Shape

252 II Seeing How It Works
Two secondary lines, moving around an imaginary main line.
What a marvelous way to start—adding complementary forms in spatial relations, and
likewise adding lines and subtracting one that’s then imaginary.
But decomposing shapes isn’t the only way to define spatial relations to use in
my schemas. I can start from scratch. I can begin with shapes—a vocabulary—and
enumerate apt configurations. There’s reason to be combinatorial after all. How things
feel when I pick them up, move them around, and line them up with my hands is a
natural way to approach this. It’s similar to what Frederick Froebel emphasizes in the
kindergarten. His famous building gifts and tablets (play blocks and plane figures),
and categories—forms of beauty (symmetrical patterns), knowledge (arithmetical and
geometrical facts), and life (buildings, furniture, monuments, etc.)—provide some
nice material for experiments. Spatial relations determine rules to augment the categories,
so that I can calculate. The spatial relations are neatly defined in kindergarten
rhymes—
Face to face put. That is right.
Edges now are meeting quite.
Edge to face now we will lay,
Face to edge will end the play.
For pieces in the gifts, it might look like this
It feels right. But let’s try it for geometrically similar polygons as boundaries of plane
figures, instead of using play blocks. It’s still the same idea, with the difference that
everything goes together equal edge to equal edge, and vertex to vertex.
Suppose I begin with right triangles like this one
Then I get twenty-one spatial relations with distinct triangles