Dihedral Angles

This is a project of mine that’s reached a nice stopping point for the time being.

From 2020-22, I got interested in “nets,” that geometric exercise by which you unfold regular 3D shapes into contiguous but irregular 2D shapes.

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:

• Even-sided polygons, from 10 to at least 16 sides
• 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
  • A hexagon out of a dodecagon…
• Will form a ring of alternating trapezoids and triangles.
• And that if you snip those at one juncture,
• And fold this loose ring at each joint
  • Always at the same angle
  • But either left or right,
• Then you find interesting symmetries in
  • The shapes formed
  • The angles that produce these shapes
  • And the decision tree as to whether you should fold these left or right at each joint

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.

The result is a somewhat abstract interactive [NOT CHROME! AND IT’S MUCH BETTER ON A FULL WEB BROWSER] 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!

What’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 🙂

What follows are the original examples I generated.

Dodecagon

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.

Decagon

Folding along the lines between them at an angle of 2.034443043…. or acos(-sqrt(5)/5) radians connects the different segments in a pill or cradle-like shape.

Ring

A third variation on this is to take a ring-shaped cross section of a truncated tetrahedron. The folding angle for this comes to: asin(sqrt(3)/3)*2, or 1.2309594173407745…