Weiqi Gao's Observations - mathematics tag

Zhou Bi Mathematical Manual (周髀算经)

Sun, 10 May 2009 00:59:58 GMT

My last excursion into ancient Chinese mathematical books, the post about Sea Island Mathematical Manual 172 days ago, generated some interest from some of my non-traditional readers.

Today, I would like to follow-up with a brief passage from another book: the Zhou Bi Mathematical Manual (周髀算经). It is written in the Zhou Dynasty (1046 BC–256 BC) about geometry and astronomy (Here Bi (髀) is the eight chi vertical pole that astronomers used to examine the shadow cast by the Sun). The opening dialog is between the Duke of Zhou (周公, about 1000 BC) and the astronomer Shang Gao (商高):

1. In the old time the Duke of Zhou asked Shang Gao: "I heard that your greatness is proficient with numbers. May I ask you a question about the ancient Bao Xi (包犧) who established the positions of the heaven and the degrees of the calendar. One cannot climb stairs to reach heaven, and one cannot measure the earth with chi (foot) and cun (inch). May I ask from where comes the numbers."

2. Shang Gao said: "The ways of the number, comes from the circle and the square. The circle comes from the square, and the square comes from the ju (carpenter's square). Ju comes from nine nines eighty-one. Thus fold the ju, form a gou (base) of three, gu (altitude) of four, and jing (hypotenuse) of five. Having squared on the outside, half one of the jus. Turn around and form a plate, there you get the root of three, four, and five. The two jus combined is twenty-five long, called the accumulation of ju. Thus what Yu (禹) used to rule the world, these numbers generated."

Original Chinese. 商高曰：“數之法，出於圓方。圓出於方，方出於矩。矩出於九九八十一。故折矩，以為句廣三，股脩四，徑隅五。既方之外，半其一矩。環而共盤，得成三、四、五。兩矩共長二十有五，是謂積矩。故禹之所以治天下者，此數之所生也。”

If the translation does not seem to make sense to you, it's because the original doesn't make much sense to me either, except that we know Shang Gao is talking about a right triangle with sides 3, 4, and 5; he's taking squares of the sides, and transforming the two smaller ones into the bigger one. He explained it with the use of ju, which is an instrument that is used to form right angles. How he did it was not clear at all.

Fortunately, a diagram and a rigorous proof of the Pythagorean Theorem was added to the book as a commentary by Zhao Shuang (赵爽) in 222 AD. Here is his diagram and a JavaFX applet that I wrote to make the transformations from c² to a² + b² apparent.

The source code for the applet can be found here. It is the simplest kind of JavaFX application there is. I threw six triangles and three squares on the screen, and animated their rotation angles, fill colors, and opacity. Take a look at syntax of the animation:

var t = Timeline {
    keyFrames: [ at(0.0s) { col=> Color.WHITE},
        at(1.0s) { a2 => 0.0; a3 => 0.0; a4 => 0.0},
        at(2.0s) { a2 => 90.0; a3 => 90.0; a4 => 90.0},
        at(3.0s) { a3 => 90.0; a4 => 90.0},
        at(4.0s) { a3 => 180.0; a4 => 180.0},
        at(5.0s) { a4 => 180.0},
        at(6.0s) { a4 => 270.0},
        at(7.0s) { col=> Color.WHITE},
        at(9.0s) { col=> Color.LIGHTGREEN},
        at(10.0s) { b=> 0.0},
        at(11.0s) { b=> 90.0},
        at(12.0s) { c=> 0.0},
        at(13.0s) { c=> -90.0},
        at(14.0s) { o=> 0.0;},
        KeyFrame { time: 14.0s
            action: function() {
                var q = stage.scene.content[0..1];
                delete stage.scene.content[0..1];
                insert q into stage.scene.content;
            }
        },
        at(15.0s) { o=> 0.7 },
        at(17.0s) { o=> 0.7 }]};

t.play();

Now that I've updated my workstation to Ubuntu 9.04 (for people who are puzzled by Ubuntu version numbers, like me six months ago, 9.04 is short for 2009-04; if the trend continues, the next version will be 9.10) which comes with the Sun JDK 1.6.0_13, including 64-bit new plugin and web start, I can play with JavaFX applets more easily. This post marks the official return of the applet (at least for me). The applet runs fine on in my browser. However, if you encounter any problem with the applet, please let me know so that I can try to fix the problems.

Timothy Gowers: The Princeton Companion to Mathematics

Thu, 28 Aug 2008 12:48:28 GMT

Timothy Gowers: A quick post to give an update on the Princeton Companion to Mathematics. It’s been off my hands for a few weeks now. If all goes well it will be printed by the middle of September and should appear in bookshops about a month later. This is what the cover will be like. (If I work out how to do it, I’ll get a reduced-size picture of the cover to appear as part of this post rather than as a link.) If you go to this page on the PUP website then you will find a podcast interview that I gave, which contains information that does not appear on this blog.

Timothy Gowers is a Fields Medal winning British mathematician. This volume seems to be this generation of mathematician's attempt at explaining what they do, just as previous generations of mathematicians have done, for example, with Mathematics, Its Content, Methods, and Meaning by the Russian school and the Bourbaki series by the French school. At 1000 pages and nearly 200 entires, I imagine it would be more like an encyclopedia than anything else. However, a glance at the contributors list will convince any mathematician that this volume is a worthy addition to the bookshelf.

Reclusive Russian Turns Down Fields Medal

Wed, 23 Aug 2006 13:09:46 GMT

Lucy Sherriff, writing for the Register: A Russian mathematician has turned down one of the discipline's most prestigious awards because he doesn't want to involve himself in self promotion. He was due to have been presented with the Fields Medal by King Juan Carlos of Spain on Tuesday this week.

Grigory Perelman published an outline of a proof for the Poincare conjecture back in 2003 as part of his work on the Geometrisation Conjecture, proposed by American mathematician William Thurston in the 1970s. This seeks to characterise all three dimensional surfaces.

So far, other researchers working to check and flesh out his idea have not found any flaws. Perelman himself has not spoken publicly about his work, saying that before the checking is completed it would be premature to do so.

John Ball, retiring president of the International Mathematical Union, told the BBC that he had gone to visit the reclusive mathematician in St Petersburg to discuss his reasons for declining the award.

He said that he wouldn't disclose Perelman's statements beyond saying that Perelman said he felt isolated from the mathematical community, and therefore had no wish to appear to be one of its leaders.

"He has a different psychological makeup that makes him see life differently," he added.

I mentioned Poincaré Conjecture on this blog once before when Cao Huaidong and Zhu Xiping announced their completion of Perelman's proof. For the truly interested, their article is available online now.