The real number -1 = -1+0i = (1,180°) has angle 180 degrees (mod 360
degrees) and length 1. The purely imaginary number [0,1] = 0+i1 =
(1,90°) has angle 90 degrees and length 1. Multiplying this point or
number by itself, that is, squaring it, gives the point with length 1
×1 = 1 and angle 90°+90° = 180°. So the product equals -1+0i = -1.
We call i, the principal square root of -1.

A second square root of -1 is obtained as follows. The imaginary number
(0,-1) = 0+i(-1) = [1,-90°] has angle -90 degrees and length 1.
Multiplying this point or number by itself, that is squaring it, gives
the point with length 1 times 1 =1 and angle (-90°)+(-90°) = -180° =
180° (mod 360°). So this product equals -1+0i = -1 as well.

This provides two square roots of -1 as both (1,+90°)2 = (1,+180°) =
-1 and (1,-90°)2 = (1,-180°) = -1.

I cannot reproduce a diagram. But what you get is four points on a
graph. Set your knives to those points, and you can cut a straight cut
on moving stock. Add other calculations, and you can graph other cuts
to minimize waste. Remember, this was pre-desk-top-computer... well
over 30 years ago and I was NOT the one making the calculations. The
above are unashamedly cribbed from a website that also cribbed from
another website... but also is dedicated to topology. Items like the
Klein Bottle and the Mobius strip can be described mathematically. As I
remember, both also use ' i ' as there are 'imaginary' conditions to be
described as points in space or points on a plain.

Peter Wieck
Wyncote, PA