Chapter 7.5, Problem 35MS

Making dough. Many delicious French desserts, such as Napoleons, are made of numerous thin layers of pastry. Once upon a time a French chef began, as usual, to make a masterpiece with a segment of dough one unit long. This time, however, he noticed that three grains of colored sugar-one red, one white, and one blue-were embedded in the pastry. He stretched the dough to twice its length and then folded it in half to produce a double layer of thinner pastry still one unit long. He again stretched the dough to double its length and, folded it to produce four layers again one unit long. He repeated, the process: stretch to twice its length, fold to produce eight layers. He stretched and folded again and again. As our chef stretched and folded, he became fascinated with the movement of those three grains of sugar. In fact, he noticed that the red grain always returned to its original location. The white grain began somewhere, went to a different location after a stretch and fold, and then returned to its original location after one more stretch and fold. The blue grain rotated among three different places. The chef was so fascinated with the infinitely recurring itineraries of these grains of sugar that he stretched and folded the dough to such an extreme thinness that his dessert creation lived up to the high expectations for light, puffy French desserts and became the Napoleon of pastries.