Shape

213 Euclidean Embeddings
The algebras U ii have an interesting property. A transformation of every shape in
each algebra is part of every nonempty shape. This is trivially so in the algebra U 00
for a point in zero dimensions. Both the empty shape and the point are parts of the
point. But when i isn’t zero
the possibilities multiply. In this case, infinitely many transformations of every shape
are parts of every nonempty shape. This is obvious for the empty shape. It’s part of
every shape under every transformation. But for a nonempty shape, there’s a little
more to do. Because i equals j, the basic elements in the shape are coembedded. And
as a result, a single basic element contains them all. But infinitely many transformations
of any basic element can be embedded in any other basic element. I can make
the former smaller and smaller until it fits in the latter and can be moved around. So
there are infinitely many ways to make any shape part of any other shape that has at
least one basic element. The triangle
in the algebra U 22 is geometrically similar to parts of a square
and in fact, to parts of itself