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# Schlafli Tessellations

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Age 11 to 18

Challenge Level

- Problem

Although the three regular tessellation (using squares, triangles and hexagons) are well known, they do pose a demanding challenge in terms of 'elegant' LOGO procedures.

This problem considers the eight semi-regular tessellations, which geometrically are more stimulating and are shown below, along with their respective Schlafli symbols.

Look carefully at each diagram in turn. What is the meaning of the symbols?

The symbols are used in two and in three dimensions in classifying the regular and semi-regular solids. You can see an account of this in this NRICH article by Warwick Evans.

Can you construct some elegant procedures to replicate these images?

A triomino is a flat L shape made from 3 square tiles. A chess board is marked into squares the same size as the tiles and just one square, anywhere on the board, is coloured red. Can you cover the board with trionimoes so that only the square is exposed?

Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.