ILLUSTRATIONS FOR TOPOLOGY

From the book *Introduction to Topology* by *Yu. Borisovich*,* N. Bliznyakov*, *Ya. Izrailevich*, *T. Fomenk*o. The book was published by Mir Publishers in 1985.

ILLUSTRATION TO CHAPTER I

The central part of the picture presents the standard embedding chain of crystalline groups of the three dimensions of Euclidean space: their standard groups embedded into each other are depicted as fundamental domains (Platonic bodies: a cube, a tetrahedron, a dodecahedron). The platonic bodies are depicted classically, i.e., their canonical form is given, they are supported by two-dimensional surfaces (leaves), among which we discern the projective plane (cross-cap), and spheres with handles. The fantastic shapes and interlacings (as compared with the canonical objects) symbolizes the topological equivalence.

At the top, branch points of the Riemann surfaces of various multiplicities are depicted: on the right, those of the Riemann surfaces of the functions w=5z√ and w=z√; on the left below, that of the same function w=z√, and over it, a manifold with boundary realizing a bordism mod 3.

ILLUSTRATION TO CHAPTER II

The figure occupying most of the picture illustrates the construction of a topological space widely used in topology, i.e., a 2-adic solenoid possessing many interesting extremal properties. The following figures are depicted there: the first solid torus is shaded, the second is white, the third is shaded in dotted lines and the fourth is shaded doubly. To obtain the 2-adic solenoid , it is necessary to take an infinite sequence of nested solid tori, each of which encompasses previous twist along its parallel, and to form their intersection.

Inside the opening, a torus and a sphere with two handles are shown. The artist’s skill and his profound knowledge of geometry made it possible to represent complex interlacing of the four nested solid tori accurately.

ILLUSTRATION TO CHAPTER III

The canonical embedding of a surface of genus g into the three-dimensional Euclidean space is represented 0n the right . A homeomorphic embedding of the same surface is shown on the left . The two objects are homeomorphic, homotopic and even isotopic . The artist is a mathematician and he has chosen these two, very much different in their appearance, from an infinite set of homeomorphic images.

ILLUSTRATION TO CHAPTER IV

Here an infinite total space of covering over a two-dimensional surface, viz., a sphere with two handles, is depicted. The artist imparted the figure the shape of a python and made the base space of the covering look very intricate. Packing spheres into the three-dimensional Euclidean space and a figure homeomorphic to the torus are depicted outside the central object. The mathematical objects are placed so as to create a fantastic landscape.

ILLUSTRATION TO CHAPTER V

A regular immersion of the projective plane RP2 in R3 is represented in the centre on the black background. The largest figure is the Klein bottle (studied in topology as a non-orientable surface) cut in two (Moebius strips) along a generator by a plane depicted farther right along with the line intersection; the lower part is plunging downwards; the upper part is being deformed (by lifting the curve of intersection and building the surface up) into a surface with boundary S1; a disc is being glued to the last, which yields the surface RP2. The indicated immersion process can be also used for turning S2 `inside out’ into R3.

On the outskirts of the picture, a triangulation of a part of the Klein bottle surface is represented.

A detailed explanation of this picture may serve as a material for as much as a lecture in visual topology.

# art

# Michael Faraday

# Cutting-chai-glass Pyramid

# Heaven and Hell

Circle Limit IV

Heaven and Hell

by M C Escher

Yesterday I have put up Escher’s Circle Limit IV – Heaven and Hell on my new desk. The Circle Limit series of drawings was drawn by Escher are essentially what are known as his hyperbolic tesselations. The new computer table that I have got has an odd shape. On one end the side is circular and it smoothly metamorphises into rectangle on the other side. Though it is not at all comparable to what Escher has accomplished, I feel bad even when I use the word metamorphosis for this, but I have not found anything better. The table is designed for use with a desktop. So it has sections for different parts of the desktop like the monitor, CPU keyboard etc.

