michel123456

Lattice geometry of Graphene- Split from Today I Learned

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On 1/2/2019 at 1:20 PM, Strange said:

Today I learned that graphene can self repair:

https://en.wikipedia.org/wiki/Graphene

Hexagonal pattern works well for surfaces (2D).

However in 3D I wonder what kind of "hexagonal pattern" it really is. The "chicken wire" pattern shown in Wiki is a mix of hexagons & pentagons.

Fullerene-C60.png

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28 minutes ago, michel123456 said:

Hexagonal pattern works well for surfaces (2D).

However in 3D I wonder what kind of "hexagonal pattern" it really is. The "chicken wire" pattern shown in Wiki is a mix of hexagons & pentagons.

Yes, you can only tile a flat surface with hexagons. For curved surfaces you either need a mixture of different, possibly irregular, polygons or you can use triangles

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22 minutes ago, Strange said:

Yes, you can only tile a flat surface with hexagons

 

I'm sorry run that past me again.?

 

 

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1 hour ago, studiot said:

I'm sorry run that past me again.?

Ah. I see the ambiguity now. Hexagons can only tile a flat surface, is what I meant. 

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7 minutes ago, Strange said:

Ah. I see the ambiguity now. Hexagons can only tile a flat surface, is what I meant. 

I think you should have added: "...of the same size".

G%C3%A9ode_V_3_1_duale.gif                                                      latest?cb=20131231013931&sp=690b61d8a661

Edited by StringJunky

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8 minutes ago, StringJunky said:

I think you should have added: "...of the same size".

Those use a mixture of hexagons and pentagons

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10 minutes ago, StringJunky said:

I think you should have added: "...of the same size".

Doesn't your pictures mix pentagons and hexagons? So that:

14 minutes ago, Strange said:

Hexagons can only tile a flat surface

should be something like: using equally sized hexagons only one can only tile a flat surface :-)

 

Edited by Ghideon
x-posted with @strange

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12 minutes ago, StringJunky said:

I think you should have added: "...of the same size".

G%C3%A9ode_V_3_1_duale.gif                                                      latest?cb=20131231013931&sp=690b61d8a661

 

Screen Shot 01-21-19 at 01.30 PM.JPG

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2 hours ago, Strange said:

Those use a mixture of hexagons and pentagons

 

2 hours ago, Ghideon said:

Doesn't your pictures mix pentagons and hexagons? So that:

should be something like: using equally sized hexagons only one can only tile a flat surface :-)

 

Duh!  :)

2 hours ago, michel123456 said:

 

Screen Shot 01-21-19 at 01.30 PM.JPG

 

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Hexagons can only tile an infinite flat surface.

They can't tile say for instance the top of a chocolate box (yum).

What I was referring to was that triangles and squares and some rectangles can (and of course straight lines, though that is usually referred to as ruling not tiling, though the process is identical)

Of course if you allow more than one shape onto the pitch (or bend the pitch) you are into Escher and Penrose tiling.

Edited by studiot

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I learned something today. I knew about Escher but I didn't know about Penrose tiling.

 

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19 hours ago, studiot said:

Hexagons can only tile an infinite flat surface.

Or a a finite surface with uneven edges ...

Arar_graphene_graphene_.jpg.c3553e548a6e7183fc36a56174d70a1c.jpg

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On 1/21/2019 at 3:09 PM, studiot said:

Hexagons can only tile an infinite flat surface.

Is the "opposite"* possible? By that I mean is it possible to tile a flat, finite, surface with infinitesimally* small hexagons?

 

On 1/21/2019 at 3:09 PM, studiot said:

They can't tile say for instance the top of a chocolate box (yum).

Is it still called "tiling" if only one tile is required?

Hexagonal-Box.jpg.2f412bd0255e04e988e4371dda0999ef.jpg

 

 

*) I do not know if this is the correct term (or if the question makes sense, I haven't yet studied more than entry level limits)

Edited by Ghideon
2nd & 3:rd questions corrected

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52 minutes ago, Ghideon said:

Is the "opposite"* possible? By that I mean is it possible to tile a flat, finite, surface with infinitesimally* small hexagons?

 

Is it still called "tiling" if only one tile is required?

Hexagonal-Box.jpg.2f412bd0255e04e988e4371dda0999ef.jpg

 

 

*) I do not know if this is the correct term (or if the question makes sense, I haven't yet studied more than entry level limits)

 

Well yes if the hexagons were infinitesimal, I don't see it would matter what shape they were, you would need an infinity of them(even for a finite area) and be into topological continuity and one of the analytical covering theorems.

https://www.google.co.uk/search?source=hp&ei=HpdHXKWnJ6mCjLsPj5eXiAM&q=mathematical+covering+theorem&btnK=Google+Search&oq=mathematical+covering+theorem&gs_l=psy-ab.3..33i160.2921.10711..11155...0.0..0.205.2192.24j4j1......0....1..gws-wiz.....0..0i131j0j0i10j0i22i30j0i22i10i30j33i21.LOeUaUKhbHI

 

Yummy, Belgian Chocolate.

I admit here I was just thinking of straight lines and boring British rectangular boxes.

But yes one tile is sufficient. 

 

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9 hours ago, studiot said:

Well yes if the hexagons were infinitesimal, I don't see it would matter what shape they were, you would need an infinity of them(even for a finite area) and be into topological continuity and one of the analytical covering theorems.

Good answer, +1. It also shows where further studies are required on my part, if I want to undersand more of the details.

 

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On 1/22/2019 at 11:26 PM, Ghideon said:

Is the "opposite"* possible? By that I mean is it possible to tile a flat, finite, surface with infinitesimally* small hexagons?

 

Is it still called "tiling" if only one tile is required?

Hexagonal-Box.jpg.2f412bd0255e04e988e4371dda0999ef.jpg

 

 

*) I do not know if this is the correct term (or if the question makes sense, I haven't yet studied more than entry level limits)

Arar_graphene_graphene_2.jpg.c821abe7ed49cb7f2b88209f330212af.jpg

You cannot completely fill an hexagon with smaller hexagons. As stated above by Strange, the shape has uneven edges.

When you reach infinity the blue areas get infinitely small but they are also infinitely many. Intuitively their area should reach zero.

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