Why can't you build continuous tiles out of shapes beyond the hexagon?

[deo]

It occurred to me, while thinking about Civ 5, that you can't build tiles fit in perfectly with each other in a surface out of shapes beyond the hexagon, only the triangle, square, pentagon and the hexagon can do that.

So is there any mathematical explanation of why this is so?

 

[warpus]

It occurred to me, while thinking about Civ 5, that you can't build tiles fit in perfectly with each other in a surface out of shapes beyond the hexagon, only the triangle, square, pentagon and the hexagon can do that.

So is there any mathematical explanation of why this is so?

You can do it with a pentagon?

My initial knee-jerk response to this is that every single shape you listed (except the pentagon) has opposite sides that are parallel. Also except the triangle, but if you join 2 triangles together you get a parallelogram. You are also forgetting about the rectangle, which works too.

How do you know you can't do this with other shapes with more sides?

Disclaimer: this is only a knee jerk response, I haven' tthought about this too much, nor am I a shapeologist

 

[ParadigmShifter]

You can only tile a hyperbolic plane with a pentagon



All quadrilaterals (even non-regular ones) tessellate the plane, if the tiles are all the same shape and size of course. I haven't got a proof.

But you are correct that the hexagon is the only other type of shape that will tile a plane. Haven't got a proof of that either.

 

[Newbunkle]

A hexagon is just six triangles.

 

[deo]

You can do it with a pentagon?

My initial knee-jerk response to this is that every single shape you listed (except the pentagon) has opposite sides that are parallel. Also except the triangle, but if you join 2 triangles together you get a parallelogram. You are also forgetting about the rectangle, which works too.

How do you know you can't do this with other shapes with more sides?

Disclaimer: this is only a knee jerk response, I haven' tthought about this too much, nor am I a shapeologist

Sorry, but just to be clear, I meant if you took only that shapes that have equal lengths on all sides.

And yes, you can do it with a pentagon (think of an old soccer ball). And no, you can't do it with shapes beyond the hexagon, you can try it out but obviously you can't try every shape beyond the hexagon, that's why I'm asking for a general proof (if it indeed is not possible)

 

[ParadigmShifter]

Sorry, but just to be clear, I meant if you took only that shapes that have equal lengths on all sides.

And yes, you can do it with a pentagon (think of an old soccer ball). And no, you can't do it with shapes beyond the hexagon, you can try it out but obviously you can't try every shape beyond the hexagon, that's why I'm asking for a general proof (if it indeed is not possible)

Well you can tile a dodecahedron with pentagons.... you certainly can't tessellate a plane with them.

A soccer ball has hexagons as well as pentagons:



EDIT: If you are asking why the only regular convex solids are the 5 platonic ones (tetrahedron, cube, octahedron, dodecahedron, icosahedron) there is a proof of that I can find.

EDIT2: 2 proofs of that here

http://en.wikipedia.org/wiki/Platonic_solid

 

[deo]

Ok, I understand that. So I'll rephrase again, why can't you blla blla blla on a 'flat' surface out of equal sided, equal sized shapes.

 

[deo]

A hexagon is just six triangles.

That's a start

 

[Mise]

Not only can you perfectly tile a plane with hexagons, but the distance from the centre of one hexagon to the centres of each adjacent hexagon is the same for all hexagons. Can't do that with other shapes!

EDIT: I'm counting "touching at the vertices" as being "adjacent".

 

[Martacus]

A hexagon is just six triangles.

Are you sure? I could have sworn it was two quadrilaterals.

 

[onedreamer]

Ok, I understand that. So I'll rephrase again, why can't you blla blla blla on a 'flat' surface out of equal sided, equal sized shapes.

beacuse such is geometry?

 

[ParadigmShifter]

If adjacent means "shares an edge" and not "shares an edge or a vertex" then that applies to regular triangles too Mise.

EDIT: Meh Mise, smart alec

 

[Mise]

If adjacent means "shares an edge" and not "shares an edge or a vertex" then that applies to regular triangles too Mise.

If it means "shares an edge" and not "shares an edge or a vertex" then it applies to squares too

 

[ParadigmShifter]

Ok, I understand that. So I'll rephrase again, why can't you blla blla blla on a 'flat' surface out of equal sided, equal sized shapes.

Because the angles at a vertex have to add to 360 degrees. Only regular triangles (60 degrees), squares (90 degrees) and hexagons (120 degrees) can be placed side by side and have the angles sum to 360 degrees.

 

[deo]

beacuse such is geometry?

Yes but it has a reason, a relationship or something...

 

[ParadigmShifter]

See post #14

 

[deo]

Because the angles at a vertex have to add to 360 degrees. Only regular triangles (60 degrees), squares (90 degrees) and hexagons (120 degrees) can be placed side by side and have the angles sum to 360 degrees.

Ok at last! Now I get it, that's perfectly clear now.

 

[StittsvilleJame]

maybe it only works with triangles, and because rectangles and hexagons are both really multiple triangles stuck together in different ways, they work?

 

[Earthling]

Because the angles at a vertex have to add to 360 degrees. Only regular triangles (60 degrees), squares (90 degrees) and hexagons (120 degrees) can be placed side by side and have the angles sum to 360 degrees.

This is the best answer, accurate and understood easily by all, and a reason why we still need a like button.

 

[dwaxe]

Because the angles at a vertex have to add to 360 degrees. Only regular triangles (60 degrees), squares (90 degrees) and hexagons (120 degrees) can be placed side by side and have the angles sum to 360 degrees.

Why do they have to add up to 360 degrees in order to have tiles?