# BO 127 – Robert Lang on ‘The Corner Sacrifice’

It was called “the chess game of the century.” In 1956, at the Lessing J. Rosenwald Trophy Tournament, Donald Byrne, one of the finest players in America, faced his opponent, a thirteen-year-old boy. As the game progressed, Byrne sensed a weakness. Carefully, he isolated the boy’s queen – the most powerful piece on the board – and, springing his trap, closed in and captured it. The boy did not even blink. He made his move; Byrne countered; and as the game progressed, he realized that a trap had been sprung, but he, Byrne was in it. With the superior position his game had bought him, the young Bobby Fisher drove deep into Byrne’s territory to win. Fischer went on to become the world champion of chess while Byrne remained one of the better American players, but the queen sacrifice” became one of the classic scenarios of chess.

So what has chess to do with origami, besides an unusual fascination with squares? They both have fixed rules: bishops move diagonally; no cuts allowed. There are also rules of thumb. “Thou shalt not tamper with king-side pawns;” “Thou shalt protect thy queen;” “Thou shalt make points of thy square’s corners.” Yes, in origami, we are taught to conserve our corners (and to a lesser extent, edges) almost as much as a chess player protects his king and queen. The reason is clear. To make most subjects, we need several points for appendages. The difference between using a corner or edge to make a point, and using the interior of the paper, can make a four fold difference in the amount of paper required and the thickness of the point. For that reason, the notion quickly becomes ingrained in every inventor that for efficiency’s sake, all four corners must eventually wind up as major features of the model.

Corners are precious, and so we must ration them out as heads, tails, arms, legs and wings. Alas, a square possesses but four corners. That allows us to have a head and two front legs, but only one back leg. The annals of the three- legged animals are richer for this restriction. For all its pleasant symmetry, the square is a pretty lousy shape from which to make an animal.

Consider the symmetry of a lizard, in figure 1. Besides head and tail, it has four appendages, two fore and two aft. Now, consider the square in figure 2. It has two “appendages,” centered along its length. It is more suited to a two-legged lizard like that in figure 3.How sad it is, that such a creature doesn’t exist. It if did, it would be a more popular subject than the elephant.

Obviously, to make a four-legged animal with the same amount of “stuff” fore and aft, we need a hexagon. Let us begin with the requirement of six points. How can we get six equal points from a square in the most efficient manner? Mathematically, the problem boils down to putting six points on the edges of a square as far apart as possible. It is not hard to find the solution, which is shown in figure 4. (The diagonal lines are angle bisectors.) The interesting thing is that two of the corners – the shaded regions – go completely unused; yet this is the most efficient solution for getting six equal points from the edge of a square. The corners might as well be cut off.

Herein lies the beauty of the sacrificial corner; by not using the corner, we get longer points (and thus, a more efficient structure) than if we had required four of the points to come from corners. Now, the conceptual leap from the crease pattern of figure 4 to a finished model is large, but the efficiency of the sacrificial corner can be illustrated by a fold from the literature.

Montroll’s Dog Base (Animal Origami for the Enthusiast, Dover, 1985) is shown with its crease pattern in figure 57 Again, I have shaded unused regions of the paper. The useful area is unmistakably hexagonal, albeit with some distortion, which shifts extra paper to the head. Like the hexagon of figure 3, the Dog Base is a maximally efficient structure for the number and proportions of its points. Therefore, the loss of the two corners was inevitable and could be predicted even without knowing exactly how the Dog Base goes together. The corners have been sacrificed, and while the result is not nearly as unexpected as Fisher’s it is at least as effective.

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