Linear Algebra, 2020a

390 Chapter Four. Determinants
axes, xy-axes. These axes appear in the picture at both antipodal spots, one in
the northern hemisphere at Q 1 and the other in the south at Q 2 . Observe that
in the northern hemisphere the positive x axis points to the right. That is, a
person who puts their right hand on the sphere, palm down, with their thumb
on the y axis will have their fingers pointing with the positive x-axis.
Q 1
Q 2
The sequence of pictures below show a trip around this space along the projective
line: Q 1 moves up and over the north pole, ending on the far side of the sphere,
and its companion Q 2 comes to the front. (Be careful: this trip is not halfway
around the projective plane. It is a full circuit. The antipodal spots at either
end of the dotted line form a single projective point. So by the third picture
the trip has pretty much returned to the same projective point where it started
from.)
Q 1
Q 1
Q 2
=⇒
Q 2
=⇒
Q 1
Q 2
At the end of the circuit, the x part of the xy-axes sticks out in the other
direction. That is, for a person to put their thumb on the y-axis and have
their fingers point positively on the x-axis, they must use their left hand. The
projective plane is not orientable — in this geometry, left and right handedness
are not fixed properties of figures. For instance, we cannot describe a spiral as
clockwise or counterclockwise.
This exhibition of the existence of a non-orientable space raises the question
of whether our universe orientable. Could an astronaut leave earth right-handed
and return left-handed? [Gardner] is a nontechnical reference. [Clarke] isa
classic science fiction story about orientation reversal.
For an overview of projective geometry see [Courant & Robbins]. The approach
we’ve taken here, the analytic approach, leads to quick theorems and
illustrates the power of linear algebra; see [Hanes], [Ryan], and [Eggar]. But
another approach, the synthetic approach of deriving the results from an axiom
system, is both extraordinarily beautiful and is also the historical route of
development. Two fine sources for this approach are [Coxeter] or[Seidenberg].
An easy and interesting application is in [Davies].