Last month I used the “Spring Into Action” model to illustrate several principles for designing geometric art by studying and extrapolating from existing sources. Since that post I’ve continued to explore the avenues I identified in that post, as well as another that occurred later, and today I’m going to share the results! (If you missed the last post or want a refresher, I encourage you to go check it out now, since I’ll assume everyone has read it as a starting point for today.)

## The First Generalization: Number of Sides

I had identified two directions in which to extrapolate from the existing model: number of segments, and number of sides. Since number of segments is pretty obvious and wouldn’t change anything except the length of the “spring”, I decided to work on number of sides. Here is the chart I derived last time:

I decided to attempt the 16-sided model, since it had an easy and accurate approximation (1:5). I generated a crease pattern exactly analagous to the original 12-sided pattern:

Note that, as I had concluded before, the proportion of the overall rectangle is the same for both (8:15). Once I started folding, though, I ran into an unexpected problem. After pre-creasing the grid and collapsing the first spiral,

the next step is to move the inside part of the spiral to the outside before continuing.

This was by far the most challenging step in the 12-sided model, and for the 16-sided model I have yet to succeed in doing it! The problem is that because the triangle segments all pass directly through the center, the junction at the center is extremely tight, making it tough to reverse that flap. For the 12-sided model I was able to “cheat it” without too much bending. But the whole problem is even worse with more sides.

So far, I am stuck here:

However, the paper when not under pressure is now kind of pretty in its own right:

so in one way perhaps I’ve already succeded in creating a new piece of art. However, it clearly seemed like greater numbers of sides was going to be a dead end in terms of creating other springs. This got me to wondering whether there was any way to make things less tight in the center, i.e., to make a bit of a “hole” in the middle. This led me to…

## A NEW Generalization: Non-Central Triangles

Polygons have a rich and fascinating inner geometry, and perhaps I’ll devote a post to this in the future. But for now, it’s enough to realize that there are many, many ways to connect the various points of a polygon. For example (returning now to the original 12-sided polygon), the combination of all possible connections between vertices looks like this:

Beautiful, but very complex! One way to think about and make sense of it is that from any vertex, you can draw a line to another vertex that is 1 away (i.e. the one next door, which will give you an edge of the polygon), 2 away, 3 away, etc. Each choice, when done around the whole polygon, yields a different 12-pointed “star” (the 1 star is the polygon itself, and the N/2 star consists of all the diameters of the polygon):

Combining all the stars above yields the original figure. So, how does any of this help our quest for a spring with a “hole in the middle”? Well, our original triangle was formed by an outer edge plus one “5” line and one “6” line, which I’ll now call a “5,6” triangle for convenience. What I realized was that there were several other triangles that could be formed that still contined an outer edge (which is necessary to create the outer edge of the spring) but did NOT contain a “6” line going through the middle. In the new notation they would be “4,5”, “3,4”, “2,3”, and “1,2”. Here they are all shown together, each in a different color:

The original spring was formed using 5,6 triangles (in blue above), but I realized that springs could also be made from 4,5 (teal), 3,4 (green), 2,3 (light green), or 1,2 (yellow-green) triangles as well. Here is what each triangle’s strip and collapsed spiral look like:

And here is what 6-segment crease patterns for each variation look like:

## Finally! A New Original Model!

After all of that, and after folding a few test strips:

I was able to successfully fold the 4,5 crease pattern above. Here are the original spring and my new creation, side by side:

From the side, the springs look almost identical, although if you look closely you can see that the new spring has a “softer” (less angular) appearance. The view from the top really shows the difference:

Sure enough, there’s a hole in the middle – just what I was looking for! And, as I expected, it was **much** easier to fold!

## Comparisons and Next Steps

As far as the original “spring” concept goes, I have to say that my new model doesn’t fare as well. I think that tightness in the center is part of what makes the model want to spring apart so much. The new model does some of that, but not nearly as much, and I expect that the other models would get progressively less springy as their holes widen and they are less “twisty”. However, having the space in the middle leads to some interesting possibilities. I am already imagining making, for instance, making a lamp out of one of these spirals, or even combining segments of different spiral types. So, as so often happens, following an exploration like this doesn’t end up where you expected, but still lands somewhere interesting.

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