John Stillwell - Naive Lie Theory.pdf - Index of

1.6 Discussion 21
√
where the absolute value of u =(x 1 ,x 2 ,...,x n ) is |u| = x 2 1 + x2 2 + ···+ x2 n.
As we have seen, for n = 2 this property is equivalent to the Diophantus
identity for sums of two squares, so a multiplicative absolute value in general
implies an identity for sums of n squares.
Hamilton attacked the problem from the opposite direction, as it were.
He tried to define the product operation, first for triples, before worrying
about the absolute value. But after searching fruitlessly for 13 years, he
had to admit defeat. He still had not noticed that there is no three square
identity, but he suspected that multiplying triples of the form a + bi + cj
requires a new object k = ij. Also, he began to realize that there is no hope
for the commutative law of multiplication. Desperate to salvage something
from his 13 years of work, he made the leap to the fourth dimension. He
took k = ij to be a vector perpendicular to 1, i, and j, and sacrificed the
commutative law by allowing ij= − ji, jk = −kj,andki = −ik. OnOctober
16, 1843 he had his famous epiphany that i, j, andk must satisfy
i 2 = j 2 = k 2 = ijk= −1.
As we have seen in Section 1.3, these relations imply all the field properties,
except commutative multiplication. Such a system is often called
a skew field (though this term unfortunately suggests a specialization of
the field concept, rather than what it really is—a generalization). Hamilton’s
relations also imply that absolute value is multiplicative—a fact he
had to check, though the equivalent four-square identity was well known
to number theorists.
In 1878, Frobenius proved that the quaternion algebra H is the only
skew field R n that is not a field, so Hamilton had found the only “algebra
of n-tuples” it was possible to find under the conditions he had imposed.
The multiplicative absolute value, as stressed in Section 1.4, implies
that multiplication by a quaternion of absolute value 1 is an isometry of
R 4 . Hamilton seems to have overlooked this important geometric fact, and
the quaternion representation of space rotations (Section 1.5) was first published
by Cayley in 1845. Cayley also noticed that the corresponding formulas
for transforming the coordinates of R 3 had been given by Rodrigues
in 1840. Cayley’s discovery showed that the noncommutative quaternion
product is a good thing, because space rotations are certainly noncommutative;
hence they can be faithfully represented only by a noncommutative
algebra. This finding has been enthusiastically endorsed by the computer
graphics profession today, which uses quaternions as a standard tool for