Lines and Circles

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).

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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.

1. I love this. Is it strange to be confronted with how separate we are as readers on the internet when seeing the scan of your hand drawn circles and lines? This small personal artifact that is made without the computer as mediator makes the mathematics feel very intimate and touchable, a strange and unexpected contrast to how abstract complex geometry seems to be. I am now intrigued by Penrose’s Road to Reality–the link you posted was nice.

I completely agree with Robert Frost’s view of work as you included it in your post. I think that due to Covid’s disruption of our regular routines, I’ve become much more aware of the importance of my solitude and work. I miss having days to weed the garden without interruption, or to clean the house and not converse with others. Everyone is home together, which is nice, too. But I am not made for small talk, which most conversation degenerates to when there is nothing new to talk about. Being quite different from me, my husband deems it a kindness to keep me company when I am cleaning up. So he stands on the edge of what I’m doing and talks at me continuously. I have noticed again how much I like work, especially when I can do it in solitude, and let the rhythm of movement carry my thoughts deep into some problem of interest.

1. elkement says:

Thanks, Michelle! I am also not made for small talk – and I could not imagine to work in an office with several other people (and with office small talk :-)) ever again.

It seems the COVID crisis has amplified so many things. In the first wave, everybody was in panic mode, I found it comforting to sew a mask. With only needle and yarn, and some patches cut from old clothes. Digital stuff did not help at all. It seems that manual work is key, moving and walking.

It was also interesting to observe how manual things are kept in memory. I had loved my descriptive geometry classes in high school, but I forgot all the technical terms. It was hard to google something. But when I picked up a pencil and compass, I knew what to do again. I think working only with digital items, we miss out on something that makes us “whole”.

1. For sure, the physical does give us connection to so much more than our minds.

I found that most of my teachers this past year were very keen to explore versions of the mind-body connection in thought and creativity. By the end of a year in lock-down, what seemed to be missing for everyone was the freedom to go out and walk, ride a bike, hike, or just be in nature, and feel the mental renewal that comes with these things. Lots of sharing of childhood experiences, and how they remembered to feel free to move and work and play in their environment. Then my daughters began talking a lot of about being kids, and how they were practically living in the backyard garden at pre-school age, and the time they spent with their games and play. It must have been something we all were feeling.

I can relate to what you say about math, even with algebraic manipulation. I started reviewing high school topics before my first university math class began. I felt at the beginning that I had no idea what anything was. But if I started a problem, it seemed that the pencil had a way of just guiding me, and then the brain would wake up and see what was going on.

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