以上照片是sarah-mario belcastro女士用毛线编制的第一个克莱因瓶。她有专门的网站介绍她如何编织克莱因瓶以及其他“数学物品”的，见 http://www.toroidalsnark.net/mkkb.html 。 她在《美国科学家》杂志2013年3－4月号撰文《数学编织奇遇》，非常有趣。以下是文章的开头部分，想读全文的可点击 http://www.americanscientist.org/issues/feature/2013/2/adventures-in-mathematical-knitting/1 。 Adventures in Mathematical Knitting Rendering mathematical surfaces and objects in tactile form requires both time and creativity sarah-marie belcastro I have known how to knit since elementary school, but I can’t quite remember when I first started knitting mathematical objects. At the latest, it was during my first year of graduate school. I knitted a lot that year, because I never got enough sleep and needed to keep myself awake during class. During the fall term I made a sweater for my dad, finishing the seams right after my last final, and in the spring I completed a sweater for my mom. Also that spring, during topology class, I knitted a Klein bottle, a mathematical surface that is infinitely thin but formed in such a way that its inside is contiguous with its outside (see Figure 1) . I finished the object during a lecture. It was imperfect, but I was excited, and at the end of class I threw it to the professor so he could have a look. Over the years I’ve knitted many Klein bottles, as well as other mathematical objects, and have continually improved my designs. When I began knitting mathematical objects, I was not aware of any earlier such work. But people have been expressing mathematics through knitting for a long time. The oldest known knitted mathematical surfaces were created by Scottish chemistry professor Alexander Crum Brown. (For more about Crum Brown's work, click the image at right). In 1971, Miles Reid of the University of Warwick published a paper on knitting surfaces. In the mid-1990s, a technique for knitting Mbius bands from Reid’s paper was reproduced and spread via the then-new Internet. (Nonmathematician knitters also created patterns for Mbius bands; one, designed to be worn as a scarf, was created by Elizabeth Zimmerman in 1989.) Reid’s pattern made its way to me somehow, and it became the inspiration for a new design for the Klein bottle. Math knitting has caught on a bit more since then, and many new patterns are available. Some of these are included in two volumes I coedited with Carolyn Yackel: Making Mathematics with Needlework (2007) and Crafting by Concepts (2011) . You might wonder why one would want to knit mathematical objects. One reason is that the finished objects make good teaching aids; a knitted object is flexible and can be physically manipulated, unlike beautiful and mathematically perfect computer graphics. And the process itself offers insights: In creating an object anew, not following someone else’s pattern, there is deep understanding to be gained. To craft a physical instantiation of an abstraction, one must understand the abstraction’s structure well enough to decide which properties to highlight. Such decisions are a crucial part of the design process, but for the specifics to make sense, we must first consider knitting geometrically.
作者：蒋迅 Source: wikipedia.org 数学领域中， 克莱因瓶 ( Klein Bottle ) 是指一种无定向性的平面，比如2维平面，就没有“内部”和“外部”之分。它最初的概念提出是由德国数学家 菲利克斯·克莱因 ( Felix Klein ) 提出的。在三维世界里，克莱因瓶是不可实现的。因为我们必须到第四维空间去实现它。这就好比在二维空间里的 莫比乌斯带 ( Moius strip )，它只有到三维空间里才能实现粘合。 Source: nobrowcartoons.com by Mark Heath Mark Heath 创作了“一幅”漫画“克莱因瓶回收中心”(Recycling Center for Klein Bottle) 。当然，他的这幅其实是“两幅画”，因为克莱因瓶是数学家想象出来的。从漫画中可以看到，一位绅士抱着一袋克莱因瓶走进“克莱因瓶回收中心”却发现他永远 走不到头。 美国人的浪费是惊人的。单是塑料矿泉水瓶每天就会被丢弃三千多万个。1971年俄勒冈州率先立法回收空瓶子。1986年加州也通过了类似的法规。还有其它一些州跟进。虽然他们浪费惊人， 回收方面的努力 也给人深刻的印象。在加州，现在回收空瓶的中心很多，许多美国人把空瓶子攒起来拿到那里去卖，一般是5美分一个。这幅漫画就源于此景。 有人说，我们的宇宙就是在一个黑洞之中。这似乎很像克莱因瓶，整个我们所居住的宇宙都在一个黑洞中，而这个黑洞又在另一个黑洞里。这个过程永无止境。 下面再来几个克莱因瓶。尽管克莱因瓶在三维空间是不可能实现的，仍然不能阻止人们试图去实现它。 Source: Futurama screencaps Source: Drinking Mug Klein Bottles - for the Thirsty Topologist Source: Klein Bottle Opener Source: Acme's Baby Klein Bottle Source: The Klein Bottle -- After Paul Chang
标签: 克莱因瓶
