Curvature allows us to comprehend views of reality that can't otherwise be seen, or appreciated without an understanding of a few basic principles. Curvature allows connections and interrelationships and juxtapositions that you may find it impossible to see if you don't have the necessary mind-set.
This doesn't just apply to theoretical physics, mathematics and abstract logical structures. it also applies to real life.
Let's suppose that we're signwriters, and we have a famous department store as a client. They'd like an impressive curved sign over their main entrance, proudly displaying their name. Wouldn't it be awful if we neglected to take into account how that curved set of letters looked from different angles, and accidentally built a sign that said a Very Rude Word?
At this point you're probably remembering the fictional Great Big Sign in Douglas Adams' "Hitchhiker" series ... the one built by Sirius Cybernetics that when collapsed to half its original size, spelt out the message "Go stick your head in a pig" ... you're probably thinking that I'm about to describe some tortured hypothetical example that would never really happen in real life ... some crazy laboured combination of store name and typeface and sign geometry that would be so improbable that it'd never ever happen.
And so, sceptical reader, I invite you to examine this real-life department store sign:
It's for a store called T J Hughes. Naturally, above the store's entrance we see the words T J HUGHES proudly displayed, in large red capitals. It's on a corner, and the letter sequence follows the curve.
If we turn the corner, cross the road, and look back at the sign, we still see the final “S” facing us, and to its right we see in white, slightly shrunken by perspective, the reversed white backsides of the letters H, J and T. Unfortunately, the letter J is very narrow, and the curved base of the letter is out of sight. So the J looks like an I, and although the H and T are seen reversed, they're symmetrical and still look like a perfectly normal H and T.
At this point you should be able to take a wild guess at the problem.
Here's the photo:
That's right. Seen from the right, the sign above their storefront really does say
This isn't a doctored photograph. The shop is real, and the sign has been there for an awfully long time. Here's the store's website, its location on Google Maps and their wikipedia entry. This really happened.
Like the title says, curvature is important. Ignore it at your peril.