Let's discuss Mathematics

[Silv Something]

There's the classic natural fractal.

The Triangle's cool, but the Pyramid cooler.

 

[Spoonwood]

I could see something shaped "like" the Menger Sponge standing (maybe I'm wrong though). I couldn't see anything shaped "like" that pyramid standing (unless, of course, there's something else going on).

 

[peter grimes]

Ummm, isn't that a sunchoke?.. aka jerusalem artichoke?

 

[ParadigmShifter]

I don't think so...

Roots

Flower

 

[ThinkTank]

So here is a civ related problem (maybe quite easy, I don't know).

In this post ZzarkLinux described a size 13 tile that covers the plane. Picture:

Spoiler :

I marginally changed that to a 13 tile that has more symmetries and also covers the plane; a picture:

Spoiler :

My 13 tile has 3 symmetries: X-axis mirroring, Y-axis mirroring, and 90-degree rotation.

Problem: exactly describe all n such that there is an n-tile that covers the plane satisifying the same 3 symmetries.

 

[ParadigmShifter]

That looks antisymmetric with respect to x/y axis mirroring.

 

[ThinkTank]

I was thinking of the symmetry of the tile. So we ignore transformations of the form x+a, y+b.

 

[IdiotsOpposite]

Does Fubini's Theorem extend to functions in the complex n-plane?

Yes or no?

 

[ParadigmShifter]

What's a complex n-plane?

A vector in n complex numbers?

Presumably yes, because those are measurable sets.

 

[ThinkTank]

All squares and all tilted squares (1, 5, 13, 25, ... = n^2 + (n+1)^2) will do; are these all?

 

[ParadigmShifter]

S-shaped tetris pieces have the same symmetry group as the picture too I believe.

EDIT: Whoops, no 90degree rotational symmetry.

 

[IdiotsOpposite]

What's a complex n-plane?

A vector in n complex numbers?

Presumably yes, because those are measurable sets.

Yeah, that may have come out wrong. I mean like the complex plane is a complex number in 1 variable, so I thought the complex n-plane would be a complex number in 2 variables.

 

[ParadigmShifter]

That's the vector space Cⁿ

 

[IdiotsOpposite]

So you're still correct that Fubini's Theorem applies in Cⁿ, right?

 

[ParadigmShifter]

Not sure Integration and measure spaces aren't my strong point. Atticus should be able to help.

 

[Spoonwood]

How many squares are in the grid ThinkTank? Your picture has edges with lines which don't exactly describe squares. I also don't understand what you mean by symmetry of the tile. For that matter also, what do you mean by covering the plane?

I would say that we have several sets of thirteen tiles which will cover a plane with n tiles. So, we can't cover a plane with m tiles using that schematic where m is not a multiple of 13. It follows that the schematic should only work out purely as it does in theory in a game in rare cases.

I can see that we can change the sets of tiles which all look red, and consequently they come as symmetric in that respect. The same goes for the green tiles. If that's what you mean by symmetry, then several other sets can swap with each other. Also, if by symmetry you mean swapping sets of tiles like that, I think how many squares in the plane ends up mattering.

Sorry, I'm at a loss here. Could you explain?

 

[LulThyme]

That looks antisymmetric with respect to x/y axis mirroring.

The tiles have the symmetry, not the tiling.

 

[LulThyme]

How many squares are in the grid ThinkTank? Your picture has edges with lines which don't exactly describe squares. I also don't understand what you mean by symmetry of the tile. For that matter also, what do you mean by covering the plane?

I would say that we have several sets of thirteen tiles which will cover a plane with n tiles. So, we can't cover a plane with m tiles using that schematic where m is not a multiple of 13. It follows that the schematic should only work out purely as it does in theory in a game in rare cases.

I can see that we can change the sets of tiles which all look red, and consequently they come as symmetric in that respect. The same goes for the green tiles. If that's what you mean by symmetry, then several other sets can swap with each other. Also, if by symmetry you mean swapping sets of tiles like that, I think how many squares in the plane ends up mattering.

Sorry, I'm at a loss here. Could you explain?

He has an infinite square grid which he wants to tile with identical tiles. He also wants each individual tile to admit dihedral symmetry.

In other words, if you take a single tile, it is is preserved under some horizontal reflection, some vertical reflection and under a quarter-turn rotation around the intersection of the two reflection axis.

He conjectures that the only possibility is if an individual tile is either a nxn square, or what he calls a "tilted" square, which is the set of grid squares which are at most a given distance from a given square.



EDIT: here is a counter-example.
Take a "cross" made up of 5 squares. Then subdivide each square in 4, to get a tile made up 20 squares. This satisfies your hypothesis, but not your conclusion.

Of course, this yields a new family of examples, by taking a tilted square and subdividing it, which is no longer a tilted square by your definition. I am not sure if there are any more examples, I might think about it.

 

[LulThyme]

Obviously, starting with such a tile and subdividing squares into smaller squares will always yield such a tile.
So if n is one of your numbers, so is nm^2 for any m>0.

Call a tile "basic" if it cannot be subdivided in equal squares.

The only basic examples you have were the "tilted" squares, which give you (1,5,13,25,41...)

Here are some more basic examples.
First, the "even" tilted squares, for example,a 4x4 square with corners missing, or a 6x6 square with an l-shaped of size 3 missing in each corner, etc...
These give you integers of the form 2(n^2-n) : (4, 12, 24,40,...)

Here is another more interesting example. Take a 5x5 square, remove the 4 corners and the 4 squares that are exactly halfway between two corners.
That leaves a 17-square tile with dihedral symmetry which can tile the plane (not hard to see).

I think I found one with 29 (take a 3x3 square, and add a 5-cross touching each side).

So the list of basic n contains at least (1,4,5,12,13,17,24,25,29,40,41...)

As noted above, you can multiply by any square, so you get at least : (1,4,5,9,12,13,16,17,20,24,25,29,36,40,41,45,48,49...).

Finally, note that, because of the symmetry, any n will be either 0 or 1 mod 4, so there are not that many possible n's missing from the list already.

Using a bit of clever brute-force, it is not hard to see that there are none with 8, 21, 28 or 32, so the list above is correct up to at least 33.

I think if you assume that the packing is a lattice packing, you can probably make a bit more headway.


EDIT: note that neither 1,4,5,12,13,17 (the start of the basic sequence) 1,4,5,9,12,13,16 (the start of your sequence) appears in the online encyclopedia of integer sequences.

 

[Atticus]

Does Fubini's Theorem extend to functions in the complex n-plane?

If you use R^2 measure in C, it should, since then C^n=R^2n.