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LOGO Challenge 5 - Patch

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

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LOGO Challenge - Triangles-squares-stars

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.

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LOGO Challenge - Tilings

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.

Napoleon's Theorem

Age 14 to 18 Challenge Level:

You might like to make a tessellation with copies of $\Delta ABC$ and three triangles drawn on the sides of $\Delta ABC$ and prove your conjecture from the tessellation. This uses only elementary geometry. An alternative method is to use the Cosine Rule.

Alternatively you can use vectors or complex numbers.