the segments of matching slope. The remaining segments form the paths in darker colors in Figure 5 (left). Now if we place a horizontal mirror at the tip of each V, the reflections in lighter colors show clearly that the shoe-store forms two sides of a triangle and the European forms the third (Figure 5 (right)). So again by the triangle inequality, the shoe-store is longer than the 向的鏡，以淺色表示的反射影像清楚顯示鞋店綁法 的線段成了三角形的兩條邊，而歐洲綁法的線段則構 成第三條邊（圖五（右））。所以再一次根據三角不等 式，鞋店綁法比歐洲綁法長；而三者之中交叉綁法必 為最短。事實上，交叉綁法亦已被證實是所有可行交 錯綁法中最短的一款 [2]。 至於要找出所有綁法中最短的一款，記得穿鞋帶 的目的是要把鞋的左右兩半拉在一起。打交叉是用 最少鞋帶就能把鞋水平拉緊，而又能垂直地向下一 行孔眼發展的方法。雖然包括加拿大五線譜綁法在 內的一些綁法只用水平和垂直線段，但是這些方法 都耗用極長的垂直線段，因此最短的綁法需要包含 交叉。也有綁法包含由多個重疊交叉組合而成的複 雜圖案，但最短的綁法明顯地只需在兩行孔眼之間 打出簡單交叉，所以最短的綁法只能包含簡單交叉。 然而根據三角不等式，兩條垂直線段長度之和總是 短於任何交叉（見圖六），所以我們要儘可能增加垂 直線段的數目，同時亦要減少簡單交叉。 Figure 5 (Left) “Unfolded” lacing paths of the European (blue) and the shoe-store (green) lacings and their reflections (lighter colors) after subtracting common segments [3]. (Right) The triangle to determine the length of both lacing methods. 圖五 （左）減去共同線段後歐洲綁法（藍色）和鞋店綁法（綠 色）經「拆解」後的路徑及其反射影像（以淺色表示）[3]；（右） 判斷兩者長度需考慮的三角形。 European, and of these three methods the crisscross is always the shortest. In fact, it can be shown that the crisscross is the shortest among all possible alternating lacings [2]. To find the shortest lacing in general, consider that we must keep the two sides of the shoe pulled together. The shortest way to do this while also advancing vertically is to use crossings; notice that lacings with only horizontal and vertical parts, such as the Canadian straitlacing, use excessively long vertical segments. Hence it is necessary for the shortest lacing to contain crossings. There are many possible complex crossings with overlapping crosses, but the shortest is clearly a simple cross between the pairs of eyelets in two adjacent rows. So the shortest lacing must contain only simple crosses. But by the triangle
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