What is the ratio of the blue area to the area of the whole regular star in this picture?
This problem has a very beautiful answer, but it's hard for most people to visualize, unless they have a lot of geometric intuition. You might want to try playing with it for a while before visiting the discussion in the NYC Math Circle Facebook page.
The Pentablocks I showed to the New York Math Circle Teachers summer workshop last week provide a very easy way to visually construct and/or confirm the solution to the problem above.
Pentablocks are a very nice way to develop your geometric intuition in understanding the relationships in pentagons, pentagrams, and related polygons, the golden ratio (phi), Penrose tilings (quasicrystals), and more. As their name suggests, Pentablocks are based on pentagons and pentagrams (five-pointed stars). All their angles are multiples of 18 degrees, and all their side-lengths are related to one another by the golden ratio, phi.
The more traditional (and easier to find) Pattern Blocks are based on hexagons. All their components have angles which are multiples of 15 degrees, and all their dimensions are related to one another by 1, 2, and the square root of 3.
You can use either set to tessellate, but the Pentablock tessellations are far more complex than the traditional pattern block tessellations.
I'll bring sets of both types of blocks to our Labor Day weekend picnic so you can explore more relationships with them.
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Here are three puzzles for our middle school students (I did not check the literature to see whether these puzzles have appeared in the past - last question was inspired while reading Euclid's Elements)
1) Divide a triangle into two triangles of equal area (using compass and straight edge).
2) Divide a triangle into two triangles of areas p/q and 1-p/q (assuming that the original area of the triangle is 1).
3) Divide a triangle into two triangles whose ratio of areas is a golden ratio.
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