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Triangle $ABC$ has equilateral triangles drawn on its edges.
Points $P$, $Q$ and $R$ are the centres of the equilateral
triangles. Experimentation with the interactive diagram leads to
the conjecture that $PQR$ is an equilateral triangle.
There are many ways to prove this
result. Here we have chosen two methods, one which uses only the
cosine rule and one which uses complex numbers to represent
vectors, and multiplication by complex numbers to rotate the
vectors by 60 degrees.

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.
Three examples of particular tilings of the plane, namely those where  NOT all corners of the tile are vertices of the tiling. You might like to produce an elegant program to replicate one or all of these.