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April 28th, 2014, 07:09 AM   #1
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imagining instead of drawing

Hi,
I'm new here and I registered because of my today's idea.
I've never liked geometry, and I think the reason is that in geometry you need to draw pictures, either making it without any things (often it is hard to see anything) or with ruler, compass. And that means for me that I need something more than only my mind. My dream is possibility of thinking about problems even while I'm doing something else, having all that I need in my head and solving it in my head. Making geometrical proofs using only language of math won't be effective because it won't be intuitive. But my idea and question is: is it possible to make pictures in my head? Have you ever heard about anybody who could just imagine geometrical problems, draw them in his head? Maybe there are people with such a great spatial intelligence, maybe any famous mathematician? I strongly believe that if I practiced hard enough I would develope that skill and I think it would improve my intuition and make me more creative so I'm just asking what do you think about it?
Maybe to remember names of points and figures I should connect letters with
pictures, for example imagining square ABCD as Arab's house, beach, Cindirella Castle, Dog's home and path between these objects. Like graph. That's just additional idea and probably it is nonsense.
There are big odds that you won't take my question seriously, but I'll try.
My english isn't perfect so sorry for mistakes if there are any.
Thank you.
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June 26th, 2014, 03:30 PM   #2
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Quote:
Originally Posted by twix View Post
But my idea and question is: is it possible to make pictures in my head? . . .

I strongly believe that if I practiced hard enough I would develope that skill and I think it would improve my intuition and make me more creative . . .
You answered your question.

Quote:
Originally Posted by twix View Post
Have you ever heard about anybody who could just imagine geometrical problems, draw them in his head?
No, I've never heard of any who could do that.

Quote:
Originally Posted by twix View Post
Maybe there are people with such a great spatial intelligence, maybe any famous mathematician?
I think you watched too many movies. Movies are make believe, wishful thinking. A classic example would be Limitless (2011).

It is true you CAN do anything, but it takes A LOT of TIME do DEVELOP those abilities.
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June 26th, 2014, 07:49 PM   #3
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If you want to be a great mathematician, I would concentrate on learning the maths. If you happen to be able to do some of it in your head, that's great. But I doubt you'll make any telling contribution to research without writing things down.

The good news is that geometry is only a small part of maths, so you can focus on something else later in your career.
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June 27th, 2014, 05:21 AM   #4
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You can definitely solve geometry problems in your head. It's true, if you want to stay in math you'll eventually hit a wall where you have problems to work on that are too complex to store entirely in your head, but you can put that off for a while.

A fun theorem from much more advanced mathematics shows that elementary geometry (which has a very technical meaning, but it includes essentially every high-school geometry problem) is decidable, that is, there is an algorithm to solve all such problems in an organized fashion. (This is not the case with, e.g., calculus.) It works by translating geometric problems into purely algebraic ones, and this connection between apparently unrelated fields is interesting to me. Despite the difference in 'feel' the two are (almost) the same.
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June 27th, 2014, 05:44 AM   #5
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That sound like a theorem I should look at, given my awful geometry skills.
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June 27th, 2014, 06:21 AM   #6
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Quote:
Originally Posted by v8archie View Post
That sound like a theorem I should look at, given my awful geometry skills.
Tarski–Seidenberg theorem - Wikipedia, the free encyclopedia

Tarski's original paper is here:
http://www.rand.org/content/dam/rand.../2008/R109.pdf
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