| Escher died before fractals were well-known. A natural speculation is
what Escher would have done if he had been familiar with fractals. |
| Enclosing infinite
complexity within a bounded space is certainly something one can do with fractals.
Peter Raedschelders did such an experiment, and reports
on it here. |
| Raedeschelders
is an engineer interested in art and mathematics. He has made about 40 prints,
mostly inspired by, but often going further than, themes developed by Escher. His
work has been shown in numerous exhibitions, and featured in books, including the
covers of two mathematics texts. |
| Making the fractal tessellation print "Butterflies I" |
| Peter Raedschelders |
| email peter.raedschelders@scarlet.be |
| webpage http://home.scarlet.be/~praedsch |
| As an admirer of the work of M.C. Escher, the famous Dutch artist
who made several beautiful tessellations, I tried to find out how
tessellations could be made. |
| There are several methods which can be used for making simple regular
tessellations. Today lots of these methods can be found in books or
on the internet. Another method which you will not find in books is
just look at the prints of Escher and do a little bit of try and error.
This method is great fun, and that is what I did. |
| Of course one must start with the easy ones, regular tessellations
that cover the complete plane. |
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| But covering the complete plane is a lot of work, and very soon, one
is no longer satisfied by this type of tilings. Again I looked at the
work of the master and I got fascinated by tessellations with limits. |
| Escher showed us that it was possible to make tessellations with border-limits,
so it was no longer necessary to cover the complete plane. |
| He used hyperbolic geometry for his woodcuts Circle-limits
(pages 429, 432, and 434 of Locher),
and Circle Limit III at the top of this page.
Escher made several other prints with limits, so I tried to find out if there are
other types of border-limits. |
| The first trial, below, was a tessellation with one border-limit. |
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| Now we use a limit but we still need the complete plane. |
| Afterwards we used a very simple construction to make a tessellation with a
real border-limit. Take a square, divide it by drawing a diagonal so you get
two isosceles triangles. On each of the shortest sides place smaller
isosceles triangles. If we continue this process we get a tessellation of
smaller and smaller triangles and a border-limit. After deforming the straight
lines into curves, (and with a little bit of imagination) seals appear. |
 |
| Seals |
| What we did with a square we can also try to do with a regular hexagon. Divide
the hexagon and use smaller versions of the pieces to build a some kind of
tessellation with a border-limit.
The result is shown below. Dr. Robert Fathauer also discovered
this fractal together with several others, all can be found at |
| http://members.cox.net/
fractalenc/encyclopedia.html |
 |
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| The first step |
The second step |
|
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| The basis drawing (Robert Fathauer, with permission) |
| An interesting thing about this drawing is that it has not only a
border-limit but also several limits within the figure. |
| Now the next problem was to find some kind of animal or other recognizable
figure that fits into this tessellation. We had segments which had all
the same shape but that shape had a good correspondence to no animal.
It is always nice to have only one recognizable figure in a tessellation,
but sometimes this makes it also really hard to imagine such a figure.
Therefore it was considered better to take two different recognizable figures. |
 |
| Left: Butterfly type 1. Right: Butterfly type 2 |
| When two segments are joined together in two different ways, two different
kinds of butterflies could be drawn. In the center another limit was
created, although this part of the print could also be filled by three
butterflies of type 2. |
| Now the complete basis drawing was filled in with hundreds of butterflies,
all drawn by hand. (At that time I didn't have a computer.) |
 |
| Butterflies I |
| We admit that the fitting is not perfect, there are some gaps between the
butterflies, but we believe that now the print looks better than with a
perfect fitting. |
| The print was made about 1984 and at that time fractals were unknown to me.
For me it was just a good looking print with a several limits, just another
print to show that mathematics can not only be very interesting but also be beautiful. |