本帖最后由 paperart 于 2012-4-14 09:36 编辑 1.第一章 引言
Origami Design Secrets SecondEdition
折纸设计的奥秘第二版
Mathematical Methods for an Ancient Art
一项古老艺术的数学方法
1．Introduction 第一章 引言
In 1988, a French artist namedAlain Georgeot prepared an exhibition of 88 elephants. They were made of foldedpaper, each different, and each one an example of origami, the Japanese art ofpaper folding. An art exhibition devoted entirely to origami is rare; onedevoted to elephants is extremely unusual; and one devoted entirely to origamielephants was entirely unprecedented.
1988年法国艺术家阿兰·若尔若筹备了一场八十八只大象的展览。令人惊讶的是，这些大象都是用纸折成的，而且每只都各不相同，可以说是日本折纸艺术（注1）活灵活现的实例。当时专门的折纸展览还比较罕见，大象的展览也极其独特，这场全部都是折纸大象的展览更是史无前例的。（注1：此处开始用日语ORIGAMI来代替folded paper。Origami现在是折纸界对折纸的统一称谓。）
A display of 88 paper elephantsillustrates both the remarkable attraction origami has for some people—afterall, how many people would take the time to fold 88 versions of the samething?—and the remarkable versatility of the art. Georgeot’s collection ofelephants represented only the tiniest fraction of the modern origamirepertoire. Tens of thousands of paper designs exist for animals, plants, andobjects, a regular abecedarium of subject matter. There are antelopes, birds,cars, dogs, elephants (of course), flowers and gorillas; horses, ibexes, jays,and kangaroos; lions, monkeys, nautiluses, octopi, parrots, quetzalcoatls,roses, sharks, trains, ukuleles, violinists, whelks, xylophones, yaks, andzebras, the last complete with stripes.
以八十八只纸象的展示为例，折纸和这项艺术的多样性深深地吸引了人们的关注。（毕竟有多少人愿意花时间折叠一样东西的八十八个版本呢？）若尔若的纸象集只是所有现代折纸中极少的一部分。如今已经出现了数以万计的折纸设计，其题材包含动物、植物和各种物体：有羚羊、鸟类、轿车、狗、大象（那是当然）、花、猩猩、马、巨角塔尔羊、松鸦、袋鼠、狮子、猴子、鹦鹉螺、章鱼、鹦鹉、羽蛇神、玫瑰、鲨鱼、火车、夏威夷四弦琴、小提琴手、峨螺、木琴、牦牛和带完整条纹的斑马（注2）。（注2：原文这26个作品是按照字母表顺序排列，原文为复数，其中多数本书作者都有相应作品。）
Innumerable innovations have beenwrought upon the basic theme of folded paper. There are action figures: birdsthat flap their wings, violinists who bow their violins, inflatable boxes,clapping monkeys, snapping jaws. There are paper airplanes that fly—one won aninternational contest— and airplanes that don’t fly, but are replicas of famousaircraft: the space shuttle, the SR-71 Blackbird, and the venerable SopwithCamel. In some models, a single piece of paper is folded into several figures(a bull, bullfighter, and cape, for example) and in others, many identicalpieces of paper are assembled into enormous multifaceted polyhedra. If you canthink of an object either natural or man-made, someone, somewhere, has probablyfolded an origami version.
由于基本主题被无数的创新所影响，出现了许多活灵活现的折纸作品：展翅高飞的鸟、演奏中的小提琴手、可充气的盒子、拍手的猴子、能发出啪啦声的松鸦。有可以飞行的纸飞机，其中一架还曾在国际比赛中获胜。还有不可以飞行但能够以假乱真的：航天飞机、SR-71黑鸟式侦察机和古老的索普维斯骆驼战斗机（注3）。一些模型是用一张纸折叠成各自的形状（例如公牛、斗牛士和斗篷）和其他部分。还有一些模型是用相同大小的纸张组合成巨大的多层的多边形。只要能想到的物体，不管它是自然的还是人工的，无论何人何地都有可能用折纸表现出来。（注3：索普维斯骆驼战斗机为一战时英国最著名的战斗机，单座单发双翼。）
The art of origami was originallyJapanese, but the 88 elephants and the tens of thousands of other designs comefrom all over the world. Many figures originated in Japan, of course, but theU.S.A., England, France, Germany, Belgium, Argentina, Singapore, Australia, andItaly are major centers of origami activity. The designs range from simplefigures consisting of only two or three folds to incredibly complex “testpieces” requiring hours to fold. Most of these thousands of designs have onething in common, however: Nearly all were invented in the last 50 years.
