Similarity has bothered me for a long time. Why do all circles have the same formula for area — how do we know nothing sneaky happens when we make them larger? In physics, don’t weird things happen when you scale things (particles, insects, small children) to gargantuan sizes? You’re saying that every circle has the same formula, yet a 300-foot honeybee cannot fly?
Things get really interesting when you start to ask about why shapes are seen as similar in general eg when a child sees ‘M’ as a seagull in flight etc
I ask the question, "If I move closer to this page, will the words change?"
And the answer gives me more insight on similarity than any geometry class I’ve taken.
Similarity is in fact just a particular case of projective geometry (hence the link with perspective), and precisely the part which leaves angles unaltered.
In fact, nearly any kind of geometry can be seen as a particular case of projective geometry, that’s the “erlangen program”…
Quite a beautiful thing, and sadly one that is largely unspoken for in school!
@Prudhvi: Thanks, that’s what helped it click for me too – it doesn’t matter how far away we are, or whether it “looks” big or small :).
@Johann: Thanks for the background! I hadn’t thought of that, but you’re right – and the neat aspect is really that it’s all from the observer’s viewpoint (does this shape look similar to another one from the eyes of this other person?). Quite true, there’s so much beyond what we learn in school :).
[…] scaled triangle (2x) and plop on another scaled triangle (times 3i). Even though it’s larger, similar triangles have the same angles — they’re just bigger (but don’t ask about its size, […]
