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@ 9f30bdc2:318356ee
2025-03-01 11:47:23
Origami pentakis dodecahedron!
I decided a few weeks ago that I wanted to make a truncated isocahredron out of sonobe units. I made some and started assembling them into triangles and then (what I now know to be) a hexapentakis truncated icosahedron. I soon decided to actually calculate how many sonobe modules I would need and decided 270 was far too many so rethought my plan and settled on a pattern that only requires 90 modules.
I then set various rules for the colours and spent a lot of last week playing with different options and trying to find one which satisfied my conditions. I was working on colouring the graph of a truncated isocahredron and wanted:
(1) all edges around a face to be a different colour to each other and
(2) all edges coming out of a face (spokes?) to be different colours to each other too
(and of course any adjacent edges must be different but this is covered by condition (1)).
After many failed attempts of doing this by hand and beginning to doubt it was possible, I eventually numbered the edges and made this into a graph colouring problem on a graph with 90 vertices and stuck it into Sage which confirmed the chromatic number of the 90-vertex graph was in fact 6 and gave me a possible colouring.
Then several hours later at the end of last week, the modules had been put together. Due to the way each edge from my diagram was rotated, the resulting shape is made of triangular faces (if you imagine filling in the gaps) and is in fact the dual of the truncated icosahedron - i.e. the pentakis dodecahedron.
#MathsArt #MathArt #Origami
https://media.mathstodon.xyz/media_attachments/files/114/086/984/176/742/261/original/8fba7af2c167984b.jpg
https://media.mathstodon.xyz/media_attachments/files/114/086/984/979/313/144/original/15f9db76ae9b2ce0.jpg