**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 **

**Dear Jinny: **

**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.) **

**-Birdman **

**[Jinny did not respond.]**

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