折纸艺术起源于日本，但是八十八只大象和数以万计的折纸设计却来自于全世界。许多折纸作品诞生在日本，当然美国、英国、法国、德国、比利时、阿根廷、新加坡、澳大利亚和意大利都是活跃的折纸中心（注4）。通常很多设计一般只表现一件事物，从两三步的简单折叠到复杂得难以置信的“神品”无所不有。然而所有这些都是近五十年内才出现的（注5）。（注4：近年来俄罗斯、越南、乌拉圭……等国家折纸艺术发展也较为迅速。）（注5：本书第一版于2003年出版，其中所述的年代请依据当前时间自行增加。）
Thus, origami is both an old artand a young art. Its youth is somewhat surprising. After all, folded paper hasbeen an art form for some 15 centuries. It is ancient; one would not expect 98percent of the innovation to come in the last 2 percent of the art’s existence!Yet it has. Fifty years ago, all of the different origami designs in the worldcould have been catalogued on a single typed sheet of paper, had anyone had theinclination to do so. No model would have run over about 20 or 30 steps. Mostcould be folded in a few minutes, even by a novice. This is no longer the case.Today, in books, journals, and personal archives, the number of recordedorigami designs runs well into the thousands; the most sophisticated designshave hundreds of steps and take several hours for an experienced folder toproduce. The past 60 years in Japan, and 40 years worldwide, have seen arenaissance in the world of origami and an acceleration of its evolution.
折纸是门既古老又年轻的艺术，它年轻得多少有点让人惊讶。它非常的古老，作为一门艺术，折纸已经有一千五百多年的历史；然而它又那么的年轻，因为这门艺术里百分之九十八的革新无不来自最后百分之二的时间里。五十年前，如果将世界上所有不同的折纸设计编个目录的话都不会超过一个单篇儿，当然也没人愿意这么做。那时候没有哪个模型的折制会超过二十步或三十步，哪怕是一个新手也会在短短几分钟内将它折好，没什么吃力的地方。可是今天，在书籍、学报以及个人档案中，被记录在案的折纸设计正在成百上千的增长，最复杂的设计需要数百个步骤和几小时的时间才能够顺利完成。在日本过去的六十年里、世界范围过去的四十年内，折纸世界已经迎来了复兴并且正在高速的发展。And this has happened in the faceof stringent barriers. The traditional rules of origami—one sheet of paper, nocuts— are daunting. It would appear that only the simplest abstract shapes arefeasible with such rules. Yet over hundreds of years, by trial and error, twoto three hundred designs were developed. These early designs were for the mostpart simple and stylized. Complexity and realism—insects with legs, wings, andantennae—were not possible until the development of specialized design methodsin the latter part of the 20th century. 为什么会这样呢？因为折纸需要面对一些严格的障碍。折纸的传统规则是：一张纸，不可以剪切（很可怕吧）。在这一规则下，只可能出现一些简单而抽象的形状。几百年来，经历了不断的尝试与失败，只出现了两三百种折纸设计，这些早期设计大都比较简单而且程式化。折一个有腿、有翅膀还有触角的昆虫这样复杂生动的作品根本无法想象。直到二十世纪后期随着专业折纸方法的发展，这才成为可能。
Although there are now manythousands of origami designs, there are not thousands of origami designers. Infact, there is only a handful of designers who have gone beyond basics, only ahandful who can and do design sophisticated models. Although there is far moreexchange of completed designs now than there used to be, there is not a similarexchange of design techniques. 虽然现在已经有数以千计的折纸设计，但是却算不上是数以千计。即使完成后的设计与过去的模型有着极其丰富的改变，甚至设计技巧上毫无类似之处。但事实上，只有极少数设计者能够超越折纸的基础形态，也只有极少数人能够设计出超级复杂的模型。This imbalance arises because it ismuch easier to describe how to replicate an origami figure than how to designone. Origami designs spread through publication of their folding sequence—a setof step-by-step instructions. The folding sequence, based on a simple code ofdashed and dotted lines and arrows devised by the great Japanese master AkiraYoshizawa, transcends language boundaries and has led to the worldwide spreadof origami.
