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Diagramming
tessellations
Dorigami asks (20/21 August 97) just exactly what is meant by "tessellations"
and asks if what is meant is Escher-like constructions.
Escher-like constructions have, indeed, been created in Origami, especially
by Mick Guy in the 1970s. Escher got his inspiration about tiling patterns from
the Moorish tiling patterns in the Alhambra Palace in Grenada in Southern Spain.
But tiling patterns made from sepate tiles are not what is meant in the present
context of Shuzo Fujimoto, Chris Palmer, Tom Hull, Alex Bateman, Paulo Taborda
Barreto et al.. It is necessary to distinguish between first, actual physical
tiling laid on a floor or wall and second, the total abstract tiling PATTERN which
can be analysed mathematically. A tiling pattern acts as a blueprint for laying
out tiles on the floor. But in this context, the tiling pattern is studied in
its own right by mathematicians and also used itself as a basis of folding by
the Origami Tessallators.
To enlarge on this: in the Escher kind of tessellation, many separate paperfolds
are made, like pottery tiles, (but usually in more complex shapes: sometimes abstract,
sometimes in the shape of animals or fishes) and they are then fitted together
physically to make a repetative pattern, just as you would lay tiles of one shape
or different shapes on a floor.
In the case of Chris Palmer et al. the word "tessellation" does not relate
to separate folded "tiles". It is used in the first instance for a regular PATTERN
in the paper which resembles possibly a simple, but more often, a more elaborate
tiling pattern. Quite apart from their interest to paperfolders, tiling patterns
today form a major branch of recreational mathematics and, indeed, of more serious
mathematics concerned with such subjects as that of symmetry, which pervades everything..
Such tiling patterns are usually regular, in which the same pattern repeats itself
a periodical intervals over the whole pattern and indeed to infinity if the paper
had no edges and could be extended without limit. But in some instances of tiling
patterns, such as those discovered by Roger Penrose, the patterns are irregular
and do not repeat periodically, (I may add that I, personally, have yet to see
an irregular tessellation pattern in Origami. I doubt if it's possible, but I
certainly wouldn't be dogmatic about it. Perhaps it's a challenge to the experts).
In this kind of paperfolded tessellation, the crease pattern of maountain
and valley folds may first be drawn with a pencil on the paper. Or a network of
parallel creases in several directions may made as preliminary guidelines. Then
mountain and valley folds are made along the lines in the particular order required
by the pattern of the tessellation being folded. Some people prefer to make their
creases without preliminary drawn lines, but, except in very simple instances,
this is much more difficult. Again, some people, like Paulo Taborda Barreto have
discovered how to create their crease patterns by computer.
Having made creases across the whole of the sheet of paper, the pattern is
then "collapsed", by folding in the respective valley and mountain crease, doubling
the paper, making twist folds and generally cajoling the paper until it is reduced
to a flat pattern. I think that "cajoling" is an apt word in this context! Nothing
happens quickly. It is usually necessary to work inwards from the edges of the
paper, but in my experience, one does whatever seems easiest and most appropriate
in the circumstances. I can't pretend that I have ever found it easy, but I expect
that practice makes perfect and I have never had the time to practise. Chris Palmer
has certainly mastered the art.
Having managed to "collapse" the paper, you will have created your finished
work of art. It will be seen that the creases you have made form a pattern of
a NEW or secondary tessellation, a sort of bas relief which arises out of, but
is different, often unexpectedly different, from the original pattern drawn on
the paper before it was creased. Some folded tessellations are three dimensional,
with such phenomena as boxes or mountains rising above the tessellated plain at
regular intervals.
Then, if you have used semi-transparent paper, you can hold your creased pattern
up to the light and you will see yet another hidden pattern painted in light and
shade accoring to where the paper is doubled or trebled in thickness.
For more about tessellations, please see my posting to
Origami-L dated 16th July, 97, where I give more general information and also
say something about Maurits Escher and also Mick Guy's Escher-like tessellations.
I added a little more, (mainly about Paolo Taborda Bareto) in my posting dated
19th July.
Tessellations are a long way from mainstream paperfolding, but they are a
glorious aspect of the wonderfully diverse kaleidoscope that is Origami.
David Lister Grimsby, England.
