Science Focus (issue 26)

23 在顧及不可連續綁出垂直線段的前提下,我們 可以在交叉綁法中儘可能加入垂直線段代替部分 交叉,而得出的綁法不就是領結嗎?因為只有領結 綁法符合這些描述,所以它必定是最短的綁法 [6]。 Figure 6 Comparison of the length of a simple cross (blue) and two vertical segments (green). 圖六 簡單交叉(藍色)和兩條垂直線段(綠色)之間的長 度比較。 inequality, the sum of two vertical segments is shorter in length than any crossing (as visualized in Figure 6), so we want to maximize the number of vertical segments and minimize the number of simple crosses. This can be obtained from the crisscross by adding in the maximum number of vertical lines to replace some crosses without making consecutive vertical lines, which exactly describes the bowtie lacing! Since only the bowtie has these exact characteristics, it must be the shortest possible lacing [6]. The Strongest Lacing Now, what is the strongest lacing? When you pull on your shoelaces to tighten them, the lacing at each eyelet acts like a pulley, and the question can be restated in terms of finding the strongest pulleys. When tied, the tension along a shoelace is some constant force T. The important thing to measure is the horizontal tension Th, the direction in which the two sides of the shoe are being pulled together by the lacing. For horizontal segments, Th = T; for vertical segments, Th = 0; and for diagonal segments, Th is the horizontal component of T which can be calculated by trigonometry. Then the total sum of Th across the eyelets of a 最穩妥的綁法 那麼,哪種才是最穩妥的綁法?當你索緊鞋帶 時,孔眼和鞋帶的運作原理就像滑輪(pulley), 因此問題可以改為如何打造最強的滑輪組。穿好 鞋帶後,其張力為恆定的 T;而我們關注的是水平 張力Th,因為水平是把鞋的左右兩邊拉在一起的 方向。在水平線段中,Th = T;在垂直線段中,Th = 0;在斜向線段中,Th 是 T 的水平分量(horizontal component),數值可以藉簡單的三角函數計算。 一個綁法中每個孔眼間Th 的總和被稱為滑輪和 (pulley sum),而最穩妥的綁法應擁有最大的 滑輪和。 再進一步討論之前,讓我們設兩行孔眼之間的 間隙g為1個單位。這能簡化之後的計算,因為