FANTASTIC COMBINATIONS OF JOHN CONWAYS NEW SOLITAIRE GAME LIFE
FANTASTIC COMBINATIONS OF JOHN CONWAYS NEW SOLITAIRE GAME LIFE
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DOI：
10.1038/scientificamerican1070-120
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发表时间：
1970-01-01
影响因子：
3
通讯作者：
GARDNER, M
中科院分区：
文献类型：
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作者：
GARDNER, M
人们可能会认为，古希腊几何学家对这个三角形进行了如此彻底的研究，以至于在后来的几个世纪里，对这个只有几条最短边和几个最长角的多边形没有多少重要的知识。这与事实相去甚远。当然，关于三角形的定理的数量是有限的，但是超过了某个点，它们就变得如此复杂和乏味，以至于没有人能称之为优雅的。乔治·波利亚曾将一个几何定理的优美程度定义为”与你从中看到的观点的数量成正比，与你看到它们所花费的努力成反比。“近几个世纪以来，人们发现了许多优美的三角形，它们既美丽又重要，但读者不太可能在初等平面几何课程中遇到。这个月我们将只考虑这类定理的一小部分，重点是那些提出难题的定理。正如詹姆斯·乔伊斯（James Joyce）在《芬兰人的觉醒》（Finnesans Wake）的数学部分所说，”费斯特，建造一台干燥的脚踝探针织机！“我们开始与三角形，ABC，任何形状[见插图下面]。在每一条边上都有一个等边三角形向外[左]或向内[中]构成。在这两种情况下，当三个新三角形的中心（两个高度的交点）由直线连接时，我们发现我们已经构建了第四个等边三角形。(The该定理有时是通过构造三个底角为30度的等腰三角形，然后连接它们的顶点来给出的，但由于这些顶点与等边三角形的中心重合，因此这两个定理是相同的。）如果初始三角形本身是等边的，则向内的三角形给出一个”退化”的等边三角形，即一个点。这是一个很好的定理，即使原来的三角形退化为如右图所示的直线，这个定理仍然成立。我不知道是谁首先想到的--据说是拿破仑--但近几十年来，已经印出了许多不同的校样。一个不寻常
One might suppose that the hum≠ ble triangle was so thoroughly investigated by ancient Greek geometers that not much significant knowledge of the polygon with the few≠ est sides and angles could be added in later centuries. This is far from true. The number of theorems about triangles is in≠ finite, of course, but beyond a certain point they become so complex and sterile that no one can call them elegant. George Polya once defined a geometric theorem's degree of elegance as" directly proportional to the number of ideas you see in it and inversely proportional to the effort it takes to see them." Many elegant triangle discoveries have been made in recent centuries that are both beautiful and important but that the reader is un≠ likely to have come across in elementary plane geometry courses. This month we shall consider only a minute sample of such theorems, emphasizing those that have suggested puzzle problems. c" Ferst," as James Joyce says in the mathematical section of Finnegans Wake," construct ann aquilittoral dry≠ ankle Probe loom!" We begin with a tri≠ angle, ABC, of any shape [see illustra≠ tion below]. On each side an equilateral triangle is constructed outward [left] or inward [center]. In both cases, when the centers (the intersections of two alti≠ tudes) of the three new triangles are joined by straight lines [shown in color], we find we have constructed a fourth equilateral triangle.(The theorem is sometimes given in terms of constructing three isosceles triangles with 30-degree base angles, then joining their apexes, but since these apexes coincide with the centers of equilateral triangles, the two theorems are identical.) If the initial tri≠ angle is itself equilateral, the inward triangles give a" degenerate" equilateral triangle, a point. It is a lovely theorem, one that holds even when the original triangle has degenerated into a straight line as shown at the right in the illus≠ tration. I do not know who first thought of it-it has been attributed to Napoleon-but many different proofs have been printed in recent decades. An unusual