| When beginning to discuss how regular divisions of the plane can help our
perception of infinity, Escher discussed the shape of the tiles. |
"What kind of figures? ... abstract, geometrical, rectilinear figures, squares or hexagons,
which at most remind us of a chessboard or a honeycomb? No, we aren't blind, deaf, and dumb.
We consciously observe the forms that surround us and in their great variety speak a clear and
fascinating language to us. That's why the forms we use to compose our plane division must
be recognizable as tokens and clear symbols of the living and dead matter around us.
When we make a universe ... it can become a universe of stones, stars, plants, animals, or
people." |
| Another approach that captures the complexity of the world around us is to use
fractal tiles. |
| For example, starting with a tessellation by hexagons, replace
each line segment by three line segments as shown on the left. |
 |
| This is a version of midpoint modification, so at
each stage the resulting shapes tessellate the plane. |
| The limiting fractal shapes are called Gosper tiles. |
| Other approaches to fractal tilings have been explored by Peter Raedschelders |
| http://home.planetinternet.be/~praedsch |
| and Robert Fathauer |
| http://members.cox.net/fractalenc/encyclopedia.html |