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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,626

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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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

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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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,626

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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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

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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**Devantè****Real Member**- Registered: 2006-07-14
- Posts: 6,400

Noticed one mistake;

Likewise if you included another vertex (say

half wayalong 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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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,626

mathsyperson wrote:

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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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

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.

Devanté wrote:

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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**Devantè****Real Member**- Registered: 2006-07-14
- Posts: 6,400

mathsyperson wrote:

... 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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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,626

UPDATE!

Added the first (much simpler) explanation. Does it read fine? Any mistooks?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,251

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.****No great discovery was ever made without a bold guess. **

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,298

Hi MathsIsFun,

The explanation is clearer. Thanks!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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