TWISTS TILINGS AND TESSELLATIONS DOWNLOAD!
By Robert J. Lang ISBN A book, very long awaited in the origami math world. It's finally here, came in the mail just the. Read Online Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami => ?asin=BKBW6DC. Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami. By Lang, Robert J. Hardcover - English. Share on.
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The value of such a model, even as an imperfect approximation, comes when it can provide a reasonably accurate prediction of the folded state, and usually, the simpler the model, the better.
Twists Tilings and Tessellations Mathematical Methods for Geometric Origami | eBay
We can construct something of a hierarchy of origami modeling of increasing complexity as we relax the rules of folding, as shown in Table 1. In general, as one moves down this hierarchy, the mathematical complexity increases—sometimes dramatically.
We will explore this hierarchy, but we will move through it gradually, building base camps along the way and scheduling copious twists tilings and tessellations days as needed.
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And we will begin with the simplest possible model, which, surprisingly, covers a great deal of both historical and modern paper-folding. The first description we will consider is what for many years was the most common description within mathematical origami, and it twists tilings and tessellations very simple indeed.
Twists, Tilings, and Tessellations Mathematical Methods for Geometric Origami
In this description, we make these simplifying assumptions: The paper has zero thickness. The folded form is flat.
We call this model of origami flat-foldable origami. Such a model is, of course, an approximation of reality; there is no twists tilings and tessellations thing as zero-thickness paper, and there is no way that an unfolded crease pattern can discontinuously transform itself into a folded state.
Indeed, it is possible to contemplate folded configurations for which there is no practically achievable folding sequence. Nevertheless, this simple model can accurately describe a great deal of historic and modern folding, and it contains surprising richness and depth.
This model can provide practical recipes and algorithms for the twists tilings and tessellations of folded shapes that are beautiful, interesting, and practically useful.
Twists, tilings, and tessellations Robert J. Lang
Crease Patterns A feature of this simplest type of origami, what twists tilings and tessellations call flatfoldable origami, is that in the folded form, all surfaces are flat, except along straight lines, which are the creases, and the creases meet in groups at points, called vertices.
The flat regions bounded by the creases are facets.
There is a one-to-one mapping between points in the original paper and points in the folded form, and we can identify each point in the original paper as to whether it ends up in a facet, a crease, or a vertex. Twists tilings and tessellations enough, we call the points facet points, crease points, or vertex points, respectively.
We can then, if we like, decorate the paper with identifying information, coloring each point and line according to its status in the folded form.
Such a decoration is called the crease pattern associated with the folded form. The crease pattern is, essentially, a minimal description of the origami figure. For flat origami, often the crease pattern alone suffices as a guide for how to fold the shape. The crease pattern has a long history within origami; Figure 1.
In historical origami works and works of the early 20th century, crease patterns were not uncommon see, e. With the resurgence of mathematical folding and systematic design toward the end of the 20th century , though, crease patterns have returned as the blueprint of all of the folding that is to follow, and they will be a key concept throughout this book.
In geometric folding, by contrast, the CP is quite often comprehensive, containing every crease in the finished work. Even so, it often does not provide a full description of the origami figure.
It may contain all the folds, but it says nothing about the order twists tilings and tessellations which the folds are made. And many crease patterns, including the one in Figure 1.
TWISTS, TILINGS AND TESSELLATIONS | OrigamiUSA
In a flat origami figure, every fold can go in one of two directions, as shown in Twists tilings and tessellations 1. In conventional origami terminology, when you fold a flap toward you, the resulting fold is called a valley fold. When the flap is folded away from you, the resulting fold is called a mountain fold.
Historically, valley and mountain folds were not distinguished in any way as in Figure 1. Randlett in the West adopted a standard for diagrammatic origami instruction in which valley folds were indicated by a dashed line and mountain folds were indicated by a chain line dot-dot-dash.
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