这种失衡的发生是因为设计者们过于强调了如何去折叠一个折纸作品而不是去设计它。折纸作家们又通过各种折序图（一步接一步的介绍）出版物传播着这一方式。我不是想反对折图，当初日本折纸泰斗吉泽章发明的这种由实线、虚线与箭头构成的简单的符号系统，它超越了语言的限制，使折纸在世界范围内得以传播。
While thousands of foldingsequences have been published in books, magazines, and conference proceedings,a step-by-step folding sequence does not necessarily communicate how the modelwas designed. The folding sequence is usually optimized for ease of folding,not to show off design techniques or the structure of the model. In fact, someof the most enjoyable folding sequences are ones that obscure the underlyingdesign of the model so that the appearance of the final structure comes as asurprise. “How to fold” is rarely “how to design.” Folding sequences arewidespread, but relatively few of the design techniques of origami have everbeen set down on paper.
尽管已经出版了成千的折序图的书籍、杂志和会刊，但是这些有步骤的折序图并未有效的传达出它的设计过程。折序图只是将折叠过程进行了优化，它没有展现出设计技巧或模型的结构。实际上折序图的很多乐趣就来自于没有理解模型隐含的设计，到了折出成品时那一刹那的惊喜。“如何折”掩盖了“如何设计”。折序图铺天盖地，可是黑字白字的关于折纸设计技巧的书籍在哪里呢？
Over the last 40 years I havedesigned some 500+ original figures. The most common question I am asked is,“How do you come up with your designs?” Throughout the history of origami, mostdesigners have designed by “feel,” by an intuition of which steps to take toachieve a particular end. My own approach to design has followed what I suspectis a not uncommon pattern; it evolved over the years from simply playing aroundwith the paper, through somewhat more directed playing, to systematic folding.Nowadays, when I set out to fold a new subject, I have a pretty good idea abouthow I’m going to go about folding it and can usually produce a fairapproximation of my subject on the first try.
过去四十年我设计了五百多种折纸作品，总会被问到这样一个问题：“你的设计是怎么弄出来？”纵观折纸的历史，很多折纸作家会依靠“感觉”来设计，那是一种通过哪些步骤能达到最终效果的直觉。我本人的设计方式是设想出一个不同寻常的式样，而这一方式是通过多年来有目标、成系统的实践才逐渐形成的。现如今，当我想折叠一个新的题材的话，在首次尝试的时候，我就会想出一个不错的主意来实现它，而且还能折的挺像。
Hence the perennial question: Howdo you do that? The question is asked as if there were a recipe for origamidesign somewhere, a cookbook whose steps you could follow to reliably produceany shape you wanted from the square of paper. I don’t think of origami designas a cookbook process so much as a bag of tricks from which I select one ormore in the design of a new model. Here is a base (a fundamental foldingpattern) with six legs: I’ll use it to make a beetle. Here is a technique foradding a pair of points to an existing base: I’ll combine these to make wings.Some designers have deeper bags of tricks than others; some, like JohnMontroll, have a seemingly bottomless bag of tricks. I can’t really teach theway to design origami, for there is no single way to design, but what I can andwill try to do in this book is to pass on some of the tricks from my bag.Origami design can indeed be pursued in a systematic fashion. There are nowsimple, codified mathematical and geometric techniques for developing a desiredstructure. This book is a collection of those techniques. It is not astep-by-step recipe book for design. Origami is, first and foremost, an artform, an expression of creativity, and it is the nature of creativity that itcannot be taught directly. It can, however, be developed through example andpractice. As in other art forms, you can learn techniques that serve as aspringboard for creativity.
