The Fitful Flog

May 20, 2010

Temple Mathematics

Three Lozenges Kamon

About a year ago, I read a book on Japanese temple mathematics that I found in the local libraries. Well, I didn’t read it completely — there was a great deal of it I couldn’t follow. But the pictures were beautiful and what I understood, I enjoyed. During the Edo period, that is, after the Japanese had been exposed to Dutch traders, but before they’d met the Americans, there was something of a renaissance in Japanese math. Part of it was that mathematicians would make wooden tablets with strange geometry puzzles on them, sangaku, and hang them in Shinto temples throughout the land. Other visiting math-heads would attach solutions to the problems. This was once a common practice, but the modernization that followed the opening to the West led to its decline and eventual abandonment. Most of the tablets were lost, but those that survived are now museum pieces and carefully conserved. They’re beautiful to look at — some of them resemble crease patterns. (That’s where I came in.)

It’s not really clear to me why temples — you’d think schools would be a more likely place for this activity. (You remember how your high school math teacher would post puzzles on his bulletin board?) Were the Japanese scholars trying to amuse the gods with their puzzles? Were the tablets thank-offerings for a moment of mathematical clarity? The book never quite satisfied me on that point.

So, Saturday, I had this idea that I wanted to make an octahedral box. There are roughly a gazillion octahedral origami boxes, but I wanted mine a little different, with a pockets and an iris closure on one side. I thought about where I wanted things to go, drew a crease pattern and made the box — it worked well and I was satisfied with the results.
Octahedral Dibs Box Octahedral Dibs Box

I figured I’d do the folding sequence later — it’s usually easier, after the fact.

Then Daniel Kwan innocently asked me if there was an elegant way to get to the grid for the model. I said, sure, no problem, and quickly discovered it was a much more complicated problem than I had thought. The center triangle is concentric with the circle and that sounds as if it should be pretty simple, but it’s not foldable in the way we usually do triangular grids. After a couple days, I came up with a triangle I could fold that would give me the right angles, but by then Daniel and Andrew Hudson were already making discoveries of their own.

Daniel came up with a very nice folding sequence:
Oschene's Octahedral Dib's Box

And this works very well, though it might need some fleshing out for the general folding public.

Andrew noticed that the grid not only uses a 6-pointed star, but also divides the circle by ten:

Division by Ten by Andrew Hudson

And then he generalized, proved the idea and started riffing upon it:
Andrew Hudson on Flickr Andrew Hudson on Flickr

Andrew wonders “how much we’re missing by restricting ourselves to squares.” A very great deal, I should imagine.

And all of this is good stuff — all of this can be used elsewhere, in tessellations, representational works, whatever. I would encourage you the folding public, to try it out. I was, as I said, satisfied with my box, but rather better satisfied with the conversation that followed it.

Maybe the Japanese mathematicians hung their puzzles in the temples because they were there and open to the public, a place travelers would pass through and where locals would visit regularly. Maybe in Edo Japan, the temples were not just places of devotion, but also served as a marketplace for ideas. Sort of like, you know, using a photo site for the exchange of origami techniques.

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6 Responses to “Temple Mathematics”

  1. 1
    Tom Hull Says:

    Do you remember which sangaku book you looked at? I know of two — one made by a very small press in Canada, paperback with a green cover, authored by Fukagawa and Pedoe (circa 1989) and a more recent & readable book by Fukagawa and Rothman published by Princeton Univ. Press (like 2008 or something). The Princeton book is more lovely and explains the history a bit more, but the older book contains more examples, including two that are actual origami-math problems! (One of which shows Haga’s Theorem, about 100 years before Haga came up with it himself!)

    These triangle subdivisions are quite nice! I wonder if they could be considered as alterations to (or isomorphic to?) one of the methods for dividing the side of a square into 3rds. (See here: at the bottom of the page.)

  2. 2
    Andrew Hudson Says:

    Thanks for the writeup, Philip! Flickr is an excellent marketplace for ideas– my best ideas come from following up on things I see on flickr…

    @Tom Hull: Yes, in a way. Both methods seem to be applications of Robert Lang’s Crossing Diagonals algorithm, which he wrote about here: (fig. 8, page 14) It’s a very powerful and elegant tool.

  3. 3
    David Hobson Says:

    I have very much appreciated your unique origami and curved folds! When I saw this, I thought of you. I’m sure you are aware of it, but I thought I would toss it out there none-the-less.


  4. 4
    Bradford Hansen-Smith Says:

    I appreciate your effort to come up with what you have done, I suspect in the old Japanese tradition of individual isolated problem solving.

    There is an easier approach to folding the circle. Check out my website. I think you will find it interesting.

  5. 5
    oschene Says:

    Very nice. I don’t do a lot of modulars, myself.

  6. 6
    Anabelle González Sáenz Says:

    I love all of your designs, their pure and simple lines and all the math that lies behind. I’m a physicist myself and every model you blog about becames my own personal dare.

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