The geometry of these folding fish is very interesting.
I have found two different folding fish. (square based and non-square based)
They can be connected together in many fascinating ways.
I named them folding fish because they look like fish and fold into a solid.
Below are the first five folding fish found in the whole number spiral.
The whole number spiral has two triangles for every triangle in the spiral of Theodorus.
The two yellow triangles are directly related to the trusty 3-4-5 (blue) triangle.
Examples of (non-square based) fish above and (square based) fish below.
The (square based) fish all fold into a solid 1/8th the volume of a pyramid.
Mathematics Dude 