但我脑中一直萦绕着这样一个问题：“你是如何做到的？”折纸设计哪些地方会有秘诀呢？能不能像烹饪书那样，将从一张方纸到想得到的形状这些步骤完美地呈现呢？能不能设想折纸设计会有烹饪书里那样的锦囊妙计——抽出一个两个设计就能变成新的模型。看，这有一个六条腿的基底（折纸的基本型），我来折个甲虫；这有一个在已知基底上加一对支点的技巧，我结合它来加对儿翅膀。一些折纸作家已经拥有比这还要好的锦囊，像约翰·蒙特利尔等人，他们的锦囊更像个无底洞。我不可能将折纸设计完整的讲解清楚，因为它并非只有一种方式，但是在这本书中，我会（我也正要这么做）将我自己的锦囊中的妙计取出来与大家一一分享。折纸设计必将变为一种时尚而风行。现在有一种简单的以数学、几何学为理论基础的折纸技巧，可以折出随心所欲的各种结构。这本书恰恰收集整理了这些技巧，可它不是一招一式的折纸秘诀。首当其冲，折纸是一种艺术形式，是创造力的表现，创造力天赋不可能直接传授，然而它可以通过实例和练习来不断提高。和其他艺术形式一样，你能够学习到一些技巧，像跳板一样来启发你的创造力。
The techniques of origami designthat are described in this book are analogous to a rainbow of colors on anartist’s palette. You don’t need a broad spectrum, but while one can paintbeautiful pictures using only black and white, the introduction of other colorsimmeasurably broadens the scope of what is possible. And yet, the introductionof color itself does not make a painting more artistic; indeed, quite theopposite can happen. So it is with origami design. The use of sophisticateddesign techniques—sometimes called “technical folding,” or origami sekkei—makesthe resulting model neither artistic nor unartistic. But having a richerpalette of techniques from which to choose can allow the origami artist to morefully express his or her artistic vision. That vision could include elements ofthe folding sequence: Does it flow naturally? Is the revelation of thefinished form predictable or surprising? It could include elementsof the finished form: Are the lines harmonious or jarring? Does the use offolded edges contribute to or detract from the appearance? Does the figure usepaper efficiently or waste it? The aesthetic criteria to be addressed arechosen by the artist. Any given technique may contribute to some criteria (andperhaps degrade others). By learning a variety of design techniques, theorigami artist can pick and choose to apply those techniques that bestcontribute to the desired effect.
本书介绍的折纸技巧就像是画家调色板上多彩的颜色。你不需要一个完整的色谱，但是要完成一副漂亮的图画除了黑色白色，还需要引入无限多的可能的颜色。然而引入颜色本身不可能使绘画更加的艺术，往往会适得其反。折纸设计也是如此。使用复杂的折纸技巧——有时称为“折叠工艺”或“折纸设计”——并不能使最终的模型更具美感或者相反，折纸艺术家们选择丰富的技巧来充分的表达他（或她）的艺术构想。在一些折叠过程中：折序是否运转自然？作品的最终形态是与预期一致还是出人预料？在一些成品形态的呈现上：线条是否和谐？折边是否有利于外观的表现？作品在纸张的利用上是节约还是浪费？这些审美标准在艺术家的选择中被处理妥当。一些技巧有助于一些标准，或许会降低其它。只有通过各种折纸技巧的学习，折纸艺术家们才能够精挑细选，决定应用哪些技巧才能够最好的表达渴求的艺术效果。
These techniques are not alwaysstrict; they are sometimes more than suggestions, but less than commandments.In some cases, they are vague rules of thumb: “Beyond eight flaps, itis more efficient to use a middle flap.” But they can also be asprecise as a mathematical equation. In recent years, origami has attracted theattention of scientists and mathematicians, who have begun mapping the “laws ofnature” that underlie origami, and convertingwords, concepts, and images intomathematical expression. The scientific fields of computerscience, number theory, and computational geometry support and illuminate theart of origami; even more, they provide still more powerful techniques fororigami design that have resulted in further advances of the art in recentyears. Many design rules that on the surface apply to rather mundane aspects offolding, for example, the most efficient arrangements of points in a base, areactually linked to deep mathematical questions. Just a few of the subjects thatbear on the process of origami design include the obvious ones of geometry andtrigonometry, but also number theory, coding theory, the study of binarynumbers, and linear algebra as well. Surprisingly, much of the theory isaccessible and requires no more than high school mathematics to understand. Iwill, on occasion, bring out deeper connections to mathematics where they arerelevant and interesting, and I will provide some mathematical derivations ofimportant concepts, but in most cases I will refrain from formal mathematicalproofs. My emphasis throughout this work will be upon usable rules rather thanmathematical formality.
