Alex's Adventures in Numberland - Alex Bellos [44]
Haga decided to be different. What if he folded a corner on to the midpoint of a side? Ker-azee! He did this for the first time in 1978. This simple fold had the effect of opening the curtains on to a sublime new world. Haga had created three right-angled triangles. Yet these were not any old right-angled triangles. All three were Eygptian, the most histct eggsand iconic triangle of them all.
Haga’s theorem: A, B and C are Egyptian.
Feeling the thrill of discovery, but having no one to share it with, he sent a letter about the fold to Professor Koji Fushimi, a theoretical physicist known to have an interest in origami. ‘I never got a reply,’ said Haga, ‘but then all of a sudden he wrote an article in a magazine called Mathematics Seminar referring to Haga’s Theorem. That was the substitute for his reply.’ Since then Haga has given his name to two other origami ‘theorems’, although he says he has another 50. He tells me this not as evidence of arrogance but as a measure of how the area is so rich and untapped.
In Haga’s Theorem a corner is folded on to the midpoint of a side. Haga wondered if anything interesting might be revealed if he folded a corner on to a random point on the side. Deciding to demonstrate this to me, he took a blue piece of square origami paper and marked an arbitrary point on one of the sides with a red pen. He folded one of the opposite corners on to this mark, leaving a crease, and then unfolded it. He then folded the other opposite corner on to the mark to make a second crease, leaving the square now with two separate intersecting lines.
Haga’s other theorem.
Haga showed me that the intersection of the two folds was always on the middle line of the paper, and that the distance from the arbitrary point to the intersection was always equal to the distance from the intersection to the opposite corners. I found Haga’s folds totally mesmerizing. The point had been chosen randomly, and was off-centre. Yet the process of folding behaved like a self-correcting mechanism.
Haga wanted to show me a final pattern. The name he chose for this discovery sounded like a haiku: an arbitrarily made ‘mother line’ bears eleven wonder babies.
Step 1: Make an arbitrary fold in a square piece of paper.
Step 2: Fold each edge along that fold separately, always unfolding to leave a crease, as demonstrated below in A to E.
Mother line showing seven of her eleven wonder babies.
Again, this is very simple to carry out, and reveals a beautiful geometric order. Every intersection is on the primary creases, as F shows. (The diagram shows the seven intersections that are inside the original square; the other four are on extensions of the folds.) The first fold was random, yet all the folds meet with perfect concision and regularity on diagonals and midlines.
It struck me that if any man can be said to embody the soul of Pythagoras in the modern world, it is surely Kazuo Haga. Both men share a passion for mathematical discovery based on a wonder for the simple harmonies of geometry. The experience seemed to have touched Haga spiritually, the same way it did Pythagoras two millennia ago. ‘Most of the Japanese are trying to create new shapes in mi,’ said Haga. ‘My aim was to escape from the idea that you have to create something physical, and instead discover mathematical phenomena. That is why I find it so interesting. You find that in the very, very simple world you can still discover fascinating things.’
CHAPTER THREE
Something about Nothing
Every year the Indian seaside town of Puri fills with a million pilgrims. They come for one of the most spectacular festivals in the Hindu calendar – the Rath Yatra, in which three chariots the size of carnival floats are pulled through the town. When I visited, the streets were crowded with cymbal-crashing, mantra-chanting devotees, barefoot holy men with long beards and middle-class Indian tourists with fashionable