I think that the polygonal progression set would make a fascinating children's toy! You could cut it out of the hard spongy material that some of those puzzles are made of, and have the triangle be all one color, let's say yellow. Then the square could be two pieces, the red that shows in the figure which looks like a mouth, and then a red triangle just the same size as the yellow triangle. That way the child could see that the triangle is part of the square, but also that we can build a square using the edge of the triangle. The pieces would continue like that, with the square/ triangle duo fitting into the pentagon, but the pentagon also having it's own green square, so the child learns to see that as a whole shape.
I also think that it's important to introduce children (boys and girls) early on to shapes that are regularly geometric. Puzzles are good for older children, but for toddlers blocks are great. It would be nice to have some blocks that could fit into a matrix, so that s/he could see that triangles come together to make a tetrahedron, that triangles or triangles and a square can make an octohedron, and so on. I think that there are a lot of toys that show how shapes can stack, like Legos. There are not, however, enough toys to show how shapes relate to each other, or can be seen within each other. For shaped toys for infants, I envision geometric mobiles and dolls whose body parts are different polygons.
I know I got introduced to shapes early as a child because I lived in Washington, D.C.. I recall my father explaining the different shapes in the monuments to me as we flew in from Atlanta. He talked about the rectangle in the Lincoln Memorial, the dome of the Capitol and the Jefferson Memorial. He even taught me how to remember which was the Jefferson and which was the Lincoln by looking at the sides of the building tops. The Lincoln has an "L" as two edges of the rectangle and the "J" of the Jefferson is signified by the curve of the dome. My mother took me to see the rectangular Reflecting Pool between the Lincoln and the Washington Monument, and the sliced out "V" of the Vietnam memorial cut into a hillside.
I think it would be interesting to see slices of more things that are regular but not linear, like braiding and weaving. Also on an unconnected note, the dual pictures remind me of beehives.
For a three dimensional exercise, construct a hypercube with boxes or origami cubes. For a less three dimensional exercise, investigate Dali in the library or on the web, and note his use of and twisting of three space in other works.
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