A great source of inspiration for how to fill that graph paper are nineteenth century Froebelian kindergarten text books. Froebel proposed a curriculum based on manipulating and creating using basic forms, realized in the form of "gifts" that were provided to students at various stages of their learning. The gifts were blocks, sticks, squares of paper, drawing tablets and other objects that were used to build, weave, cut, fold, and draw combinations of basic forms. Some old textbooks include nice illustrations of how the gifts were used. For example, the seventh gift was "parquetry tablets," very much like the pattern blocks that are found in many classrooms. The illustrations for these provide some nice inspiration for what you can do when presented with a blank sheet of graph paper.
These illustrations, showing the richness of what can be done with the 45-90 triangle, are from The Kindergarten Guide: An Illustrated Hand-book, Designed for the Self-instruction of Kindergartners, Mothers, and Nurses (1877), by Maria Kraus-Boelte (on google books, here).
The similar set below are taken from The Paradise of Childhood: A Manual for Self-instruction in Friedrich Froebel's Educational Principles, and a Practical Guide to Kinder-gartners (1869), by Edward Wiebe (on google books here).
For example, starting with motif 72/232, I found one way to fill the page.
a pattern based on motif 72/232
Of course, it wasn't long before I felt compelled to re-create it using some dynamic geometry software (GSP):
As you get into the flow, a host of associations and observations may come to mind. One thing I noticed about this pattern was that the gaps look like something you would make from an origami windmill base (almost a pajarita or cocotte, a popular "European" origami model). Not all things that pop into your head at this point are legitimate observations, of course, and upon investigation I found that the gaps are not quite a proper pajarita:
the pajarita
But seeing origami birds flying around in that pattern seems appropriate: the eighteenth Froebelian gift was folding papers, which were used to explore patterns that could be obtained from the windmill base:
Back to doodling, I tried another variation using the same basic motif:
another pattern based
on motif 72/232
Is it just a coincidence that the gaps in both patterns have the same area? Maybe, but there must be a minimal gap for any pattern based on this motif, perhaps three squares is what it is (more doodling required).
gaps in patterns based on 72/232 have the
same total area, maybe
same total area, maybe
I have to put this aside now, but you should get started. Here are a few more panels from The Paradise of Childhood that might inspire some grid paper doodles:
