A Birthday Puzzle.

It's my birthday. I told someone I was 69 but of course I’m 68, just entering my 69th year or the last year of my seventh decade.

My gift to myself was a new perspective on the platonic solids. I've been fooling around with three-dimensional shapes for a long time. I drew a three circle overlapping Venn diagram the other day and noticed it creates seven spaces.

One in the center where all three circles overlap, three where each of the three pairs of circles overlap and the remaining three spaces where there is no overlap.

But I should call this eight spaces--one for the background.

And then considered what space it diagrams. How these lines would divide a sphere in three-dimensional space. It's an octahedron. [of course. three circles at right angle inscribed on a sphere.]

What had I not noticed or discovered this before?

So here is a beautiful flower produced by an octahedron.

Next I am looking at the star of David inscribed in a hexagon. These of course are the lines the shadow of an octahedron will make or--if you have an octahedron hanging from your ceiling fan--a pattern you can see with your own eyes if you look in the right direction.

What I was trying to do, having colored the Venn diagram was trying to imagine how the octahedron itself looked in these colors. To my surprise the octahedron can be viewed somewhat as the icosahedron appears --a structure on top that mirrors the structure on the bottom with a belt of isosceles triangles around the middle. In the case of the octahedron it's two triangles facing each other with six triangles in a belt around the middle.

So while I'm thinking of all these things I have in mind a project or at least the question: what is the Venn diagram for the other platonic solids?

The Venn diagram produced by the octahedron is beautiful --more beautiful than the the solid itself:

Sitting by the fire
under the twilight sky at
her oceanfront home

thinking of
her lover
also alone.

Zphx