John Stillwell - Naive Lie Theory.pdf - Index of

18 1 The geometry of complex numbers and quaternions
R
L
θ/2
P
N
ϕ/2
M
Q
Figure 1.5: Reflections in great circles on the sphere.
1.5.5 Adapt the argument of Exercise 1.5.3 to great circles L , M ,andN shown
in Figure 1.5. What is the conclusion?
1.5.6 Explain why there is no exceptional case analogous to Exercise 1.5.4. Deduce
that the product of any two rotations of R 3 about O is another rotation
about O, and explain how to find the axis of the product rotation.
The idea of representing isometries as products of reflections is also useful in
higher dimensions. We use this idea again in Section 2.4, where we show that any
isometry of R n that fixes O is the product of at most n reflections in hyperplanes
through O.
1.6 Discussion
The geometric properties of complex numbers were discovered long before
the complex numbers themselves. Diophantus (already mentioned in Section
1.2) was aware of the two-square identity, and indeed he associated a
sum of two squares, a 2 +b 2 , with the right-angled triangle with perpendicular
sides a and b. Thus, Diophantus was vaguely aware of two-dimensional
objects (right-angled triangles) with a multiplicative property (of their hypotenuses).
Around 1590, Viète noticed that the Diophantus “product”
of triangles with sides (a,b) and (c,d)—namely, the triangle with sides
(ac − bd,bc + ad)—also has an additive property, of angles (Figure 1.6).