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The basic phenomenon is that often the best way to think about "little homotopies" is to use the geometric parts of your brain --- to use primarily your GPU rather than (geometry processing unit), with your CPU arithmetic processing unit, logic processing unit and your lexical processing unitunits all in the background, so to speak. However, when writing down a proof, it's customary, and usually easier to transcribe it into symbolic form. This tends to be a one-way process --- it's much harder to start from an algebraic description symbolic formulas and regenerate the geometric intuiton than to start from the geometric intuition and transcribe it into algebrasymbolic formulas.

It has become much easier to create reasonable figures illustrating geometric ideas than it used to be (say 20 or 30 years ago), but it's still hard. It's especially hard to directly convey geometric intuition in higher dimensions --- word portraits of geometric ideas can be good, but most mathematical writing neglects them.

I think the best strategy for learning is to avoid reading symbolic definitions of these little homotopies until you have spent some time effort thinking about them for yourself, primarily in your head. (Sketches can be good too, but they're often another layer of difficulty. Geometric imagination is not predominantly visual; it's a learned, tricky skill to be able to draw an image on paper that adequately represents a geometric mental model.)

In my experience, the symbolic descriptions often actively interfere with geometric understanding; at first, only use them as hints, for times after you've thought hard and are stuck. It takes time and concentration to build good mental images, but geometric imagination does improve with practice, and it's worth the effort. Eventually, you learn to read the formulas and evoke the geometric images.

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The basic phenomenon is that often the best way to think about "little homotopies" is to use the geometric parts of your brain --- to use your GPU rather than your CPU and your lexical processing unit, so to speak. However, when writing down a proof, it's customary, and usually easier to transcribe it into symbolic form. This tends to be a one-way process --- it's much harder to start from an algebraic description and regenerate the geometric intuiton than to start from the geometric intuition and transcribe it into algebra.

It has become much easier to create reasonable figures illustrating geometric ideas than it used to be (say 20 or 30 years ago), but it's still hard. It's especially hard to directly convey geometric intuition in higher dimensions --- word portraits of geometric ideas can be good, but most mathematical writing neglects them.

I think the best strategy for learning is to avoid reading symbolic definitions of these little homotopies until you have spent some time thinking about them for yourself, primarily in your head. (Sketches can be good too, but they're often another layer of difficulty. Geometric imagination is not predominantly visual; it's a learned, tricky skill to be able to draw an image on paper that adequately represents a geometric mental model.)

In my experience, the symbolic descriptions often actively interfere with geometric understanding; at first, only use them as hints, for times after you've thought hard and are stuck. It takes time and concentration to build good mental images, but geometric imagination does improve with practice, and it's worth the effort. Eventually, you learn to read the formulas and evoke the geometric images.