Compound of Two Cubes with Convex Hull and Common Solid
There is a category of compounds of solid forms that I stumbled into some time ago which became one of those rabbit holes that had me mesmerized. This year I put together a family of shapes and submitted it to this year's Bridges Organization with hopes that it would be accepted into their exhibition of mathematical art. Yes it was accepted. The show opens in a couple of days.
I want to show you my piece, hoping that to convey how fun it is. Honestly, though, the best way to see it on-line is in video form, so here I am showing it to you: https://youtu.be/Bo61Vyx308A?si=vPT5NNdG9pnkN5Sh
If you don't want to click over to a video, here are some photos.
The first thing that I fell for with this form, which creates that the illusion of two cubes being caught in mid-motion of rotating at different rates around a common center, is that the whole piece is made up of a series of triangular surfaces.
The illusion itself is worthy of total delight: if you look carefully at the photo above, you will see a grey cube with colorful pieces emerging from it, or you will see a colorful cube with a gray pieces emerging from it. I find that it's hard to register seeing both cubes at once.
I passed a good bit of time staring at this form. I finally wondered what the common volume was between them. I went through all sorts of contortions to figure this out, not knowing that I could have simply done a search for the common sold of the compound of two cubes. My computer search would have told me that it was a hexagonal dipyramid, but at the time I didn't know what to ask. 's'okay, it was fun figuring it out for myself.
Next, I had to figure out how to get it inside of the the compound of two cubes.
I worked out a system of magnets inside the cubes, and paper clips inside the hexagona dypyramid that worked to keep the parts of the cubes together, but that a gentle pull would reveal the inner shape.
Somewhere along the line heard about convex hulls. Basically, this a a box that a shape can fit into perfectly. They really do snuggle together nicely. The box for the compound of two cubes is called an octahedral prism.
My most satisfying challenge of this exploration was designing how the convex hull would open to reveal the compound solid, how the compound shape would reveal the common volume, and how the common volume would open to reveal whatever it was that I chose to put inside of it.
OH, and I not only wanted these pieces to open, but I wanted them to close back up in an elegant way.
It took like five times as many attempts (at least) than is evidenced below by these rejected models that are still hanging around.
Frustrating at times to fail over and over again at getting things just right, and don't even get me started on having lost some of my files, but am happy with the the result.
Here's a link to my page in the Bridges catalogue: https://gallery.bridgesmathart.org/exhibitions/bridges-2024-exhibition-of-mathematical-art/paula-b-krieg
Here's the link to all the pieces in the Bridges 2024 Exhibition of Mathematical Art, Craft, and Design: https://gallery.bridgesmathart.org/exhibitions/bridges-2024-exhibition-of-mathematical-art
Wait, don't go! I have one more thing to say.
Back in the summer of 2022, when I first started this journey, I exchanged patterns with someone that I do know know who writes in a language whose characters that I do not recognize, This person shared what I had made with children and I got this a Facebook post link that I am sharing with you here https://www.facebook.com/permalink.php?story_fbid=pfbid029iVhPARetJrvgdHkMxpxg38FwtLUbGXQa7rCkV2X9cjDYGkdXJHpif4ofWMKSAoFl&id=100005012358812
If you can't get into the facebook post (or simply don't want to), here's a picture from that post. Was such a thrill.
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