Math 629: History of Mathematics
Hyperbolic Soccer Balls

Frank Sottile


This is intended to help guide you in your construction of the hyperbolic soccerball.
It is under construction, and any comments you have will be incorporated to improve it.
First, read the webpage I have for this activity, as well as the handout, which I sent to you in a package.
You will need the following materials.
Verify that you have the handout and five pages of hexagons/heptagons as shown below, as well as a few extra heptagons.

The first task is to cut out the templates. Please take note that these are precision models. If they were printed correctly (more later), then the polygons all have the same side length. While the model is forgiving, being careful with your cutting and assembly will help you to make a better model.
    A good place to start is the template that is all hexagons (the last two). These are not absolutely necessary for the activity, but are included to allow you to make a bigger model. The hexagons are to be cut along the dotted lines; and solid lines are to be left alone. These solid lines are for purely aesthetic purposes. Resist the urge to fold along the solid lines for that will detract from the attractiveness of the model and make it harder to assemble.
    When cutting along the dotted lines, cut through their middle, for the polygon you want is defined by the middle of the dotted lines. The template on the left above has a hexagon in its centre. There is a dotted line from the periphery to the central hexagon, which is surrounded by dotted lines. You will want to cut along the dotted line to the central hexagon and cut it out intact, for it is used in the model, as is the path you cut to it.
    The solid black heptagons ideally should be cut so there is no white left on the heptagon and no black left on the scraps.
    A final note about cutting: We will be using the templates from left to right above in the construction. I would recommend keeping the cut pieces together in a folder, such as the envelope I sent everything to you in. I find it best to do this on a clean table with good light. I also find it a good idea to not keep the scraps in a single piece, for it makes it unweildy to manipulate the paper during cutting.
    When you are done, you will have the pieces shown at the right (and a few extra black heptagons). Note also the scissors, pencil, tape, and index card.
If you encounter any questions in your cutting, or have suggestions for improving these notes, please write me.

Before starting to tape, observe that each polygon has two sides, a printed side and an unprinted side. You will always put tape on the inprinted side; the other side will have the beautiful regular pattern of a hyperbolic soccer ball that gives this activity its name.
    You will be taping the polygons to each other edge-to-edge. Each edge will get two (short) pieces of tape; these are placed across and not along the edge, so that the tape is roughly perpendicular to the edge. Place the first piece of tape up to, but not over, the vetex at the end of the edge, and then the second piece of tape up to, but not over, the vertex at the other end of the edge. The pictures at right show one, and then a second piece of tape applied to one edge. The contrast is not great, but you can see that the two pieces overlap in the middle, and each goes up against a vertex. It is important that the tape not cover the vertices (where three polygons meet), for the model is not flat at the vertices. Note that you will want to use one hand to steady the paper, and the other to apply the tape. I get the tape started with one finger, and then smooth it with another. It helps to not have too large pieces of tape. Also, if you do a nice job, it will help later as you will be writing on the tape.

    Now it is time to look at the sequence of pictures and commentary on the construction of a smaller model with these same pieces.
    With these instructions, you are ready to begin. Start with the triangular-shaped piece with nine hexagons on it. Where the central hexagon was removed, you will tape in a heptagon (it is black on the other side). Laying both the piece with the nine hexagons and one heptagon face down on a table, line up an edge of the heptagon with one of the six edges (best if it is not adjacent to the path cut to the central hexagon). Tape them together along this edge, as explained above. In fact the pictures above are from this step. When you are done with this, you will have the heptagon attached to the hexagons, but able to flap freely.
    Now you are at the most crucial step. The adjacent edges do not line up. We can figure out the overlap as follows. Each hexagon subtends 120 degrees at its vertex, while the heptagon subtends 180-360/7 = 128 and 4/7 degrees. Then the sum of the angles is 120+120+128 4/7 = 360 + 60/7 degrees. There is an overlap of 60/7 degrees, which is about 8.571428 degrees. You have to open up your model this much to get the edges to match. Goto the construction page for the crucial step and the next one, and return.
    Remember to always tape an edge adjacent to one you have already taped, if there is such an edge to be taped, and to tape up to, but not over each vertex. After taping six of the seven sides of the heptagon, the slit you cut to the central hexagon has now opened up. Ideally, that hexagon (or any of the other ones you have cut out) should fit in the space left. But life may not be ideal, as we may see on the first picture at right. The corners do not match up, and the overlap is a bit more than one milimetre for each. The hexagon simply will not fit. The solution is to carefully cut a very thin triangle off each of the two hexagons on the side, you may see the result of this in the second picture at right.
    This illustrates a general principle for this model. You may perform small surgeries like this to get pieces to fit, and the model is very forgiving.
After taping in the hexagon, you have a model with a black heptagon surrounded by seven hexagons, with three further hexagons hanging off of it. It has bilateral symmetry. Look at this picture and return.
Here is where this model will start to diverge from the model on the construction page. In each of the six bays formed by three hexagons on the periphery of your model, tape in a heptagon, attaching it on three sides to the hexagons. With each heptagon, your model will become more and more three dimensional. You should note that you can alwways manipulate it a bit, making one part somewhat flat, at the expense of other parts becoming more wild. Below on the left is this model with the six heptagons in it. Notice that the two heptagons on the left are nearly adjacent in that there is one edge of one hexagon between them. Note the same for the two heptagons on the bottom. For each pair, we will attach one of the W-shaped pieces with seven hexagons on it. Review the steps for attaching the W-shaped piece and return.

The picture in the middle above is what you should see after attaching one of the W-shapes pieces (it is on the lower left), and the picture on the right above is after the second W-shaped piece has been attached (it is on the lower right). If you examine it, you will see there are now three additional bays for attaching heptagons. Go ahead and do that. This will complete your model (at least you will have enough real estate for the mathematical activity that we will do). For this activity, look at the handout and at the instructions in my comments for Week 12.
If however, you want to forge ahead, you will note that there are now five heptagons portruding on the periphery of your model. Four are attached at three edges and have four free edges and for one it is the opposite. You could now attach up to five of the configirations of four hexagons that you cut out ages ago at the start of this activity, but I recommend against doing that until we complete the mathematical activity. I managed to attach all of the pieces except for three single hexagons (I will remake my templates in the future), which gave the model below. If you have more pieces, you could make a much bigger model, for this can be repeated indefinitely. It does become hard to handle once you make it a little larger. It is possible to get more pieces, by printing the templates that I have on my website. However, it is hard to print the blacks dark enough (and not recommended on an inkjet printer), and the model works best when heavyweight (32lb. in our case) paper is used. Lastly, and I have learned this the hard way, different printers print out in slightly different magnifications, and even a one or two percent difference can make the template not fit together. (The printer at the Fields Institute has printed out in different magnifications, and you may observe that the extra heptagons did not fit so perfectly well, for they camne from templates I printed in the Autumn.)
Last modified: Tue Apr 11 20:58:59 EDT 2017