Soccer balls and sequins

By Bobby Neal Winters

I was sitting by the soccer field stupefied.  

This is not an unusual condition for me, that is to say, neither stupefaction nor watching a soccer game.  My daughter plays soccer so I am frequently at the field watching her.  And there are also frequent intervals on any particular occasion when I am astonished or shocked.  But I do find myself sitting by a soccer field in such a state more frequently than random chance would merit.  This is because I have no understanding of the game.

But on this occasion, my stupefaction had nothing, or at least very little, to do with the game at hand.  I say very little because my confusion with aspects of the game had led me to withdraw into my own mind while staring at an object in the immediate environment.  As I was by a soccer field, the object at which I was staring turned out to be a soccer ball.  I was mesmerized by it.  I was noting how its surface was covered with hexagons and pentagons; twelve pentagons and twenty hexagons.  I am not sure where my musings would’ve taken me if my reverie hadn’t been interrupted by an interloper.

The interloper was about two and a half feet tall.  Determining his age is problematic.  I figured he had to be at least two years old to gather that much dried mucous in his nostrils.  I decided he was somewhere between too old for that loaded diaper he was wearing and too young for that sharp pair of scissors he was carrying.  

It was the sight of the scissors in the young child’s hand that really woke me up.  While I usually keep my distance from the children of strangers, I make an exception to this when I believe the child is unattended and I believe the child is in danger.  I was about to intervene when the child’s mother arrived on the scene.

“There you are, you little scamp,” she said. “Did you find yourself a soccer ball?”

He is say anything, but he rubbed the soccer ball and giggled.

“Hehehehe,” he laughed.  It came out with such rapidity it sounded like a little laughing machine gun.

“So are you going to play with it?” she asked.

“Hehehehe,” he replied.

I was on the verge of thinking to myself how cute this was when the scissors plunged into the soccer ball. My eyes glanced up toward his mother to see her reaction.

“Oh my goodness,” she said.  “What a strong little boy!”

I looked side to side to note the reactions of the folks around me, and they were all looking at the game, of all things.

“What do we do now?” she asked him in a baby-talk tone.  His reaction was immediate as he began cutting the ball into pieces.

By this time, she’d noticed that I was watching, and I’d noticed that she’d noticed that I was watching, so, in an attempt to keep things from getting awkward, I thought I might start a conversation.

“So your son,” and I was sure of the son part because the loaded diaper was hanging a bit low, “your son likes to cut up soccer balls?”

“Oh, yeah,” she said. “It keeps him busy while his big brother is on the soccer field playing.”

I glanced out an the soccer field, but I didn’t see any of the twelve-year-olds running around in diapers, though there was one with a tattoo.

“I wouldn’t want him to get run over,” she continued. “Plus, it helps me with my hobby.”

“Hobby?” I echoed stupidly.

“Yes,” she said. “I glue them back together and turn it into a craft.  I’ve got one hanging from my review mirror. See?”

As she spoke she pointed toward the street where I saw an ancient Volkswagen van with what was apparently won of her re-constituted soccer balls hanging from the rear-view.

About that time, I heard laughter from the boy again.

“Hehehehe,” he giggled. “Done.”

I looked in his direction again, and there in front of him were a pile of pieces of what had been a functioning soccer ball.  He had--and in short order I must say--cut the ball into a small number of irregular polygons.

“So how do you do your craft?” I asked.

She seemed pleased by my inquiry.

“Oh,” she said. “It’s real easy.  I use a hot glue gun and glue the pieces back together along one of the cuts he’s made.  Then at places where several of those cuts come together I put a big shiny sequin to cover the hole.”

She paused for an instant as if getting an idea. Then she smiled.

“I got an idea,” she said. “I’ll just run over to the van and get one to show to you. I’ve got lots of them.”

She left her son, who was now digging in the dirt with the scissors, went to the van, and fetched back the reconstructed soccer ball.

It was just as she said.  The pieces of the ball were joined together by hot-glue (now cold of course) and there was a big sequin where the ends of the seems came together.  Absently, I started counting: nine sequins, fifteen seams, and eight pieces.  I must have been doing this out loud because the woman commented on it.

“What did you say, mister?” she asked.

“Oh, I was just counting the sequins, seams, and pieces,” I said.  “I do this because of Euler’s formula.”

“Oiler?” she said.

“That’s right, Leonard Euler, a Swiss mathematician discovered that it you slice up a sphere in the way your son did that the number of the vertices minus the number of edges plus the number of faces is always equal to two.  That is to say, sequins minus seams plus pieces is two.  This works as long as the pieces are topological disks.”

“Topo-whatical whats?”

“Topological disks,” I said.  “Though it took many years to create the mathematical machinery to put it into those terms.  The way we say it now is that the Euler characteristic of the sphere is equal to two.”

By this time, the woman and her son were both looking directly at me, looking somewhat stupefied themselves. If I let that look bother me, I would have to give up my career teaching math.  I continued undaunted.

“Euler used this formula to prove there are only five Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.”

“Eye-eye-eye-caws a head run,” the boy said.

“That’s right,” I said. “And it’s funny you should mention that one in particular, because if you snub an icosahedron, that is do some careful clipping near its vertices, you obtain a soccer ball, which is a very pretty example of what is known as an Archimedean solid, even without the hot glue and sequins.”

By this time, she had picked up her son, taken the scissors from him, and was nonchalantly moving away from me.

And that had sort of been part of my plan.

When the game ended, I took my family home and took out the pad I keep by my recliner for moments of mathematical thought.

In a Platonic solid, we assume that every face has the same number of edges; call this number m.  We also assume that the same number of edges comes into every vertex; call this number n.  In addition, every edge has two vertices and every edge is on two faces.  If we let V be the number of vertices, E the number of edges, and F the number of faces, we have the following equations:

V-E+F=2

m F=2E=nV

So

(2/n)E-E+(2/m)E=2

This can be rearranged to 1/n+1/m=1/2+1/E, and, since E is positive, this yields the inequality 1/n+1/m>1/2. Now n and m must be at least 3, so we can by work out by elimination that (n,m) can only be equal to (3,3), (3,4), (3, 5), (4,3), and (5,3).

From this information, it can be shown there are a total of five Platonic solids: the tetrahedron having four faces, the cube having six, the octahedron having eight, the dodecahedron having twelve, and the icosahedron having twenty. Each of these obeys the formula vertices minus edges plus faces equal two.

The soccer ball, with its pattern of twelve pentagons and twenty hexagons, a total of thirty-two faces, has 60 vertices and 90 edges, and also follows Euler’s formula.