Number 3 in a series of adult amusements – drawing soccer balls

Posted: 15 January 2023 | Author: sageandonions | Filed under: Drawing, Entertainment, Mathematics, Uncategorized | Tags: Drawing, mathematics | Leave a comment

It’s been five years since my last one in this series. Time for another. This is another doodling opportunity. Do you find that your doodles start to all look the same? Are you getting a bit bored with them? Well this should keep you going through at least another week or two’s worth of dull work meetings.

It’s based on your classic soccer ball. I won’t insert a picture of an actual soccer ball, for reasons that will become apparent later in this essay. The classic soccer ball design is a patchwork of interleaved hexagons and pentagons. The patches for one of those two shapes (I don’t recall which) are all black, and only a few of them occur on the ball, with no two ever touching one another – not even just at a single vertex (corner). Let’s call them “isolated patches”. The patches of the other shape are all white, and surround each of the black patches. Let’s call the white patches “common patches” because they outnumber the isolated patches.

It turns out that, if we choose carefully, we can perfectly tile the surface of a sphere – like a soccer ball – using such a combination, such that all the isolated patches are the same shape and size as each other and so are all the common patches. Also, every pentagon and hexagon is “regular”, which means each of its edges is the same length and each of its corners has the same angle.

Here are a blue-outlined regular hexagon and a red-outlined regular pentagon. Note how I’ve avoided black and white in order to preserve the uncertainty about which type of patch is black/isolated on an actual white and black ball!

Our doodles will be attempts to draw, on a flat sheet of paper, as much of a soccer ball pattern as we can, starting with the black – ie less common – patch.

One way to try is to make the pentagon the isolated patch, and to then surround it by a ring of five hexagon patches, like Figure P1:

I’ve coloured the isolated patch red here, to give greater contrast, and I’ll maintain that practice in all the drawings below, whether the isolated patch is pentagon or hexagon.

The other way to start is to make the hexagon the isolated patch and surround it by six pentagons, like figure H1.

The challenge, then, is to continue drawing more patches, gradually moving outwards from our central starting patch, and obeying the rules, which are:

1. No two isolated patches can touch, not even on a corner;
2. Include only as many common patches as you need to keep the isolated patches apart; and
3. We want the entire pattern to have the highest degree of rotational symmetry that we can achieve.

The “degree of rotational symmetry” means the number of different angles you can rotate it through and still have exactly the same pattern. For example, for a pattern consisting of just a single regular pentagon that degree is five, because rotating it by one, two, three, four or five fifths of a full turn leaves the shape unchanged. A degree of one means essentially no symmetry at all, as any rotation will change the pattern.

When the isolated shape is a pentagon, the rotational symmetry must either have degree five (same as the pentagon) or one (no symmetry). So I’d say that unless our pattern has degree five, we’ve failed. When the isolated shape is a hexagon, the rotational symmetry can have degree six, three, two or one. I think six is probably impossible and two is unlikely. Three would be good.

We don’t require all patches of the same type (hexagon or pentagon) to appear to be the same shape in our drawing, or even to be regular. On the actual ball we want that to happen, but things get distorted when you transfer patterns from a sphere to a flat surface, which is why Antarctica’s enormity gets exaggerated and aeroplane routes appear to not go straight when shown on a paper map. You’ll see this distortion happening in the patterns we draw below.

Looking at Figure P1, what do you think should be the next step? The outer ring is all hexagons, so we need to put in some pentagons, but with hexagons in between them so they don’t touch. It looks like we have a choice. We could either insert five pentagons pointing outwards with bases against the “flat bits” – see figure P1a – or we could insert five pentagons pointing inwards, nestled into each of the five “bays” in the coastline of the shape P1 – see figure P1b.

In each case we need to put a hexagon between each of the outer pentagons. I’ll leave that to you.

Note how the shapes further from the centre are more irregular. Just like how the standard Mercator map projection is most accurate near the equator ande gets increasingly distorted near the poles, this representation of a curved patchwork on a flat page gets more distorted as we move away from our starting patch. That makes it harder to work out how to draw outer rings of shapes, as they will look very different from what they’d look like on the ball. Since we can’t use similarity to an actual ball to help us, we use the above two rules as a guide. Figures P1a and P1b both obey those rules, having no two pentagons touching, and a rotational symmetry of degree five – ignoring the irregularities caused solely by my inaccurate drawing.

You might wonder – how do we finish this off? That is, how do we know when our pattern is big enough to cover the soccer ball? Well that depends on how much time we have to spend on the doodle. The quick way is to identify when we have achieved a pattern that can match up with an identical copy of itself so that when we sew them together along the edges – with the ball inside – they will cover the ball with no gaps and the pattern will follow the rules. The long way is to keep going, drawing more and more rings of patches (that get more and more irregular!) until we get to a pattern where the outside of the pattern has the same number of edges as the central patch – which will be five or six and there are no isolated patches in the outer ring. Then – imagining the pattern is drawn on a stretchy rubber sheet – we can pull it over a ball and pull it tight so that the outer perimeter of the sheet forms a shape the same as the original central patch – but at the antipodal point of the sphere. We can then sew another patch of that shape to the edges of the rubber sheet, to complete the ball’s patchwork cover.

Let’s think about the quick way first. How can we identify when it can match a copy of itself?

