For almost a century the drawings of MC Escher have stood out in the world of art as not merely uniquely beautiful, but uniquely magical. They are magical because their complexity makes it difficult to understand how the artist could have ever created them, or at least most of them.
What I am going to do in the present essay is to reveal what I consider to be Escher's basic secrets -- the fundamental techniques of how he did what he did. They are actually quite simple -- at least in principle -- and while their application may be tedious in some cases, they make clear that mortal men, and not just Escher, can do much of what this man accomplished. This is not of course to diminish the honor which goes to Escher as the discoverer and applier of these techniques, nor is it to say that these techniques cover every aspect of what he did. But I believe that this essay will remove much of the aura of mystery about Escher's accomplishments, and thereby bring us closer to understanding the truths of this often-mysterious world.
The techniques which I will be discussing refer to what is known as tessellation -- the dividing of the plane into regular repeating patterns. While this matter has often been discussed from the standpoint of mathematics (eg, how can a vertex be completely surrounded by polygons), the secrets behind Escher's work do not require such considerations. In particular, understanding Escher's work requires what might be called a 'holistic' viewpoint, ie, not one where one attempts to formulate generalizations about vertex-surrounding or plane-covering regular polygons, but instead looks at the larger and much simpler question of how to cover the plane. (Exactly what we mean will become clear in a moment.)
In approaching the secret of Escher's techniques, we need to remember that his original inspiration was the designs which one finds on Moorish architecture. Upon reflection it is unlikely that the Moorish builders resorted to fancy mathematics for their creations, but instead did something simpler. The question then is, What?
The first thing to recognize is that one does not have to be an Escher to fill the plane with regular repeating patterns -- any idiot can do it. As a simple example, draw a bunch of parallel lines on a sheet of paper. Then draw another set of parallel lines not parallel to the first set. And viola da gamba! You are a budding Escher! Or at least a budding Moorish architect.
Now you can get great distances in the tessellation business by simply superimposing sets of parallel lines on the plane, but this is not an Escher secret, which is:
First Escher Secret: Select the pattern you wish to repeat. Then repeat it in any way it suits you across the plane. Then, if any part of the plane is not covered, repeat the pattern on the plane to cover further portions of the plane, and repeat until the plane is covered.
We can see an example of this technique in the following Escher creation:
What we have here is a circle filled with designs that is repeated in vertical rows in three different variations: black, red and yellow. The circles are superimposed on one another to cover the plane, and are colored so that the nature of the design becomes apparent.
But this is the simple stuff. More complicated is a drawing like
The question which this raises is, Is it possible to take an arbitrary figure and repeat it to fill the plane without overlap? The answer is, No, but almost. This leads to
Second Escher Secret: If you want to fill the plane with an arbitrary repeating figure, repeat the figure in rows and columns, fitting each close to the next (touching is not necessary for filling the plane, but we will assume we want to touch). Then there are two ways to fill the plane. First, let the space between the figures be included in the overall picture, and Voila!, the plane is filled. We see an example of this strategy in
A second strategy is to adjust the figure on 'all 4 sides' so that the front will fit into the back and the top into the bottom. (This is apparently what was done in the winged horse figure above.) This could be a tedious operation, and would be analogous to using a computer to get a function closer and closer to an asymptote, but it can be done. It is possible that the adjustments could be done by calculation, but I am not sufficiently knowledgeable in mathematics to know how to do it. One possible way is to seek a minimum area for the background, using random numbers plugged into a formula, in an attempt to get closer and closer to a zero value.
Third Escher Secret: Use lens distortion.
A famous drawing of Escher's is his self-portrait, which is drawn using a reflection in a polished metallic ball. From this and other drawings, it is clear that Escher used distortions of this kind to produce drawings. What he probably did was to produce a drawing and then project it thru a lens or reflect it from a curved surface onto a paper, copying the distortion by means of light projection (magic lantern). The following drawing is clearly of this type:
I do not think that what has been written here exhausts all of Escher's secrets, but I believe it removes his creations from the realm of mystery. Which means that I can finally get a little sleep.
Letter to Jinny Beyer
Slightly edited from the original
Date: 7/5/03 3:59 PM
From: John Bryant
Subject: yr book
I have read thru your book Designing Tessellations (well, at least turned all the pages). It is very enlightening, and certainly does a lot better in explaining the nitty-gritty than the websites I have seen. But some unanswered questions remain, and I would appreciate your comments.
For purposes of explanation, let us make the following stipulations:
(a) The Fundamental Theorem of Tessellations is that if you chop off a part of a tessellatable figure and then stick it back on on a side of equal length, the figure will be tessellatable.
(b) We shall call the above process 'making trim equal stick-on.'
There is a basic question which you never address directly (tho you do address it indirectly), namely,
* How can you take an arbitrary drawing and turn it into a tessellation?
Another question which arises from the above is:
* If it isn't always possible to tessellate a drawing, then how can it be adjusted? (You suggest an answer to this in your discussion of Escher's Reptiles: Stick the drawing in a tessellatable form, such as a hexagon, and then adjust the drawing so that trim equals to stick-on.)
[Jinny did not respond.]
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