21 The Shortest Lacing First let’s test the dense lacings. Assuming the laces are pulled tight, we could use Pythagoras’ theorem to find formulas for their lengths in terms of h, g and n [4]; interested readers may try to derive them by referring to the diagrams: 最短的綁法 首先讓我們探討緊密綁法。假設鞋帶被拉緊， 沒有因鬆弛而耗用多餘長度，我們就能用畢氏定 理，以h、g和 n表示各種綁法所需的鞋帶長度 [4]，有興趣的讀者可以試試看著圖樣推導公式： Crisscross 交叉綁法 : European 歐洲綁法 : Shoe-store 鞋店綁法 : But the alternating property gives us a nice way to compare their lengths without doing any calculation. Say one end of the lacing lies on column A, at eyelet Ak. Imagine setting a mirror on column B so column A is reflected; then we can draw the reflected column A as a new column C with the eyelet Ak reflected onto Ck. Continue “unfolding” the lacing in this way [5]. Now instead of being laced between two columns of eyelets, our lacing moves horizontally across Figure 4 column by column, while reflecting the real length of each segment throughout the lacing. By considering the orange triangle in Figure 4, we know that the crisscross lacing must be shorter than the European lacing, because the European lacing always runs along two sides 然而它們線段交錯的性質使我們不需進行計 算就能比較三者長度，譬如鞋帶線段的一端位於 A行的Ak 孔眼，假設我們在B 行設置一塊鏡把 A 行反射，鏡中反射的A行鏡像現在就可以被畫在 圖表上，並被命名為新設的 C 行，而 Ak 的鏡像則 被標示為Ck。透過繼續重覆這個步驟「解開」鞋 帶 [5]，我們的鞋帶就不再是交錯在兩行之間，而 變成逐行向橫發展的圖四，當中每個線段都反映 其真實長度。 透過考慮圖四中的橙色三角形，我們知道以交 叉綁法穿的鞋帶一定短於歐洲綁法，因為歐洲綁法 的鞋帶反覆地在圖表裡構成三角形的兩條邊，而交 叉綁法的鞋帶則成為餘下的第三條邊，因此根據三 Figure 4 “Unfolded” lacing paths of the three alternating lacings: crisscross (red), European (blue), and shoe-store (green) [4]. The triangle shaded in orange can be used to compare the length of the crisscross and the European lacings. 圖四 三種交錯綁法經「拆解」後的路徑圖：交叉綁法（紅 色）、歐洲綁法（藍色）和鞋店綁法（綠色）[4]。橙色三角 形能被用於比較交叉綁法和歐洲綁法的長度。 角不等式（三角形任何兩條邊的長度總和必定比餘 下一條長），歐洲綁法比交叉綁法長。至於要比較歐 洲綁法和鞋店綁法，我們可以暫時忽略水平線段（兩 者同樣有n – 1個水平線段）和相同斜度的線段，把 它們移除後會得到圖五（左）以深色線表示的路徑。 現在如果我們在兩個 V 字的尖端放一塊水平方 of a triangle while the crisscross runs along the third side. By the triangle inequality (the sum of any two sides of a triangle is longer than the third side) the European is longer than the crisscross. To compare the European and the shoe-store method we eliminate the horizontal segments from both lacings (each of them has n-1 horizontal segments) and
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