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The weird world of one-sided objects

You have most likely encountered one-sided objects hundreds of times in your daily life – like the universal symbol for recycling, found printed on the backs of aluminium cans and plastic bottles.


This mathematical object is called a Mobius strip. It has fascinated environmentalists, artists, engineers, mathematicians and many others ever since its discovery in 1858 by the German mathematician August Mobius.

数学上有一种模型叫做莫比乌斯带 (Mobius strip)。自1858年被德国数学家莫比乌斯 (August Mobius) 提出以来,莫比乌斯带吸引了环境学家、艺术家、工程师、数学家等许多不同的人。

Mobius discovered the one-sided strip in 1858 while serving as the chair of astronomy and higher mechanics at the University of Leipzig. (Another mathematician named Listing actually described it a few months earlier, but did not publish his work until 1861.) Mobius seems to have encountered the Mobius strip while working on the geometric theory of polyhedra, solid figures composed of vertices, edges and flat faces.

莫比乌斯于 1858 年提出了这一单侧曲面带状模型,当时他在莱比锡大学 (University of Leipzig) 担任天文学和高等力学教授。事实上,几个月前一位名叫李斯廷 (Listing)的数学家也提出过这一模型,但一直到1861年才发表。当时,莫比乌斯正致力于研究多面体(由顶点、棱和单面构成的立体图形)的几何理论,而莫比乌斯带似乎只是他的偶然发现。

A Mobius strip can be created by taking a strip of paper, giving it an odd number of half-twists, then taping the ends back together to form a loop. If you take a pencil and draw a line along the centre of the strip, you’ll see that the line apparently runs along both sides of the loop.


The concept of a one-sided object inspired artists like Dutch graphic designer MC Escher, whose woodcut “Mobius Strip II” shows red ants crawling one after another along a Mobius strip.

单侧平面物体的概念启发了艺术家创作。比如荷兰单面设计师埃舍尔 (MC Escher),在木版画“莫比乌斯带 2(Mobius Strip II)”中,他呈现了红蚁一只接一只地沿着莫比乌斯带缓缓行进的情景。

The Mobius strip has more than just one surprising property. For instance, try taking a pair of scissors and cutting the strip in half along the line you just drew. You may be astonished to find that you are left not with two smaller one-sided Mobius strips, but instead with one long two-sided loop. If you don’t have a piece of paper on hand, Escher’s woodcut “Mobius Strip I” shows what happens when a Mobius strip is cut along its centre line.

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莫比乌斯带令人惊艳的性质可不止这一个。例如,如果用剪刀沿着刚刚画的线将带剪开,你可能会吃惊地发现,眼前并没有出现两条窄了一半的莫比乌斯带,而是一条长长的双面环。如果手头没有纸条,也可以看看埃舍尔的作品“莫比乌斯带 1(Mobius Strip I)”。这幅作品呈现了沿着中心线将莫比乌斯带剪开后的情景。

While the strip certainly has visual appeal, its greatest impact has been in mathematics, where it helped to spur on the development of an entire field called topology.


A topologist studies properties of objects that are preserved when moved, bent, stretched or twisted, without cutting or gluing parts together. For example, a tangled pair of earbuds is in a topological sense the same as an untangled pair of earbuds, because changing one into the other requires only moving, bending and twisting. No cutting or gluing is required to transform between them.


Another pair of objects that are topologically the same are a coffee cup and a doughnut. Because both objects have just one hole, one can be deformed into the other through just stretching and bending.


The number of holes in an object is a property which can be changed only through cutting or gluing. This property – called the ‘genus’ of an object – allows us to say that a pair of earbuds and a doughnut are topologically different, since a doughnut has one hole, whereas a pair of earbuds has no holes.

孔是一种只能通过切割和黏合来实现数量增减的性质,这一性质叫做物体的“亏格” 。根据这一理论,我们可以这样说:甜甜圈有一个孔,耳塞没有孔,所以耳塞和甜甜圈的拓扑学结构不同。

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Unfortunately, a Mobius strip and a two-sided loop, like a typical silicone awareness wristband, both seem to have one hole, so this property is insufficient to tell them apart – at least from a topologist’s point of view.


Instead, the property that distinguishes a Mobius strip from a two-sided loop is called ‘orientability’. Like its number of holes, an object’s orientability can only be changed through cutting or gluing.


Imagine writing yourself a note on a see-through surface, then taking a walk around on that surface. The surface is orientable if, when you come back from your walk, you can always read the note. On a nonorientable surface, you may come back from your walk only to find that the words you wrote have apparently turned into their mirror image and can be read only from right to left. On the two-sided loop, the note will always read from left to right, no matter where your journey took you.


Since the Mobius strip is nonorientable, whereas the two-sided loop is orientable, that means that the Mobius strip and the two-sided loop are topologically different.


The concept of orientability has important implications. Take enantiomers. These chemical compounds have the same chemical structures except for one key difference: they are mirror images of one another. For example, the chemical L-methamphetamine is an ingredient in Vicks inhalers. Its mirror image, D-methamphetamine, is a Class A illegal drug. If we lived in a nonorientable world, these chemicals would be indistinguishable.

可定向性的概念含义重大。以化学中的对应异构物为例:有些化合物具有相同的化学结构,但有很重要的一点不同:它们互为镜像。例如维克斯吸入器 (Vicks inhaler) 的一种成分L-甲基苯丙胺,它的镜像 D-甲基苯丙胺 是A 级违禁毒品。如果生活在不可定向性的世界中,这些化学物质将变得难以区分。

August Mobius’s discovery opened up new ways to study the natural world. The study of topology continues to produce stunning results. For example, last year, topology led scientists to discover strange new states of matter. This year’s Fields Medal, the highest honour in mathematics, was awarded to Akshay Venkatesh, a mathematician who helped integrate topology with other fields such as number theory.

莫比乌斯的发现为研究自然科学开辟了新的道路,拓扑学研究不断取得惊世成果。例如去年,科学家通过拓扑学发现了物质的新状态;数学家文卡特什 (AkshayVenkatesh)将拓扑学与其他领域(如数论)融会贯通,也因此受颁今年数学界的最高奖项——菲尔兹奖。



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