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#1 2006-10-22 22:52:24
- MathsIsFun
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Platonic Solids - Why Five?
Had some fun with graphics and explanations.
Have a read of Platonic Solids - Why Five?, and tell me if it makes sense (and if I made any mistakes 
)
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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#2 2006-10-23 00:19:20
- mathsyperson
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Re: Platonic Solids - Why Five?
Ooh, very nice. I'd never seen a proof of why there were only 5 Platonic solids before, so that was very interesting. And it all looks accurate and punctuated and stuff as well.
Why did the vector cross the road?
It wanted to be normal.
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#3 2006-10-25 14:05:51
- MathsIsFun
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Re: Platonic Solids - Why Five?
I have now added another page which talks more about Euler's Formula
I wasn't sure how to approach it, because "F+V-E=2" is often taught, so I thought "OK, let's do that, then show it isn't"
Good strategy? Or does it not flow very well?
And I am not sure if I am on solid ground when I say "that doesn't intersect itself". Convention is to just say "not concave", but a polyhedron can be concave and still have F+V-E=2. Anyone have any better knowledge on this?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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#4 2006-10-25 21:29:09
- mathsyperson
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Re: Platonic Solids - Why Five?
Wow. I'd never thought about any shape like those. I just knew that F+V-E=2 for any solids I tried it on, so assumed it was always true.
So does that mean that Euler's formula is useless now? F+V-E = something
Or can you define a set of properties so that any solid that meets them will make F+V-E=2 work?
Nice page though. Very interesting, and I couldn't find any mistakes. Although there might be some that can only be spotted by advanced topologists.
Why did the vector cross the road?
It wanted to be normal.
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#5 2006-10-25 21:39:14
- Devantè
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Re: Platonic Solids - Why Five?
Noticed one mistake;
Likewise if you included another vertex (say half way along a line)
you would get an extra edge, too.
Not really a mistake; Just that halfway, in this sense, would be a single word. 
No other mistakes are visible to me on that page. It certainly is very informative and gives the reader a huge idea of more on Platonic Solids. Nice work; Yet another document to add to the collection.
EDIT: I must be an advanced topologist, then. 
Last edited by Devanté (2006-10-25 21:40:12)
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#6 2006-10-25 23:57:58
- MathsIsFun
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Re: Platonic Solids - Why Five?
Or can you define a set of properties so that any solid that meets them will make F+V-E=2 work?
For all solids that can be "deformed" to the sphere it still holds. So that is an entire class of solids.
Another class of solids can be deformed to a torus, and for them F+V-E=1
I didn't go into on that page, but there is a 2D equivalent of F+V-E=2, and it is V-E=0
And the 1D version is V=2
In fact it goes like this for "non intersecting" shapes:
0D: 0
1D: V=2
2D: V-E=0
3D: F+V-E=2
4D: F+V-E-(something)=0
etc!
Each dimension needs a new parameter, and the sum goes 0,2,0,2,...
Perhaps I should mention that?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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#7 2006-10-26 01:20:20
- mathsyperson
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Re: Platonic Solids - Why Five?
I wouldn't. It's mightily interesting stuff, but it would probably just cause confusion. "What? But there isn't a 4th dimension! That makes no sense. 
" By all means, make a different page about higher dimensions, but talking about it here would have too many subjects on one page.
In my opinion, that is.
I must be an advanced topologist, then.
Not necessarily. Don't make me go Boolean on you.
Why did the vector cross the road?
It wanted to be normal.
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#8 2006-10-26 01:50:49
- Devantè
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Re: Platonic Solids - Why Five?
... Although there might be some that can only be spotted by advanced topologists ...
Sorry, I couldn't resist taking advantage of that statement.
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#9 2012-02-06 22:43:10
- MathsIsFun
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Re: Platonic Solids - Why Five?
UPDATE!
Added the first (much simpler) explanation. Does it read fine? Any mistooks?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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#10 2012-02-06 23:13:08
- bobbym
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Re: Platonic Solids - Why Five?
Hi MathsIsFun;
Reading the earlier posts, the update is clearer.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#11 2012-02-07 00:19:57
- Jai Ganesh
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Re: Platonic Solids - Why Five?
Hi MathsIsFun,
The explanation is clearer. Thanks!
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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