Anyways the main point that I want to tell is that the table at one end is circular. Since I had put Escher’s Three World on another table, I thought it would be a good idea to use a ciruclar print of Escher for this part of the table. Of all the prints I had, which I had taken when I had at my disposal A3 sized printers, the one which fitted the purpose seemed to be Circle Limit IV – Heaven and Hell.

Let us see what Escher himself has to say about this series of works viz. The Circle Limits:

So far four examples have been shown with points as limits of infinite smallness. A diminution in the size of the figures progressing in the opposite direction, i.e. from within outwards, leads to more satisfying results. The limit is no longer a point, but a line which border’s the whole complex and gives it a logical boundary. In this way one creates, as it were, a universe, a geometrical enclosure. If the progressive reduction in size radiates in all directions at an equal rate, then the limit becomes a circle. [1]

And he says this about Heaven and Hell:

CIRCLE LIMIT IV, (Heaven and Hell)

[Woodcut printed from2 blocks, 1960, diameter 42 cm]

Here also we have the components diminishing in size as they move outwards. The six largest (three white angels and three black devils) are arranged about the centre and radiate from it. The disc is divided into six sections in which, turn and turn about, the angels on a black background and then the devils on a white one, gain the upper hand. in this way, heaven and hell change place six times. In the intermediate, “earthly” stages, they are equivalent. [1]

Like most of Escher’s drawings this one also takes you to a different world. A world which is far away from the reality. A world of mathematics. A world of abstraction. But then as always we can make connections between this abstract world and the real world. The connections that we can make are dependent on the world view that we have. Some people fail to make the connection. They cannot `see’.

The Circle Limit series is what brought Escher to the eyes of the mathematicians. H. S. M. Coxeter used Circle Limit II as an illustration in his article on hyperbolic tesselations. Since then the other works of Escher have been examined by the mathematicians, and we find that very deep and fundamental ideaso of mathematics are embedded in them. As to how Escher did it is amazing. The kind of clear insight that Escher exhibits in his artwork is astounding. He could visualize the mathematical transformations in his head and then transform them onto the artwork he was working with. Escher has said

I have brought to light only one percent of what I have seen in the darkness. [2]

This must be certainly true, as most of his artwork is nowhere close to what we see in the light. I rate the artwork of Escher as greater than that of the renessaince artist’s as they had just beautifully drawn what one could “see.” But with Escher we go a step beyond, imagination takes the control. What interests me in Escher is that he can make you imagine the unimaginable. What you know is not possible is demonstrated just in front of your eyes. Logic is discarded. Rather it is kept in the basement which is upstairs for Escher.

Yesterday you start to believe what you thought was impossible tommorow.

The way different things merge for Escher is just unparalled in the work of other artists. What has now become known as “Escheresque” is just the typical of his style. Lot of later artists are influenced by the works of Escher, I have found one Istvaan Orosz particulary good. There are others who are equally good but I don’t remember their names now….

Coming back to Heaven and Hell. The main artwork is in a woodcut format in black and white. For me this is a kind of dyad which represents the world. The idea of two opposing forces one termed to be evil and the other good are all permeating in the Universe. Here also the bat-devils and the angels are the representative of the same. There is no part of the Universe where these two are not present. It might seem that somewhere far out there there is nothing, but it is not so. Even there, the design is the same, it is just too far for us to see. This is what harmony in the universe is about. It is the same everywhere, when you have a broad enough world-view. The cosmologists say that the Universe is homogenous and isotropic, if you choose to “see” it at the right scale. The cosmologists often use Heaven and Hell to illustrate this point. For me introduction to Escher came in a talk by a cosmologist who used The Waterfall to illustrate the idea of a perpetual motion machine. Since then I have become addicted to Escher, as has everybody else who has some sense of imagination. For those who cannot appreciate Escher, I can just pity at their miserable imagination.

References:

[1] The Graphic Work of M C Escher by M C Escher

Ballantine 1975, ISBN 345246780595

[2] M. C. Escher (Icons) by Julius Wiedemann (Editor)

Taschen 2006, ISBN 3822838691