What are some surprising/paradoxical theorems of projective geometry?
Posted Jan 26, 2009, 2:55 AM UTC
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Hyperbolic projections come to mind. Also, tilings.
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It may be too complicated or simply irrelevant but you could venture into color spaces. There are some neat tricks you could do with basic math an d color spaces.
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Interesting. I didn't know there was so much involved with quantifying colors. Hopefully I'll find time to learn more about it so that I can explain it. Thanks.
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I'm teaching a class aimed at non-math majors this semester. I have a number of art majors who are totally unexcited by math in my class. In class we're going over symmetries and tilings and that sort of thing, but I'd like to have some other examples to show them. Does anyone know of any neat results in projective geometry that are easily translated to be statements about perspective drawings in art that they might find cool?
Ideally such a theorem would be some existence result about a shape/object/transformation that has some surprising property that you wouldn't expect without using some mathematics. They should be able to understand what it means for such a thing to exist in lay-terms, but finding it without mathematical means would be quite hard.
Any interesting or cool results would be welcome!
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Showing that all conics are really the "same" would be a neat idea, especially since these students have had trouble with algebra in the past. There's a quite a bit of baggage that goes with that, but maybe I can find some way to distill it down. Thanks.
This has nothing to do with projective geometry, but it's a cool visual concept (IMO) that might pique their interest.
You could talk about graphs and the problem of finding a drawing for a graph with no edge crossings. Then you could show the Kuratowski graphs and mention (no details necessary) that you need higher genus surfaces to draw some graphs. Then you can draw some pretty pictures of toruses and stuff. Maybe mention continuous deformation and show some PG Dali slides to illustrate.
Also, I'm sure you've showed them some Escher slides. :)
On the theme of symmetry and tiling, John Conway just coauthored a book (full disclosure: one of the coauthors is my department chair) on the subject aimed at nonspecialists which may be useful.
We will be talking about basic graph theory later, so maybe I'll try to work in planar graphs and the four color theorem. Thanks for the suggestion.
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The first result that springs to mind is that geometric projection can readily demonstrate a one to one mapping between the open unit interval and the open ended infinite real line.
The fact that mathematically there are as many points in a unit interval as there are in an infinite line can rather surprise some people.
I don't know that I'd be able to get them convinced enough about bijections that they'd be all that surprised. It is a easy thing to present though. Thanks.
I don't know much projective geometry but I always thought probability was pretty crazy. For instance, the Law of Large Numbers still makes me smile. I think bringing up concepts that are applied would be best.
You could also discuss imaginary numbers and how they're useful in signal processing or some other application. I think people tend to think of numbers in a 'Euclidean space way' (for lack of a better way to describe it) so the idea of using these 'non-Euclidean space' numbers to do something in real life was pretty mind blowing for me. It's like there's another universe out there that gives us shortcuts to places in our own universe but the only way we can access it is with math.
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We'll be doing some probability and statistics later on. They get a nice selection of counter-intuitive examples: Monty Hall, Gambler's fallacy, Prosecutor fallacy (reversing conditional probabilities). I give them a plain English version of the Central Limit Theorem; some of them might find it helpful to see the Law of Large Numbers (suitably unformalized of course). Thanks for the idea.
Desargues' theorem?
edit: and Pascal's / Brianchon's