折纸技巧并非总是严格的，很多时候它们更像是建议而不是清规戒律。某些情况下，它们是模糊的经验法则：“边缘的八个纸翼就比用一个中间的纸翼更有效率。”但它们也能够像数学方程式一样精确。近年来，折纸引起了一些科学家和数学家的注意，他们开始像数学表达式那样制定关于基础折纸、用语转换、概念、图像的自然法则。计算机科学、数论以及计算几何学等科学领域都被用来支持和阐明折纸艺术，这些学科为折纸艺术提供了更加强大的方法，使得这门艺术在最近几年中取得了长远的发展。折纸的许多设计规则在其他方面得以应用。例如：在一个基基底上有效的安排支点实际上可以引申出深奥的数学问题。没有几个学科能够像设计折纸这样，能够与几何学、三角法、还包括数论、译码原理、二进制的研究、线性代数等等有着那样明显的关联。令人吃惊的是，大量的理论都很容易理解，甚至只需要中学数学的水平。在适当的时机，我会说些深一点的折纸与数学的联系，这种联系是相关的也是有趣的。同时我还会提供一些重要概念的数学来历，但大部分情况下我会克制住繁琐的数学证明，毕竟我的重点是要通过这些工作得出合用的折纸规则而不是数学形式。
As with any art, ability comes withpractice, whether the art is origami folding or origami design. The buddingorigami designer develops his or her ability by designing and seeing theresult. Design can start simply by modifying an existing fold. Make a change;see the result. The repeated practice builds circuits in the brain linkingcause and effect, independent of formal rules. Many of today’s origamidesigners develop their folds by a process they often describe as intuitive.They can’t describe how they design: “The idea just comes to me.” But one cancreate pathways for intuition to take hold by starting with small steps ofdesign. The great leap between following a path and making one’s own patharises from the development of an understanding of why: Why did the designer doit that way? Why does the first step start with a diagonal fold rather than asquare fold? Why do the first creases hit the corners? Why, in another model,do the first creases miss the corners only by a little bit? Why does a group ofcreases emanate from a spot in the interior of the paper? If you are abeginning designer, you should realize that no design is sacred. To learn todesign, you must disregard reverence for another’s model, and be willing topull it apart, fold it differently, change it and see the effects of yourchanges.
