[math-fun] Pick formulas generalized to noncubic lattices: trivial
From: Joerg Arndt <arndt@jjj.de> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Pick theorems
Pick's theorem (cf. http://en.wikipedia.org/wiki/Pick's_theorem):
The area A of a simple polygon with all corners on a square grid is A = i + b/2 - 1 where i is the number of lattice points in the interior and b is the number of lattice points on the boundary.
For the following A is the area as number of unit cells covered.
Triangular lattice: A = 2*i + b - 2
Hexagonal lattice: A = i/2 + b/4 - 1/2
Square grid, every second column shifted by a half unit: A = i/2 + b/2 - 1 where b counts only the boundary points whose local neighborhood is not concave (i.e., 90 degrees of outside, 270 degrees of inside).
I obtained these scribbling on a train ride yesterday.
Best, jj
--it seems to me that any "Pick theorem" (in any space-dimension D) can be trivially generalized to work for a lattice L which differs from the plain integer lattice Z^D by just linearly transforming. Whatever lattice L you have, it is just an invertible linear transformation of D-space of the plain Z^D lattice. Wlog L is scales so its voronoi cells have unit D-volume in which case the linear transform is volume-preserving. Now note, the polygon (or polytope) counts of lattice points within, on vertices, most geenrally on k-faces (0<=k<=D), all are preserved by the linear transformation. Also, the volume is preserved (as we just postulated wlog). Hence, any Pick formula should generalize to work for any lattice. QED
Is there a Pick formula that works in 3D? --Rich ------ Quoting Warren D Smith <warren.wds@gmail.com>:
From: Joerg Arndt <arndt@jjj.de> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Pick theorems
Pick's theorem (cf. http://en.wikipedia.org/wiki/Pick's_theorem):
The area A of a simple polygon with all corners on a square grid is A = i + b/2 - 1 where i is the number of lattice points in the interior and b is the number of lattice points on the boundary.
For the following A is the area as number of unit cells covered.
Triangular lattice: A = 2*i + b - 2
Hexagonal lattice: A = i/2 + b/4 - 1/2
Square grid, every second column shifted by a half unit: A = i/2 + b/2 - 1 where b counts only the boundary points whose local neighborhood is not concave (i.e., 90 degrees of outside, 270 degrees of inside).
I obtained these scribbling on a train ride yesterday.
Best, jj
--it seems to me that any "Pick theorem" (in any space-dimension D) can be trivially generalized to work for a lattice L which differs from the plain integer lattice Z^D by just linearly transforming.
Whatever lattice L you have, it is just an invertible linear transformation of D-space of the plain Z^D lattice. Wlog L is scales so its voronoi cells have unit D-volume in which case the linear transform is volume-preserving.
Now note, the polygon (or polytope) counts of lattice points within, on vertices, most geenrally on k-faces (0<=k<=D), all are preserved by the linear transformation. Also, the volume is preserved (as we just postulated wlog).
Hence, any Pick formula should generalize to work for any lattice. QED
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No, there are counterexamples to Pick in 3D. Charles Greathouse Analyst/Programmer Case Western Reserve University On Mon, Oct 21, 2013 at 2:15 PM, <rcs@xmission.com> wrote:
Is there a Pick formula that works in 3D? --Rich
------
Quoting Warren D Smith <warren.wds@gmail.com>:
From: Joerg Arndt <arndt@jjj.de>
To: math-fun <math-fun@mailman.xmission.com**> Subject: [math-fun] Pick theorems
Pick's theorem (cf. http://en.wikipedia.org/wiki/**Pick's_theorem<http://en.wikipedia.org/wiki/Pick's_theorem>
):
The area A of a simple polygon with all corners on a square grid is A = i + b/2 - 1 where i is the number of lattice points in the interior and b is the number of lattice points on the boundary.
For the following A is the area as number of unit cells covered.
Triangular lattice: A = 2*i + b - 2
Hexagonal lattice: A = i/2 + b/4 - 1/2
Square grid, every second column shifted by a half unit: A = i/2 + b/2 - 1 where b counts only the boundary points whose local neighborhood is not concave (i.e., 90 degrees of outside, 270 degrees of inside).
I obtained these scribbling on a train ride yesterday.
Best, jj
--it seems to me that any "Pick theorem" (in any space-dimension D) can be trivially generalized to work for a lattice L which differs from the plain integer lattice Z^D by just linearly transforming.
Whatever lattice L you have, it is just an invertible linear transformation of D-space of the plain Z^D lattice. Wlog L is scales so its voronoi cells have unit D-volume in which case the linear transform is volume-preserving.
Now note, the polygon (or polytope) counts of lattice points within, on vertices, most geenrally on k-faces (0<=k<=D), all are preserved by the linear transformation. Also, the volume is preserved (as we just postulated wlog).
Hence, any Pick formula should generalize to work for any lattice. QED
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participants (3)
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Charles Greathouse -
rcs＠xmission.com -
Warren D Smith