Twists, Tilings, and Tessellations

Mathematical Methods for Geometric Origami

Robert J. Lang

• Publisher: A K Peters/CRC Press
• Year: 2017
• ISBN:
  9781482262414
• Edition: 1
• Topic: Matemáticas
• Subtopic: Matemáticas general

Twists, Tilings, and Tessellation describes the underlying principles and mathematics of the broad and exciting field of abstract and mathematical origami, most notably the field of origami tessellations. It contains folding instructions, underlying principles, mathematical concepts, and many beautiful photos of the latest work in this fast-expanding field.

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★ 1.1. Modeling Origami

The term origami refers to something very specific: Japanese paper-folding. But mathematical origami is much broader than the traditional craft: it isn’t necessarily Japanese, it involves materials other than paper, and it involves actions other than just folding—bending and crumpling, for example, although both could be said to be a form of folding. What we will focus on in this book, though, are those aspects of mathematical folding that are characteristic of most origami: the use of a non-stretchy sheet-like material, manipulated in three dimensions, with few or no cuts. Mathematical folding doesn’t require that you use paper—in fact, in real-world applications of mathematical folding, one can use materials as diverse as plastic, Mylar, Kapton, leather, cloth, and even mats of carbon nanotubes. But throughout this work, for simplicity of language, I will generally refer to the material being folded as “paper,” and paper is often the ideal material to work with: inexpensive, widely available in diverse forms, and possessed of mechanical properties that make it particularly suited for folding.

Part of the beauty of origami in general and mathematical folding in particular is that it is tactile and visual; you can feel the paper, you can see the result, and integration of hand-eye experience builds an intuition of what is possible more effectively than any set of mathematical formulas or algebraic description. Nevertheless, there are limits to intuition, and mathematics can provide powerful tools to understand the possibilities of paper and to design specific structures and forms. And so, throughout this work, I will attempt to provide a mathematical description of the topic at hand.

There are many ways to describe folding mathematically, and the most natural way depends in large part on the level of abstraction that one chooses in the description. Is the folded form flat or three-dimensional (3D)? Are surfaces straight or curved? Are creases straight or curved? Do we care about effects of material thickness, tensile forces, mechanical yield, creep, and plastic deformation? There is no single “correct” mathematical description of folded paper; there are only various approximations that idealize, emphasize, and/or ignore different aspects of the folding process.

Two properties stand out above others as necessary to describe what is recognized as origami and that play a role in nearly all mathematical descriptions:

• Non-stretchy paper. The folded shape is a 3D deformation of a planar surface that does not appreciably stretch (or compress) in any direction.

• Non-self-intersection. The paper cannot intersect itself in the folded form, or in any intermediate stage.

Any mathematical description of paper-folding must include these two properties in some way or another. These two properties—non-stretchiness and non-self-intersection—are at the heart of the folding arts.

It is a little awkward to describe the properties of paper by what it is not; better to have a positive term. There are terms for both non-stretchiness and non-self-intersection. When we say that the paper is not stretchy, we mean that if we draw a line on the paper, fold the paper, and then measure the length of the line along the paper, that is, following the path of the paper, the length is unchanged. This property is a quality called isometry—taken from the Greek iso, meaning “same,” and -metry, meaning “measurement.” So the essence of origami folding is that it is isometric: distances along the surface of the paper are preserved going from the flat to the folded state (and, ideally, in all intermediate states).

The second property, that the paper cannot intersect itself, also has a mathematical name: injectivity. In the language of mathematics, a mapping from one set (the domain) to another (the range) is an injection if no two points in the domain map to the same point in the range. In real physical origami, we cannot have two points on the paper occupy the exact same point in space when the paper is folded. Even if you fold two layers together, one layer must lie above or below the other. If two layers switch places—here layer 1 lies above layer 2, there layer 2 lies on top—the rearrangement must happen without the paper penetrating itself, neither in unfolded layers nor at a fold. So injectivity is the quality of non-self-intersection. These two qualities are what define the mathematics that are particular to origami.

This is not to say, however, that every mathematical model of origami must strictly have these two qualities. In fact, as we will see, it is frequently convenient to model origami paper as a zero-thickness surface, in which case a stack of layers may very well violate injectivity by occupying the exact same position in mathematical space. The important thing in such cases, though, is that in such a model, we know that the mathematical idealization violates one or the other of the fundamental properties of origami. Frequently, we will patch up such an ideal mathematical model to recover the lost properties.

A mathematical description of origami must also make some assumption about the folding process, that is, the way that the paper gets from its initial flat state to its final configuration, the folded form. In standard origami books, that process is a relatively linear sequence of small steps: fold the pap...