The Magnetic 59 Icosahedra

The regular icosahedron (or 20-sided die) is a very fascinating shape. By extending the 20 faces into planes, and looking at their intersections, one can generate "stellations", or further geometric shapes with icosahedral symmetry. These have been studied thoroughly — there are 59 of them, described here on Wikipedia: The Fifty-Nine Icosahedra.

Many intricate models of these shapes have been made over the years. But for the first time, you can make a model of all 59 at once, with this magnetic set of all the cells that compose all the stellations*! These include some very beautiful shapes, such as:

• The regular compound of five octahedra
• The regular compound of five tetrahedra
• The great icosahedron
• The excavated dodecahedron
• The final stellation (cover photo), or echidnahedron

The components were all designed in Fusion 360; the .f3d files are attached. STL files are included below.

* Actually, some of the 59 mathematically defined stellations are not connected solids — you can't make those! But you can make all of them that are connected.

You will need to 3d print multiple copies of 12 types of cell, detailed below. You will also need a lot of magnets, and some superglue. The magnets must be 4x2 mm disc magnets, the stronger the better. I recommend N50 neodymium magnets, which can be obtained here. (Weaker magnets will work, but then the final stellation (large purple assembly) may not be strong enough to support its weight.)

Any superglue will work, but if you're going to glue a large number of magnets, you'll want a large quantity of superglue; this is a good option.

To make the entire set, you will need to print a total of 415 pieces. By far the most time-consuming are the 60 large h spikes, shown in purple in the cover image. However, you don't need to go all the way to layer h. You can start by printing layers a (the central icosahedron), b (20 copies of piece b), and c (30 copies of piece c), and that will let you make the compound of five octahedra. Each additional layer you add creates more possibilities. You can also experiment with different coloring options. I recommend using five colors for each piece type that has a number of pieces divisible by five (all but a and f₂), but you can also color by piece type or by layer type, or just use one color.

The number of magnets you need depends on how many layers you want to print. To go up to a given layer, you will need this many total magnets:

• a: 0
• b: 40
• c: 160
• d: 400
• e: 760
• f: 1,360
• g: 2,080
• h: 3,280

Be sure to order more magnets than you think you will need. You will make some mistakes!

You may also want a copy of the book The Fifty-Nine Icosahedra, by Coxeter, Du Val, Flather, and Petrie, available here, but if you prefer you can refer to the above Wikipedia page for background and the list of stellations you can make.

To make the full set, you will need to print this many copies of each kind of piece:

• a: 1
• b: 20
• c: 30
• d: 60
• e₁: 20
• e₂: 60
• f₁ (left): 60
• f₁ (right): 60
• f₂: 12
• g₁: 30
• g₂: 60
• h: 60

You will also want to print one copy each of the two magnet jigs, bottom-jig and top-jig; these make it easier to glue in the magnets.

I suggest starting with just the a, b, and c pieces, and adding layers on one by one after that, to whatever level you want to complete. Each new layer adds new possibilities. Especially once you reach the e, f, and g pieces, of which there are multiple kinds of each (e₁ and e₂, etc.), the possibilities really start to multiply — that's why the numbers get up to 59 in the end. The final stellation, using all the h pieces, is quite large, and weighs 10 pounds!

I printed my pieces in PLA with a layer height of 0.16 mm. Each piece has its type engraved on one of its "bottom" faces, as well as a version number for the design (currently, v4). This makes it easy to identify which piece is which (and for me, keep my prints from various versions from getting confused!). But you will still likely want to separate the pieces by type in bins or ziploc bags when not in use.

The magnet jigs make the task of gluing all the magnets into their magnet holes in the pieces much easier and less error-prone. There is a "top" jig and a "bottom" jig. Each one needs a magnet glued into the hole in its top with superglue. It doesn't matter which side of the magnet is north or south; just make sure the magnets you glue in each one have opposite orientations! If you did it right, the two should stick together (after the glue has dried) to form a double-cone shape.

Note: you can use these jigs for other projects with magnets as well, to help ensure you are always gluing magnets in with the right polarity.

Each piece has some "bottom" faces and some "top" faces. The bottom faces are those that go inward, towards the center of the stellation; you will not see them when your stellation is assembled. The top faces go outward. Bottom faces of a piece of any type will connect to a top face of a piece of the next-lower type, and vice-versa (for example, a d bottom face will connect to a c top face).

The key here is that the bottom faces are all marked with a dot. (The piece label is also always on a bottom face.) You want to use the bottom jig for gluing a magnet into a hole in a bottom face, and the top jig for a hole in a top face. First, put a single magnet onto the jig. It will stick on with the correct orientation, because it will attach to the magnet that's permanently glued there. Next, put a small drop of superglue into the hole on the piece. Then, pick up the jig and use it to place the magnet in the hole. Force it in flush, then slide the jig aside, leaving the magnet neatly in place. Quickly wipe away any excess glue with a paper towel or rag. (See the attached movie for an example.)

There is one "gotcha" here. Magnets live in the world of the magnetic field. They feel it, and they contribute to it. Sometimes, that field can be a bit annoying. What can happen is this: you've already glued some magnets into a piece, and you're gluing another one in. The new one senses the field of the ones already there. Sometimes, that field can make the new magnet want to flip. I think the best example of this is the top faces of the e₁ pieces. You have to be very careful here that when you push the magnet in and slide the jig aside, the magnet doesn't flip in place! This can happen almost too quickly to see. If the fit is not snug, you need to hold the magnet in for several seconds for the glue to harden enough to hold it against the ambient field. If the magnet does flip, you may have a chance to extract it. Quickly hold a large stack of magnets against it and pull. If you're not in time, the piece is ruined, and you'll have to print it again.

Note: the magnet counts above in Supplies assume you only put magnets in the bottom faces of the outer layer. (You wouldn't need them in the top faces.)

Each stellation has an official name, which consists of the kinds of pieces that compose it. There is a shorthand: if a letter is capitalized, that means also use all pieces that go inside that layer. For example, C means use all a, b, and c pieces. Assemble those, from the inside out, and you will have made the regular compound of five octahedra!

With this convention (fully explained in the Wikipedia link above), the excavated dodecahedron, for example (middle right photo above), is called Ef₁g₁. So just assemble all a, b, c, d, e, f₁, and g₁ pieces, from the inside out, and you will have it.

There can be a bit of a puzzle aspect here as well, in terms of color symmetries. For each layer, there can be a few different ways to place the pieces symmetrically, if you have printed them in five colors. Some ways are better than others at highlighting symmetries for a given stellation. For example, the e₂ and f₁ pieces are arranged differently when making the compound of five tetrahedra, vs. the excavated dodecahedron, in the pictures shown here. Have fun experimenting!