After reflecting on this I realised that, around any valid pattern of isolated and common shapes we can make a boundary ring of common shapes. If we couldn’t then as we continued the pattern we’d end up with at least two isolated patches touching somewhere, thus breaking the rules. The boundary ring may look zig-zaggy rather than straight, but it will continuously encircle all the isolated shapes in the pattern. In figures P1 and H1 we can easily see the boundary ring.

Let’s put rings around the patterns in P1a and P1b. Here we go:

I have put green spots in the pentagon patches making the boundary ring.

Now we could sew each of these this together with an identical patch to make complete, ball-covering sheets. But that would break rule 2. We’d have too many common patches, as we’d have two complete rings of them next to one another, which is more than we need to separate the isolated patches.

So instead, let’s do the following trick. Concentrate now, this will stretch your powers of spatial imagination!

First remove every second patch in the boundary ring, creating a cog-wheel shape. Then make a copy of that, and flip the copy upside down. By that, I don’t mean rotate it by 180 degrees on the page. I mean pick up the imaginary rubber sheet on which we’ve drawn the pattern, turn it over and lay it down again. Figure P3a shows the original next to the flipped sheet.

Now, temporarily imagining the drawing surface as being cardboard, we fold the patches marked with green spots up (away from the screen) in the original and down (towards the screen) in the flipped copy.

We then place the the flipped copy on top of the original and sew the two together, sewing together the edges with the same numbers in both picture, ie sew 1 to 1, 2 to 2, etc. Continune numbering around the perimeter of each of the two sheets and continue sewing together edges with matching numbers until complete.

That gives a shape like that of a short cylinder or disc, with the green-spot patches making the sides. By looking at the numbers above, you can see that we never have to sew two red edges together, which would make two pentagons (isolated shapes) touch and hence break rule 1.

Now all we need do is stretch the disc into a sphere, and we have a soccer ball! It obeys rules 1 and 2 and a quick look confirms that it has rotational symmetry of degree five – the maximum possible – so it also satisfies rule 3.

Similarly, we can make a complete ball from P2b by removing every second patch in the boundary ring. The ring forms a zigzag, so we can either remove all the outermost spotted patches or all the innermost ones. Either way we can make it work, although we need to distort the shapes more than for P2a in order to make the edges align for sewing. If we remove the innermost spotted patches, we get long two-patch tabs that we need to fold up or down on the two copies before sewing. If we remove outermost patches, we only have to fold up half of each outer common patch, to get things aligned for sewing.

When completing P2b we don’t need to flip over the copy before sewing. Can you think why that is?

Although the above can take us all the way to a completed soccer ball, we can have fun in taking the long approach, trying to draw the whole thing – except the final patch – on a flat sheet of paper. That gets increasingly difficult as we go further out, because the shapes get so distorted it is hard to relate it back to what it would look like on the ball.

It would spoil the fun of trying to draw them, to show diagrams here. But for those that can’t wait to know, follow these links to see sketches of completions of the patterns started as P1a and P1b2 above:

P3a – Pentagon as isolated shape, outer pentagons pointing outwards

P3b – Pentagon as isolated shape, outer pentagons pointing inwards

I’m not sure which one of those actual soccer balls use, as they both look feasible. Currently I’d guess P3a, but I’m still staying away from soccer balls, to preserve the suspense.

In those pictures, a dashed line shows the boundary between the two halves (hemispheres) of the surface. Isolated shapes have red, bold outlines on the near (starting) hemisphere and yellow, bold outlines on the far side.

What about H1, where we make the hexagon the isolated shape. Here is my attempt to complete an entire ball with that plan:

H2 – Hexagon as isolated shape

You can see from that, that shapes rapidly get really badly distorted. We can complete the ball, but the two hemispheres will be very different, some patches will be much bigger than others, and some may even be concave! Nevertheless it’s a nice challenge trying to draw it freehand.

That’s enough for now. Happy drawing!

Andrew Kirk

Bondi Junction, January 2023

PS Turns out the soccer ball shape is a “truncated icosahedron”, which means you start with an icosahedron, then slice off (‘truncate’) each of its vertices. An icosahedron is a roughly spherical 3D shape with twenty flat, triangular faces. Since five triangular faces meet at each vertex, slicing a vertex off gives you a new pentagonal face, and doing that at all twelve vertices turns the existing triangular faces into hexagons. So another way to draw the ball’s pattern would be to draw as much as you can of an icosahedron: a pattern of triangles where five meet at each vertex. Then colour in a pentagonal shape at each vertex, as per the following sequence:

PPS I wonder if there are any other patterns we could use to tile a soccer ball with two different shapes. This one has five and six-sided shapes. I tried with three and four and made a complete mess of it – probably my bad drawing. Four and five looked more promising, but the shapes soon started getting quite irregular. I enjoyed trying though. One pattern we know must be possible is the twin of the truncated icosahedron, which is a truncated dodecahedron, where we slice off each vertex of a dodecahedron. A dodecahedron is a roughly spherical 3D shape with twelve flat pentagonal faces. Since three pentagons meet at each vertex, slicing a vertex off gives a new triangular face, and doing that at all twenty vertices turns the pentagons into ten-sided faces (decagons). So that gives a tiling of three- and ten-sided shapes.

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