与任何一种艺术都一样，不管这门艺术是“折”折纸还是“设计”折纸，其才能均来自于实践。折纸设计新手依靠不断的设计和理解结果来提升他（或她）的能力。设计可以通过改造已存的折纸来简单的开始。做一个改变，看一下结果，不用依靠正式的规则，这种反复的实践就会在大脑中建立因果联系。现今的很多折纸设计者会凭着他们描述为直觉的过程来开发他们的折纸作品。他们无法描述自己如何来设计：“我是一下子就想到的。”有一个方式可以通过以较少步骤的设计开始来建立直觉的终南捷径，日积月累的不断的解读为什么，这将会得到大幅度的提升：为什么设计者会采用这种方式？为什么要以对角折开始而不是对边折？为什么第一道折痕要沿着这个角？为什么在另一个模型里第一道折痕要与这个角错开那么一点？为什么会有一组折痕在纸内的一点呈放射状分布？如果你是个设计初学者，那么就要意识到没有什么设计是神圣的。为了学习设计，你必须忽略对其它模型的尊重，要有意的将其分解、折出不同、改装它并且在你的改变中看到效果。
Small ideas lead to big ideas; theconcepts of design build upon one another. So do the chapters of this book. Ineach chapter, I introduce a few design principles and their associated terms.Subsequent chapters build on the ideas of earlier chapters. Along the way youwill see some of my own designs, each chosen to illustrate the principlesintroduced in the chapter in which it appears.
小想法启发大创意；设计思想彼此支撑。本书此章到此为止。在其他章节里，我要介绍几个设计原理以及它们之间的联系。后面的章节都建立在前面章节的想法之上。顺便，你会看到一些我的折纸设计，每个设计都是当前章节所阐述原理的实例说明。
Chapter 2 introduces thefundamental building blocks of origami: the basic folds. If you have foldedorigami before, you may already be familiar with the symbols, terms, and basicsteps, but if not, it is essential that you read through this section. Chapter2 also introduces a key concept: the relationship between the crease patternand the folded form, a relationship that we will use and cultivate throughoutthe book.
第二章将介绍折纸中最基础的构成部分：折叠基础。如果你之前接触过折纸，已经很熟悉折纸符号、术语和基本步骤，可以略过本章。否则的话，通读这个章节还是必要的。第二章还介绍了一个关键的概念：折痕图与成品的最终形态之间的关系，通过本书我们将会使用和理解这一关系。
Chapter 3 initiates our foray intodesign by examining a few designs. The first stage of origami design ismodification of an existing design; in this chapter, you will have anopportunity to explore this approach by devising simple modifications to a fewfigures.
第三章我们通过对几个设计的研究来开始折纸设计的首次尝试。折纸设计的第一阶段是对已有设计进行修改，在这章中，通过对几个作品的简单修改，你将有机会探索这个方法。
Chapter 4 introduces the concept ofa base, a fundamental form from which many different designs may be folded. Youwill learn the traditional bases of origami, a number of variations on thesebases, and several methods of modifying the traditional bases to alter theirproportions.
第四章介绍基底的概念，基底是可以折叠出许多不同设计的基本形状。你会学到折纸中传统的基底、一些传统基底的变形、以及如何改变基底的比例这一改装方法。Chapter 5 expands upon the idea ofmodifying a base by focusing upon modifications that turn a single point intotwo, three, or more simply by folding. This technique, called point splitting,has obvious tactical value in designing, but it also serves as an introductionto the concept of modifying portions of a base while leaving others unchanged.
第五章展开介绍改装基底的好点子，用简单的折叠来修饰一个纸尖儿，在上面分出一个、两个甚至更多的纸尖儿。这个技巧称为尖角分离，在折纸设计中有着明显的战略价值。除此之外，它还会引领我们学习另外一个概念——在保留其它部分不变的情况下，如何对基底的部分进行修改。
Chapter 6 introduces the concept ofgrafting: modifying a crease pattern as if you had spliced additional paperinto it for the purpose of adding structural elements to an existing form.Grafting is the simplest incarnation of a broader idea, that the creasepatterns for origami bases are composed of separable parts.
第六章介绍移植的概念：建立折痕展开图，犹如在原来基础上增加额外的纸一样，给一个已有形状添加一些结构要素。把基底的折痕展开图分解成一个个单独部分的组合，移植可以说是众多的创意里最简单的实例了。
Chapter 7 then expands upon theidea of grafting and shows how multiple intersecting grafts can be used tocreate patterns and textures within a figure—scales, plates, and othertextures. This set of techniques stands somewhat independently, as almost anyfigure can be “texturized.”
