5 are each a sum of three squares and 15 is not, so no three-square identity exists ×1
• a labelling exists on which every line reads in its own multiplication order ×1
• a left multiplication and a right multiplication commute ×1
• a product of whole octonions is whole ×1
• a product of whole quaternions is whole ×1
• a quaternion and its negative perform the same rotation ×1
• all seven units are placed ×1
• and every non-zero octonion still has an inverse ×1
• and every one of them has length one ×1
• and every one of them has squared length two ×1
• and every point of a fibre lies over the same point of the sphere ×1
• and every root sees the same profile of neighbours as every other ×1
• and in the ring with halves it is closer than one, which is a division algorithm ×1
• and its determinant is one, so it turns rather than reflects ×1
• and most of every circle is still drawn, so the breaks read as crossings ×1
• and no two circles touch ×1
• and none of them has two arguments the same, which is alternativity ×1
• and one hundred and twenty-eight have every coordinate a half ×1
• and one is its negative ×1
• and the five classes account for every root ×1
• and the other six are straight ×1
• and the quotient that achieves it is one of the halves ×1
• and the rotations they perform number twelve, two units to each ×1
• and the trace of its square gives the same two angles again ×1
• and the twenty-four sit inside the hundred and twenty five times over ×1
• and the two-square identity is the same statement in the plane ×1
• and their real parts take nine values ×1
• and they are twenty-four different quaternions ×1
• and they are two hundred and forty different vectors ×1
• and tilts within a right angle ×1
• between two and eight circles are drawn ×1
• conjugating a pure quaternion leaves it pure ×1
• doubling the complex numbers gives Hamilton's own multiplication ×1
• each circle is sampled at between 60 and 400 points ×1
• each composite is a rotation rather than a reflection ×1
• each unit has eight others at distance one, which is the 24-cell's vertex figure ×1
• every icosian is counted once ×1
• every number up to the bound is a sum of four squares ×1
• every pair of circles is linked exactly once ×1
• every pair of roots meets at an inner product of −2, −1, 0, 1 or 2 ×1
• every point of a fibre is on the unit three-sphere ×1
• exactly one of the seven lines is the drawn circle ×1
• fifty-six roots stand at sixty degrees to any given one ×1
• i j k = −1, which is the rule the rest follows from ×1
• in the whole-coordinate ring the quotient is a full unit away, so no remainder is smaller than the divisor ×1
• its centre is exactly plus and minus one ×1
• negating both quaternions gives the same rotation ×1
• negating only one of them does not ×1
• no drawn circle comes near the point the projection removes ×1
• no two vertices project onto one another ×1
• one full turn of the object leaves the quaternion at −1 ×1
• one hundred and twenty-six stand at a right angle ×1
• one of them is the identity ×1
• reflecting one root in another lands on a root ×1
• sixty-four straddle the two halves ×1
• some strand passes behind another, which is what makes the picture a diagram ×1
• some triples of units fail to associate ×1
• the associator changes sign when two of its arguments are swapped ×1
• the base points sit at a latitude strictly inside the poles ×1
• the columns of the map are orthonormal, so lengths are kept ×1
• the eight squares of the product add to the product of the two sums of eight squares ×1
• the first quaternion is four whole numbers no larger than twenty ×1
• the four squares of the product add to the product of the two sums of four squares ×1
• the highlighted line is one of the seven, or none ×1
• the highlighted product is ij, ji, squares, or none ×1
• the icosians number one hundred and twenty ×1
• the left quaternion is four numbers ×1
• the left quaternion is meant to have length one ×1
• the lift really does send i to the base point ×1
• the multiplication is associative on every triple of units ×1
• the numbers that really need four squares are the ones Legendre's condition names ×1
• the octonions are not even associative ×1
• the plane has seven lines ×1
• the quaternion has length one ×1
• the quaternions do not commute ×1
• the reals commute ×1
• the right quaternion is four numbers ×1
• the right quaternion is meant to have length one ×1
• the rotation after one full turn is the identity again ×1
• the second quaternion is four whole numbers no larger than twenty ×1
• the solid has ninety-six edges ×1
• the sums of squares are checked between 20 and 400 far ×1
• the sweep is taken in 8 to 24 steps, a multiple of four ×1
• the system is symmetric under negation ×1
• the tower stops at dimension 1, 2, 4 or 8 ×1
• the trace of the matrix is twice the sum of the cosines of the two angles ×1
• the turn is between a tenth and a half of a full circle ×1
• the two blocks are orthogonal ×1
• the two blocks are shaded or not ×1
• the two orders give different quaternions ×1
• the two rotations move at least one axis a long way apart ×1
• the units are closed under multiplication ×1
• the view is one the family draws ×1
• the view is tilted between minus ninety and ninety degrees ×1
• the view turns by a real angle ×1
• there are twenty-four units ×1
• there are two hundred and forty roots ×1
• they are closed under multiplication ×1
• thirty of them are half-turns ×1
• twenty-four in the last four ×1
• twenty-four of them are the Hurwitz units ×1
• twenty-four roots live in the first four coordinates ×1
• two different axes from i, j, k ×1
• two full turns bring it back to 1 ×1
• two-dimensional inputs give a two-dimensional answer, which is why the plane closes ×1
• which are themselves closed under multiplication ×1
• which is all of them, counted twice by two different rules ×1

The whole library · What the figures prove