The Pythagorean Theorem: What Is It About?
The Pythagorean Theorem: What Is It About?
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毕达哥拉斯定理：它是关于什么的？
DOI：
10.1080/00029890.2006.11920304
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
A. Givental
中科院分区：
文献类型：
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作者：
A. Givental
虽然已有25个世纪的历史，但毕达哥拉斯定理似乎生机勃勃，无处不在。作为笛卡尔坐标法中距离公式的关键，该定理隐含地存在于所有涉及空间关系或三角的科学模型和工程计算中。作为点积运算的无形伴侣，它是数学物理和连续介质力学方程中固有的，无论是那些以拉普拉斯、纳维-斯托克斯、麦克斯韦还是杨-米尔斯命名的方程。在线性代数和黎曼几何的公理化构造的伪装下，它为量子物理学和爱因斯坦的万有引力赋予了活力，否则它们是如此难以调和。当然，统计学家或实验学家很少有一天没有欧几里得的噩梦？将任意三个或多个点拟合成一条直线，这是由高斯最小二乘完成的，因此也让人想起毕达哥拉斯。因此，哲学家的裤子被骄傲地展示在中学教科书中，这是理所当然的，也许是唯一在公众中流传的“有证据”的科学真理。最流行的一种（如图1所示）确实非常令人信服。然而，它把整个问题描绘成一个剪切和粘贴的难题，给我们留下了一种不相称的感觉：大自然最基本的事实之一是由于巧妙的平铺技巧。绝大多数其他证明在性质上是相似的。
Although twenty-five centuries old, the Pythagorean theorem appears vigorous and ubiquitous. A key to the distance formula in Descartes's method of coordinates, the theorem is implicitly present in all scientific models and engineering computations involving spatial relationships or trigonometry. An invisible companion to the dot product operation, it is inherent in equations of mathematical physics and continuum mechanics, be those named after Laplace, Navier-Stokes, Maxwell, or Yang-Mills. Disguised by axiomatic constructions of linear algebra and Riemannian geometry, it animates both quantum physics and Einstein's gravitation, which are otherwise so hard to reconcile. And of course, a rare day for a statistician or experimenter goes by with out Euclid's nightmare?fitting any three or more points into a straight line, which is accomplished by Gaussian least squares and hence is also reminiscent of Pythagoras. Quite deservedly, therefore, the philosopher's pants are proudly displayed in middle-school textbooks to represent, perhaps, the only scientific truth circulating among the general public "with proof." The most popular one (shown in Figure 1) is very convincing indeed. Yet it pictures the whole issue as a cut-and-paste puzzle and leaves us with a feeling of disproportion: one of the most fundamental facts of nature is due to an ingenious tiling trick. The vast majority of other proofs are similar in nature.