Science Focus (issue 26)

現在同一行上下孔眼之間的高度h亦可以被想作{上 下孔眼間高度}/{左右孔眼間隙}之比。 已有理論證明最穩妥的綁法為最常見的交叉綁法 和鞋店綁法 [6]。為了決定在某一情況下哪種才是較 穩妥的綁法,我們需考慮孔眼數目n 和上下孔眼間的 相對高度h。 在n > 2的情況下,設C(n, h)為交叉綁法的滑輪 和,S(n, h)為鞋店綁法的滑輪和。每個孔眼數目n都 有一個對應的hn > 0 使C(n, hn) = S(n, hn),而: • 當h < hn,交叉綁法最為穩妥; • 當h = hn,兩種綁法同樣穩妥; • 當h > hn,鞋店綁法最為穩妥。 換言之,當孔眼上下高度比較接近,最穩妥的會是 交叉綁法;上下高度比較遠時,最穩妥的則會是鞋店 綁法。 對於數值較小的 n,hn 的近似值可見於表一 [6]。 你可以自行量度鞋上高度間隙之比找出相對高度 h,再透過比對孔眼數目n,在表上找出 hn 的相應值。 筆者寫這篇文章時穿著以交叉綁法穿的鞋,鞋子有4 對孔眼,而 h 值約為 0.4,所以我在穿鞋帶時顯然做了 正確的決定。當然,現在鞋店裡售賣的鞋都用上h值 非常接近hn 的設計,所以無論你選擇哪款綁法都會 非常穩妥 [6]。 lacing is called its pulley sum, and the strongest lacing has the largest possible pulley sum. Before we go further, let’s set the gap g between columns to 1 unit. This will simplify any calculations because now h, the height between successive eyelets in the same column, can also be thought of as the ratio of {height between eyelets}/{gap between columns}. There is a nice theorem that the strongest possible lacings are also the two most commonly used: crisscross and shoe-store [6]. To determine which of the two lacings is stronger in a given case, we need to consider the number of eyelets n, and the relative height between eyelets h. For n > 2, let C(n, h) be the pulley sum of the crisscross lacing and S(n, h) be that of the shoe-store lacing. Then for a given number n of eyelets, there exists exactly one hn > 0 such that C(n, hn) = S(n, hn); and furthermore: • when h < hn, the crisscross lacing is strongest; • when h = hn, both lacings are strongest; • when h > hn, the shoe-store lacing is strongest. In other words, the strongest lacing is the crisscross when the eyelets are close together in a column, and the shoe-store when they are farther apart. The approximate values of hn for small n are listed in the following table [6]. n 3 4 5 6 7 8 9 10 hn 0.9029 0.7412 0.6450 0.5794 0.5309 0.4931 0.4625 0.4372 Table 1 The approximate values of hn for small n. 表一 n 在數值較小的情況下 hn 的近似值