Linked Ring Crafts

Today we're going to look at some arts and crafts projects related to great (and not-so-great) circles found on polyhedra and links. A few of these models were shown to me by Mircea Draghicescu of ITSHPUN fame and the rest are riffing on the theme.

Setup

For this activity, I've been using strips of paper, so that's how I'll write it up. I've done similar things with pipe cleaners in the past, but paper is nice because you can choose different length to width ratios that help reinforce the structure of particular models. The ratio is the important part, so starting size of the stock paper isn't as important. However, larger scales will typically be easier to work with. I've got some origami paper that I can slice up pretty easily, so I've gone with \(15\) cm long strips of various widths cut from \(15\) cm \(\times \, 15\) cm squares.

Models

\(3\)-Strip Model

The easiest way to think about this model is as a set of Borromean rings:

To make it, start by taping two of your strips into rings and then nesting one (blue) orthogonally inside the other (red).

Then take the third (yellow) strip and thread it inside the inner (blue) ring and outside the outer (red) ring.

The model pictured above uses \(2\) cm \(\times \, 15\) cm strips and feels a little too loose. The model below is built from \(3\) cm \(\times \, 15\) cm strips, which is about as high as I would let the ratio run, since it's pretty snug. I feel like \(2.5\) cm \(\times \, 15\) cm is probably a nice compromise.

If we think of the strips as supplying edges, their intersections vertices, and the triangular gaps as faces, then this model resembles an octahedron. In the language of spherical polyhedra, we can think about it as a spherical octahedron with edges formed by great circles.

\(4\)-Strip Model

Sticking with the interpretation of spherical polyhedra create by great circles, this model is based on the cuboctahedron, as illustrated by Tilman Piesk:

To make it, start by taping three of your strips together so that each interlocks with the other two. It can be helpful to lay them out in a weaving pattern before your start linking them:

One they are linked, you should have two poles where the three strips swirl together. One on top (pictured below), and one on the opposite side of the burgeoning sphere. If you drop the sphere, it can be hard to tell where these poles are, but one way to recover is by holding the three rings mutually orthogonal to one another to create the eight triangular windows like in the 3-strip model. Two opposite triangles will have the right swirling pattern, so make those your poles.

If we think of the poles as being the north and south poles, the fourth strip will be the equator. Thread it around the equator, weaving over-under. To start this weave you will first want to find two intersecting strips that it can form a triangular swirl with, which will tell you whether you're starting your weave with an over or an under.

In the image above, for example, you can see the swirl of the red, blue, and yellow strips left of center. You can think of the fourth green strip as making another swirl with red and blue to the right side of the image, starting under blue, over red. The full weave will then be under blue, over red, under yellow, over blue, under red, and over yellow before the loop is closed. (These swirls help keep the structure intact, since otherwise the rings could slide around more freely.)

The model pictured above uses \(1.5\) cm \(\times \, 15\) cm strips and feels pretty good. The model below is built from \(2\) cm \(\times \, 15\) cm strips and is a bit on the snug side.

\(5\)-Strip Model

This model was inspired by a relative of the Borromean rings, another sort of Brunnian link (image by Alain Esculier):

The way we'll interpret the image is that each ring contains the two rings counterclockwise from it and is contained in the two clockwise from it. Using this as our instructions, here's what we get if counterclockwise we add red, orange, yellow, green, and blue:

The model pictured above uses \(1.5\) cm \(\times \, 15\) cm strips and is far too loose. The model below is built from \(2\) cm \(\times \, 15\) cm strips and feels pretty good:

\(6\)-Strip Model A

This model is based on the spherical icosidodecahedron, is illustrated below by Tilman Piesk:

This one is pretty tricky. While advanced students might like to figure out how to create it from scratch, I'll give some step-by-step color-based instructions here. We'll use red (R), orange (O), yellow (Y), green (G), blue (B), and violet (V) strips.

First take the red, blue, and yellow strips and interlink them as rings as you would when starting the 4-strip model.

Using the a red, blue, yellow triangular swirl pole as a point of reference, here's how we'll build:

• Take the violet strip and place it at the vertex of the pole opposite the yellow strip, weaving it under red and over blue. The weaving pattern is then:

  under R, over B, over Y, over R, under B, under Y

• Take the green strip and place it at the vertex of the pole opposite the red strip, weaving it under blue and over yellow. The weaving pattern is then:

  under B, over Y, under V, over R, over B, under Y, over V, under R

• Take the orange strip and place it at the vertex of the pole opposite the blue strip, weaving it under yellow and over red. The weaving pattern is then:

  under Y, over R, under G, over B, under V, over Y, under R, over G, under B, over V

The finished product looks like below. (Note: my model has swapped the roles of green and orange, but I gave the instructions the way I did since it is more neatly color-coded in terms of primary and secondary colors.)

The model pictured above uses \(0.75\) cm \(\times \, 15\) cm strips and feels pretty good. I tried to make a model using \(1\) cm \(\times \, 15\) cm strips and it was too snug to build, but there's some wiggle room between \(0.75\) and \(1\).

\(6\)-Strip Model B

This model is easier than the other 6-strip model since it's more rectilinear, so advanced students should be able to work it out from a photo. The premise is to start with the three pairs of not-so-great circles of the rhombicuboctahedron and then interweave them in a checkerboard pattern.

To build it, we start with a pair of rings standing parallel to one another:

Then we weave another parallel pair of strips around the rings, with one weaving outside when the other weaves inside:

Then we weave the final pair of strips around both, alternating under-over with one strip and over-under with the other. The only thing to be mindful of is that each corner should look like a three-strip triangular swirl, so that should decide which of the strips weaves over-under and which weaves under-over.

The model pictured above uses \(1\) cm \(\times \, 15\) cm strips and is a bit too loose. The one below uses \(1.5\) cm \(\times \, 15\) cm strips. It felt a little snug while building it, which is why it looks a little sloppy, but it doesn't feel too snug when finished.

Notes

There's a bit of a tug-of-war between using narrower or wider strips. When you use narrower, you have more wiggle room and it's usually a little easier to see what you're doing while building. If the final structure is rigid, it also looks better to have gone more narrow. On the other hand, wider strips tend to provide a little more structural integrity and often help scaffold the build as you go. For that reason, I think erring on the wider side is the better option for beginners or students with less-developed fine motor skills.

Extra Challenges

• A few of my designs were based on links and I didn't explore all of them. Advanced students might like to explore what can be built with strips based on other models. Some of Alain Esculier's diagrams of links with low numbers of components might give some inspiration!

• Our \(6\)-Strip Model B is based on parallel pairs of rings organized with the same orientations as our \(3\)-Strip Model. There might be some interesting \(8\)-, \(10\)-, and \(12\)-strip models worth exploring based on our \(4\)-, \(5\)-, and \(6\)-strip models. It's also not hard to imagine what \(9\)-strip and \(12\)-strip models might look like with triplets and quadruplets of parallel rings.

This post was sponsored by the Julia Robinson Mathematics Festival