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I approached the problem from the other side, though, and asked whether we could fold regular 2D shapes into well-formed 3D shapes. It turns out that you can! Specifically:<\/p>\n\n\n\n
- Even-sided polygons, from 10 to at least 16 sides<\/li>
- With the polygon of half its number of sides cut out from the middle of it, e.g.,
- Cut a pentagon out of a decagon, or<\/li>
- A hexagon out of a dodecagon…<\/li><\/ul><\/li>
- Will form a ring of alternating trapezoids and triangles.<\/li>
- And that if you snip those at one juncture,<\/li>
- And fold this loose ring at each joint
- Always at the same angle<\/li>
- But either left or right,<\/li><\/ul><\/li>
- Then you find interesting symmetries in
- The shapes formed<\/li>
- The angles that produce these shapes<\/li>
- And the decision tree as to whether you should fold these left or right at each joint<\/li><\/ul><\/li><\/ul>\n\n\n\n
This blog post used to have a few examples of my early experiments. However, I more recently formalized the steps that I used to produce these one-offs, and set the supercomputers loose on testing many different possible combinations.<\/p>\n\n\n\n
The result is a somewhat abstract interactive [NOT CHROME! AND IT’S MUCH BETTER ON A FULL WEB BROWSER]<\/strong> that allows you, by clicking the nodes on the graph, to call up the animations that the my algorithm determined are interesting. Some are more interesting than others!<\/p>\n\n\n\n
![\"\"](\"https:\/\/i1.wp.com\/www.johncmulligan.net\/blog\/wp-content\/uploads\/2023\/07\/Screenshot-2023-07-24-at-2.53.58-PM.png?fit=819%2C532&ssl=1\")
Base URL: http:\/\/johnconnor.pythonanywhere.com<\/a>
Codebase here: https:\/\/github.com\/JohnMulligan\/truncated_tetrahedron<\/a><\/figcaption><\/figure>\n\n\n\nWhat’s particularly interesting is that I know there are more of these matches out there (I limited my parameter sweeps) — and the angles I found appear to conform to known dihedral angles of regular polyhedra. Comments more than welcome \ud83d\ude42<\/p>\n\n\n\n
What follows are the original examples I generated.<\/p>\n\n\n\n
Dodecagon<\/h2>\n\n\n\n
A dodecagon with a hexagon, formed by connecting every third vertex cut out of it, leaves a ring composed of trapezoids and triangles. Folding along the lines between them at an angle of 1.4154719912951197… or (acos(2*sqrt(3)\/3-1)) radians connects the different segments in a butterfly shape that approximates half of a truncated tetrahedron.<\/p>\n\n\n