# topology

The trick to learning math is to complain constantly.
—  Topology professor

When people ask me how I can be a math major and still say I’m not good with numbers, I’m like ‘here, let me draw you a picture.’

In mathematics, it’s not like your professors understand everything. It’s just at some point they get comfortable with not understanding.
—  Algebraic topology professor

math textbooks are the best

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In an important field of mathematics called topology, two objects are considered to be equivalent, or “homeomorphic,” if one can be morphed into the other by simply twisting and stretching its surface; they are different if you have to cut or crease the surface of one to reshape it into the form of the other.

Consider, for example, a torus — the dougnut-shape object shown in the intro slide. If you turn it upright, widen one side and indent the top of that side, you then have a cylindrical object with a handle. Thus, a classic math joke is to say that topologists can’t tell their doughnuts from their coffee cups.

On the other hand, Moebius bands — loops with a single twist in them — are not homeomorphic with twist-free loops (cylinders), because you can’t take the twist out of a Moebius band without cutting it, flipping over one of the edges, and reattaching.

Topologists long wondered: Is a sphere homeomorphic with the inside-out version of itself? In other words, can you turn a sphere inside out? At first it seems impossible, because you aren’t allowed to poke a hole in the sphere and pull out the inside. But in fact, “sphere eversion,” as it’s called, is possible.

Incredibly, the topologist Bernard Morin, a key developer of the complex method of sphere eversion shown here, was blind.

Livescience.com

Who needs numbers when you can have doughnuts?
—  Topology professor
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August Möbius – Scientist of the Day

August Möbius (Moebius), a German mathematician, was born Nov. 17, 1790 (see fourth image above for a portrait).