I poked at complex function 1/z, and its real and imaginary parts look like magical towers. When you look at these towers from above or below, you see sections of perfect circles. This is hinting at some underlying simplicity.
Using the map 1/z, another complex number – w=1/z – is mapped to z. Four dimensions would be needed to fully display this mapping, or you can display real and imaginary part separately. You can also envisage the action of 1/z by thinking about what happens to the grid lines of constant real and imaginary part, constant x and y in z = x + iy. A one-dimensional line is a curve described by a single parameter: For a line parallel to a co-ordinate axis, one co-ordinate stays constant, while the other one takes on all the numbers from minus to plus infinity. When following this curve, z is expressed by a function of this free parameter. If you connect all the w’s corresponding to the z’s along the curve, you see a corresponding curve in the w-plane. The free parameter is now in the denominator of the complex function, so that cannot be straight lines anymore. If you plug in the parametrization, you notice that straight lines are mapped to circles.
There is a another, geometrical way to think about 1/z as a map: it is closely linked to stereographic projection, the method of creating a distorted image of the surface of the earth on a plane. Points on the sphere are connected with one of the poles, and the projecting rays intersect the equatorial plane. The curved surface is projected down to the plane, and the plane is projected up to the sphere. Infinitely remote points on the plane become a legitimate point on the sphere, as “Infinity” is mapped to the other pole.
You obtain the curves that the grid lines are mapped to, by projecting twice: First, you project the straight lines “up” to the sphere, connecting every point to the north pole. All connecting rays from a straight line to a chosen point lie in a plane that is defined by the pole and the straight line. Thus, the projection up to the sphere is the intersection of a plane and a sphere, which is a circle.
In a second step, you use the other pole to project the circle on the sphere “down” to the equatorial plane again. It is less obvious, but circles remain circles. Even straight lines follow this rule, as they can be seen as circles with infinite radii. Stereographic projection is conformal: Angles are preserved. If all the right angles between all the diameters of a circle are preserved, a circle has to be mapped onto a circle. In addition to the projection, you have to flip the plane over by 180°. Imaginary number i is mapped onto 1/i which is equal to -i. Would the plane remain unflipped, everything on the equator would be mapped onto itself (including i being falsely mapped to i).
My inspiration was Roger Penrose’s book The Road to Reality. It is not a popular science book, but more of a textbook where the detailed proofs are offloaded to problems at the end of each chapter. All illustrations in the book are beautiful “old-school” pencil drawings by Penrose himself. You can see some of these drawings on this Pinterest account. The Riemann sphere makes an appearance in many chapters.
I think that Penrose drew pictures not only for the sake of illustrating his books. He explains his own diagrammatical notation for tensors and other multi-dimensional objects living in abstract spaces – an aesthetically appealing yet useful tool he had once invented for himself.
Drawing by hand seems to activate some part of the brain that contributes to understanding, in a subconscious way. Maybe this is common sense: We are three-dimensional beings in a three-dimensional world. We have evolved to process things with our hands and eyes, in a natural speed.
Nicholas Carr alluded to this in his book on automation, The Glass Cage, especially in the last chapter, that borders on the poetic. Carr features an architect who abandoned software to become more creative, and a photographer who returned to the darkroom. He quotes a physics professor who undertook a self-imposed program to learn navigation through environmental clues inspired by Inuit hunters. This professor describes his experience as primal empiricism, akin to what people describe as spiritual awakenings.
About Robert Frost’s poem Mowing, and the hypnotic effect of labor, Carr writes:
Only through work that brings us into the world do we approach a true understanding of existence, of “the fact.” It’s not an understanding that can be put into words. It can’t be made explicit. It’s nothing more than a whisper. To hear it, you need to get very near its source. Labor, whether of the body or the mind, is more than a way of getting things done. It’s a form of contemplation, a way of seeing the world face-to-face rather than through a glass. Action un-mediates perception, gets us close to the thing itself.