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Sepak takraw is a sport native to the Malay-Thai Peninsula. A few days ago, a friend of mine taught me how to make a sepak takraw ball. The ball is related mathematically to a $32$-face semi-regular polyhedron, known as a truncated icosahedron.

We can construct a ball with the same form as the sepak takraw ball using $6$ simple packing tapes. The ball has $12$ pentagonal holes and shows a weaving pattern with $20$ intersections.

(This page (PDF) shows some helpful figures, especially figure $4,5,6$. This page shows how to make a 'sepak takraw ball' ornament from a plastic bottle.)

Then, I got interested in the following question.

Question : If we use $6$ packing tapes with distinct colors, how many balls with distinct patterns can we make?

Though I've been thinking about this question, I'm facing difficulty. I would like to know not only an answer but also how to find and calculate an answer with some explanation. Can anyone help?

mathlove
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1 Answers1

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Use color ${\bf 0}$ for the equator strip. Then around the north pole $N$ you see a hollow pentagon surrounded by five pieces of strips. Starting with color ${\bf 1}$ and going counterclockwise you have $4$ possibilities for the color of the adjacent strip, then $3$, and finally $2$, making a total of $24$ different circular arrangements of the colors ${\bf 1}$ to ${\bf 5}$. After having made these choices, say ${\bf 12345}$, and looking at the south pole you will find out that you see the same pattern, but clockwise arranged. This means that in fact so far you have just $12$ different balls, since ${\bf 12345}$ and ${\bf 15432}$ give rise to the same ball. But there is one last point: At the very beginning we can choose whether near $N$ the strip ${\bf 1}$ should go "over" or "under" the next left strip, and this then decides the complete over-under-pattern of the ball. Altogether we can conclude that there are $24$ different balls.

Christian Blatter
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