Hexaflexagons will blow your mind, man. Also, fantastically narrated.
This is an old trick for finding the centre of a circle (this particular example being drawn from a Russian book, published in 1986).
How does it work? It’s based on Thales’ Theorem, which states that if points A and C (below) are the ends of a diameter, then joining them to a third point on the circle, B, will always form a right angled triangle, no matter where B lies.
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The formula 1+2+3+…+n = k2 holds for only certain values of n and k.
The first solution n=1 k=1 is trivial.
The next solution is n=8 k=6. The following pictures show polyominoes of length 1-8 tiled inside a 6 x 6 square, one with L polyominoes, and the other with W polyominoes:
For more information on the fascinating world of polyominoes, see:
http://mathworld.wolfram.com/Polyomino.html
And don’t even get me started on the r-pentomino from Conway’s Game of Life.
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Fractal Federation Façade.
A mathematical landmark in the heart of Melbourne.
Listen above, or to delve more deeply, read this:
http://www.springerlink.com/content/k11885r713151508/fulltext.pdf
The Plastic Brain 