CROSSING THE DIVIDE                 In 1936, the Dutch artist M. C. Escher visited the Alhambra, the fourteenth-century Moorish palace in southern Spain, and experienced a revelation. Until that time, Escher, who lived from 1898 to 1972, had directed his gaze toward the natural world. His work had consisted of portraits, plant and figure studies, and renderings of Italian hill towns and the Mediterranean coastline. An extraordinary craftsman who worked primarily in woodcutting and lithography, Escher had painstakingly studied natural form and explored techniques for transforming three-dimensional objects into two-dimensional graphic designs. He had not yet devised the tile patterns, geometric solids, impossible structures, and optical illusions for which he would become famous.   Escher's trip to the Alhambra gave new direction to his work. The walls and floors of the palace are decorated with colorful and intricately carved tessellations, patterns of tiles capable of covering an entire surface without leaving space between them. Escher filled sketchbook after sketchbook with pencil drawings reproducing the patterns and analyzing their geometry. Excited by his discovery, he wrote, years later:    What a pity it was that Islam forbade the making of "images."  In their  tessellations they restricted themselves to figures with abstracted  geometrical shapes.  So far as I know, no single Moorish artist ever  made so bold as to use  concrete recognizable figures such as birds, fish, reptiles, and human  beings as elements of their tessellations.  Then I find this         restriction all the more unacceptable because... it is precisely this         crossing of the divide between abstract and concrete representations,         between "mute" and "speaking" figures, which leads to the heart of         what fascinates me above all in the regular division of the plane.   Crossing the divide is a spiritual act. At its most abstract, folding an origami animal replicates both the growth of the animal from fertilized egg to adult (the early, symmetrical folds paralleling the highly mechanical process of mitosis) and the origin of life itself. In the paper, as in the primordial cosmic soup, chaos yields to order, formlessness to form, darkness to light. When Escher reflected on the origin of his tessellations, the neutral gray background from which the black-and white figures emerge, he felt transcendent:    I consider the indeterminate, misty grey plane as a means of  expressing static peace, of rendering the absence of time and the  absence of dimension that preceded life and that will follow it; as a  formless element into which all contrasts will dissolve again, "after  death."   Let us begin, then, like Escher, with the formless element into which all contrasts dissolve - the empty square.   In the beginning was the square.   To the paperfolder, the square is the origin of all form. Geometric shapes, animals, objects, and human beings arise from the square and then, unfolded, dissolve back into it. The empty square is the alpha, the genesis, and the prime mover of origami. In Taoist philosophy, the square is the First Form, the undifferentiated void from which the opposing Yin and Yang forces arise. Where others see only the void - dull, blank, meaning less - the folder sees a world already overflowing with possibilities. His mission is to discover those possibilities and bring the square to life.   Because paper is the folder's only medium - his canvas, paint, and brush - he must get to know it intimately. What is its color? Its texture? If you fold it in half and press it flat, will it hold the crease or spring open? How far will it stretch before it rips? Rub it back and forth between your fingers. How does it feel ?   There are many things you can do with an ordinary sheet of paper. You can crumple it and throw it away. You can roll it against the edge of a ruler and make it curl. You can write on it, and it becomes a letter. Then, if you put it in another piece of paper (an envelope) and fasten a smaller piece of paper onto that one (a stamp), it can be delivered to a friend. "Dear fellow folder... "   But there are some things you can do only with a square sheet of paper. The square has geometric properties that can be exploited for folding. To begin, it is regular. It has four corners, all of them measuring the same angle, 90 degrees. It has four sides, all of them the same length. And it has a vast, undifferentiated middle - as yet, unpromising. The corner of the square takes up 90 degrees of paper, the edge 180 degrees, and the middle 360 degrees.   Our tool is geometry; our purpose, to create a representation of an animal, an object, or a human being. To do so, we must transform the square into a new shape and manufacture a separate flap from the corners, edges, and middle for each feature of the figure we're trying to create: head, neck, arms, legs, wings, horns, antennae, tail. As these appendages become long and thin, the body of the animal becomes concentrated and thick, and paper that serves no function must be tucked out of sight.   For this reason, the finished model must be efficient and compact. When angles and edges line up, there is little excess paper to hide from view. The regularity and symmetry of the square mean that when you fold it, the angles and edges often align. The square is the only shape that is both a rectangle (a form with four identical angles) and a rhombus (a form with four identical sides).   Both rectangles and rhombuses exhibit a kind of symmetry called left-right symmetry or mirror-symmetry. Rectangles have mirror-symmetry along their orthogonals: If you fold the adjacent corners together, the sides will meet. Rhombuses have mirror-symmetry along their diagonals: If you fold the opposite corners together, the sides will meet. A square has both properties, which means that there are many ways of folding it so that both the angles and the edges line up.   Now that we've covered the geometry of the square, we're ready to start folding.     Information from the book "Origami from Angelfish to Zen" by Peter Engel  
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