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.

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.

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.

This article originally appeared on The Conversation, and is republished under a Creative Commons licence.

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