第七章展开介绍移植，同时揭示怎样用多个交叉的移植来为作品添加图案和纹理，例如鳞片、板块和其它纹理。仅仅独立使用这一个技法，就差不多可以将任意一个作品进行“纹理化”。
Chapter 8 generalizes the conceptof grafting to a set of techniques called tiling: figuratively cutting up andreassembling different pieces of crease patterns to make new bases. Thischapter defines both tiles and matching rules that apply to the edges of tilesto insure that the assemblies of tiles can be folded into a flat shape. Chapter8 also introduces the powerful concept of a uniaxial base—a family ofstructures that encompasses both the traditional origami bases and many of themost complex modern bases.
第八章从移植法推衍出另外一个技法——镶嵌术：就像是把折痕展开图切碎再重新组装最后做出一个新的基底。本章将详述镶嵌术和确保镶嵌中的边缘在平面中能够被折叠出来的算法。第八章还会介绍一个非常重要的概念——单轴基底：包含传统折纸基底与许多最复杂的现代基底的一个结构类。
Chapter 9 shows how the tiledecorations that enforce matching can be expanded into a design technique intheir own right: the circle/river method, in which the solution of an origamibase can be derived from packing circles into a square box. Circle/riverpacking is one of the most powerful design techniques around, capable ofconstructing figures with arbitrary configurations of flaps, and yet it can beemployed using nothing more than a pencil and paper.
第九章将展示如何基于镶嵌装饰自身来实施计算，拓展出另外一个设计技巧：圆河法，这是一个折纸基底的解决方案，好比将多个圆放进一个方盒子里面。圆河填充大约是最重要的折纸设计技巧，应用它可以随意的进行纸翼搭配，使得各种构造的作品成为可能，而最终所用到的不过是笔和纸。Chapter 10 explores more deeply thecrease patterns within tiles; those that fit within circle/river designs arecalled molecules. The chapter presents the most common molecules,which are sufficient to construct full crease patterns for any uniaxial origamibase. 第十章深层次的探讨镶嵌术中的折痕展开图；还有圆河法里的“粒子”。本章介绍最普遍的“粒子”，那些足以构造任意单轴折纸基底的折痕展开图的“粒子”。
Chapter 11 presents a differentformulation of the circle/ river packing solution for origami design, calledtree theory, in which the design of the base is related to an underlying stickfigure, and the packing problem is related to a set of conditions applying topaths along the stick figure. Although equivalent to circle/river packing, theapproach shown here is readily amenable to computer solution. It is the mostmathematical chapter, but is in many ways the culmination of the ideaspresented in the earlier chapters for designing uniaxial bases.
第十一章出场的是一个与圆河填充不一样的折纸设计方法，这就是“树论”，一种有关于作品的优先线条的基底设计，以及关于沿着作品线条进行的填充问题。与圆河填充相比，这种方法更多的要依赖于计算机来解决。与前面的关于同轴基底设计的章节相比，本章是最数学化的章节，但也是设计创意中的集大成者。
Chapter 12 then introduces aparticular style of origami called box pleating, which has been used for someof the most complex designs ever constructed. Box pleating in some ways goesbeyond uniaxial bases; in particular, it can be used to construct fullythree-dimensional figures by various combinations of box-like forms, pleats,flaps, and more.
第十二章要介绍一种特殊类型的折纸——箱式折叠。有一些极其复杂的折纸作品用到了这种方法。箱式折叠在某种程度上超越了单轴基底，它能够通过箱格、褶、纸翼以及其它元素的多方面的组合构建出完全的三维的形态。
Chapter 13 expands upon the flapconcept of box pleating to introduce a new concept in design, called polygonpacking, and a particular type of polygon packing, uniaxial box pleating, thatties together the concepts of box pleating and tree theory.
第十三章通过对箱式折叠中纸翼概念的拓展来介绍一个新的设计概念——多边形填充。还要介绍多边形填充中的一个特殊类型——单轴箱式折叠，箱式折叠与树论的完美结合。
Chapter 14 continues thedevelopment of polygon packing and uniaxial box pleating, introducing the new designtechnique of hex pleating and methods of generalizing polygon packing furtherto arbitrary angles.
