MATHEMATICS AND ORIGAMI

Mathematics and Origami

12 TESSELLATIONS
They are decorative surfaces made out of tesserae. Though strictly speaking, these are
the small square pieces used in mosaic work, we can generalise and say that a tessellation is
composed by combined figures that produce geometric drawings or traceries like in arabesques.
Though in its source, origami had nothing to do with tessellations, we have to honour
the paperfolders devoted to this kind of work because eventually they came to the association
of both arts.
We can distinguish three dissimilar fields:

• The first is, in fact, a tessellation. It deals with the construction of a unique flat piece of ori-
  gami that, in turn, fitted with many others alike, can fill a surface as large as we wish,
  showing repetitive geometrical drawings. As we can imagine it entails a double problem: to
  design the unit piece and to envisage the large geometric drawing in which the unit will be
  integrated. When we said a unique piece we did not mean that the unit cannot be made of
  different colours or even that the paper could not have different colours in either side. If we
  add that one can play with the unit by translation, turns, symmetries, revolving it upside
  down, etc. we may arrive to a very much enriched kind of a puzzle.


• The second field starts with a large tracery, all of their lines having to be mountain or valley
  folded. We can guess how difficult it may be to design such a drawing in order that all par-
  tial figures could be folded flat. The process requires gathering and hiding much of the
  available paper: thus the flattened final figure presents a smaller size than it had the paper
  we began with. Besides, the final drawing (obverse as well as reverse) has little to do with
  the original.
  As a matter of fact, this type of construction is not a tessellation strictly speaking for it
  does not use separate pieces of tesserae. Nevertheless its traceries can be used as the guide-
  lines to copy on them paper cut pieces as tessera-like units, that will fit in the great surface
  and be reproduced till the infinity.


• The third field looks more like the second rather than to the first. It has the particularity,
  though, that instead of leading to a flat figure (obtainable as well) it holds a certain volume,
  shaped as bas-relief that gives a splendid contrast of light and shadow (recall gypsum deco-
  rative works in Arab constructions).
  Let ́s see some examples of each case.

CASE 1A: Forcher ́s fish

Figs. 1 to 22 show the fish folding process starting from an equilateral triangle.

Fig. 23 is the fish we reach at, both, as it looks like and with its triangled surface in case we
wish to draw it on a piece of cardboard to be used as the unit of a puzzle. The side of those triangles is
1/12 of the big original triangle. It should be noted that all the angles in the fish are multiples of 30º
which is conclusive to fit the units with each others till the infinity.

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