第十四章继续探讨多边形填充与单轴箱式折叠的发展。将介绍到十六制分折，推论出在多边形分折中如何得出任意角度的方法。
Chapter 15 continues to move beyonduniaxial bases, introducing the idea of hybrid bases, which combine elementsfrom uniaxial bases with other non-uniaxial structures. The world of origamidesigns is enormously larger than the uniaxial bases that are the focus of thisbook, but as this chapter shows, elements from uniaxial bases can be combined withother structures, expanded, and extended, to yield evergreatervariety in origami figures.
第十五章我们将目光转移到单轴基底，将介绍混合基底的创意，将其它非单轴结构中的元素结合到单轴基底上。折纸设计的世界是如此的广阔无垠，远远超过本书所关注的单轴基底，所以本章会展示——单轴基底的元素与其它结构相联合，去延伸、去拓展，去征服无限的多样化的折纸作品。
The References section providesreferences and commentary organized by chapter with citations for material fromboth the mathematical and origami literature related to the concepts in eachchapter.
参考文献部分按照章节顺序以引文的形式提供了每章的相关概念所涉及的数学、折纸著作的参考文献与注释的出处。
Each chapter includes step-by-step foldinginstructions for one or more of my origami designs chosen to illustrate thedesign concepts presented in the chapter. I encourage you to fold them as youwork your way through the book. Most have not been previously published. I havealso, in several chapters, presented crease patterns and bases of models whoseinstructions have been published elsewhere; for many of them, you will findsources for their full folding sequences in the References section, though forsome, the discovery of how to collapse the crease pattern into the base will beleft as an exercise for the reader.
每章都附有我个人设计的一些作品的折序图，这些作品大都没有公开出版过，选择它们是为了对当前章节讲述的设计理念进行说明。通过本书的学习，我还是鼓励你能够以你自己的方式去折叠它们。个别章节里，我会插入在其它地方出版过的一些折痕展开图和模型的基底；其中大部分在参考文献部分你会了解到哪里会有它们完整的折序图。还有一些将基底到折痕展开图的探索就留给读者们作为练习吧。
The concepts presented here are byand large my own discoveries, developed over some 40-plus years of folding.They were not developed in isolation, however. Throughout the book I havepointed out sources of influence and/or ideas I have adopted. In several casesothers have come up with similar ideas independently (an event not withoutprecedent in both origami and the sciences). Where I am aware of independentinvention by others, I have attempted to identify it as such. However, theformal theory of origami design is very much in its infancy. Sources of designtechniques are often unpublished and/or widely scattered in sometimes obscuresources. This work is not intended to be a comprehensive survey of origamidesign, and if it seems that I have left out something or someone, no slightwas intended.
本书中提出的概念大体上是我个人的探索，是我四十多年来折纸生涯的发展历程。然而它们不是孤立的发展而来的。贯穿本书，我已经提出外界影响来自于哪里以及我所采用的创意的来源。其它个别案例来自于一些独立的小点子（这些点子在折纸界以及科学界都是史无前例的）。我从其他地方那里得知一些独到的见解，我都会尝试验证一下。因为折纸设计的理论还仅仅处于它的萌芽阶段，大量的折纸技巧未能发表，或者来源模糊以致过于散乱，同时对折纸设计进行综合调研整理还未能开展，请见谅我无意中忽视了部分的人或事。
Technical folding, origami sekkei,is an edifice of concepts, with foundations, substructure, and structure.Because the organization of this book mirrors this structure, I encourage youto read the book sequentially. Each chapter provides the foundation to build conceptsin the next. Let’s start building.
专业折纸，折纸设计就像是一座由各种概念建造的大厦，有地基、有基础框架、有内部结构。这本书就是参照这一结构来组织的，所以我鼓励你按照顺序来阅读本书。每一个章节都是为了之后提出的概念在打基础。现在，就让我们来建造折纸设计这座大厦吧！
