[math-fun] The bathroom square dance
To make the tiles really dance ... Danny Caligari ; https://www.google.com.au/amp/s/lamington.wordpress.com/2013/01/13/kenyons-s...
Nicely spotted --- thanks! WFL On 5/29/17, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote: Wikipedia has a good article on these tilings; Wikipedia.org/wiki/Pythagorean_tiling There are 4 names given: Pythagorean tiling Two squares tessellation Hopscotch pattern Pinwheel pattern (not to be confused with pinwheel tiling) The only other thing about these tilings I am aware of that is not mentioned in the article is that these tilings can tile a torus. See Geoffrey Morley's article www.squaring.net/sq/st/st.html To make the tiles really dance ... Danny Caligari; https://www.google.com.au/amp/s/lamington.wordpress.com/2013/01/13/kenyons-s... Stuart Anderson
Which suggests the question: For which N do there exist periodic tessellations of the plane using tiles that are squares of N distinct sizes? (Or else N distinct symmetry classes.) —Dan
On May 29, 2017, at 1:12 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Nicely spotted --- thanks! WFL
On 5/29/17, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote: Wikipedia has a good article on these tilings;
Wikipedia.org/wiki/Pythagorean_tiling There are 4 names given: Pythagorean tiling Two squares tessellation Hopscotch pattern Pinwheel pattern (not to be confused with pinwheel tiling)
The only other thing about these tilings I am aware of that is not mentioned in the article is that these tilings can tile a torus. See Geoffrey Morley's article www.squaring.net/sq/st/st.html
To make the tiles really dance ... Danny Caligari; https://www.google.com.au/amp/s/lamington.wordpress.com/2013/01/13/kenyons-s...
Dan, do you mean one square of each size (per fundamental domain)? Because if you can have any number, the answer is "all positive integers". Start with a single square, and subdivide it as needed. On Mon, May 29, 2017 at 4:36 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Which suggests the question: For which N do there exist periodic tessellations of the plane using tiles that are squares of N distinct sizes?
(Or else N distinct symmetry classes.)
—Dan
On May 29, 2017, at 1:12 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Nicely spotted --- thanks! WFL
On 5/29/17, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote: Wikipedia has a good article on these tilings;
Wikipedia.org/wiki/Pythagorean_tiling There are 4 names given: Pythagorean tiling Two squares tessellation Hopscotch pattern Pinwheel pattern (not to be confused with pinwheel tiling)
The only other thing about these tilings I am aware of that is not mentioned in the article is that these tilings can tile a torus. See Geoffrey Morley's article www.squaring.net/sq/st/st.html
To make the tiles really dance ... Danny Caligari; https://www.google.com.au/amp/s/lamington.wordpress.com/ 2013/01/13/kenyons-squarespirals/amp/
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Yes — when I said N distinct sizes I meant a net of N squares, of distinct sizes, per fundamental domain. Or else N squares of distinct symmetry classes (regardless of whether some or all of the squares are the same size). —Dan ---- * By a symmetry class is meant a maximal set of squares that can be taken to each other by a symmetry of the entire pattern. Even though more than one of those squares could belong to the same fundamental domain.
On May 29, 2017, at 1:55 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Dan, do you mean one square of each size (per fundamental domain)? Because if you can have any number, the answer is "all positive integers". Start with a single square, and subdivide it as needed.
On Mon, May 29, 2017 at 4:36 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Which suggests the question: For which N do there exist periodic tessellations of the plane using tiles that are squares of N distinct sizes?
(Or else N distinct symmetry classes.)
—Dan
On May 29, 2017, at 1:12 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Nicely spotted --- thanks! WFL
On 5/29/17, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote: Wikipedia has a good article on these tilings;
Wikipedia.org/wiki/Pythagorean_tiling There are 4 names given: Pythagorean tiling Two squares tessellation Hopscotch pattern Pinwheel pattern (not to be confused with pinwheel tiling)
The only other thing about these tilings I am aware of that is not mentioned in the article is that these tilings can tile a torus. See Geoffrey Morley's article www.squaring.net/sq/st/st.html
To make the tiles really dance ... Danny Caligari; https://www.google.com.au/amp/s/lamington.wordpress.com/ 2013/01/13/kenyons-squarespirals/amp/
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I think we can still get every positive N. A notched rectangle can tile the plane, regardless of the dimensions of the rectangle or the notch. I can make a notched rectangle with N distinct squares, for any N. On Mon, May 29, 2017 at 5:10 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Yes — when I said N distinct sizes I meant a net of N squares, of distinct sizes, per fundamental domain.
Or else N squares of distinct symmetry classes (regardless of whether some or all of the squares are the same size).
—Dan ---- * By a symmetry class is meant a maximal set of squares that can be taken to each other by a symmetry of the entire pattern. Even though more than one of those squares could belong to the same fundamental domain.
On May 29, 2017, at 1:55 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Dan, do you mean one square of each size (per fundamental domain)? Because if you can have any number, the answer is "all positive integers". Start with a single square, and subdivide it as needed.
On Mon, May 29, 2017 at 4:36 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Which suggests the question: For which N do there exist periodic tessellations of the plane using tiles that are squares of N distinct sizes?
(Or else N distinct symmetry classes.)
—Dan
On May 29, 2017, at 1:12 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Nicely spotted --- thanks! WFL
On 5/29/17, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote: Wikipedia has a good article on these tilings;
Wikipedia.org/wiki/Pythagorean_tiling There are 4 names given: Pythagorean tiling Two squares tessellation Hopscotch pattern Pinwheel pattern (not to be confused with pinwheel tiling)
The only other thing about these tilings I am aware of that is not mentioned in the article is that these tilings can tile a torus. See Geoffrey Morley's article www.squaring.net/sq/st/st.html
To make the tiles really dance ... Danny Caligari; https://www.google.com.au/amp/s/lamington.wordpress.com/ 2013/01/13/kenyons-squarespirals/amp/
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Can you do it with integers from 1 to N? I get stuck around 6. I recall a visual proof of the formula sum(K^2) = N(N+1)(2N+1)/6 which laid out 3 of each size square, fitting into a rectangle of dimensions N(N+1)/2 by 2N+1. It could be regarded as three tiles: two were proper, just the squares laid out consecutively, adjacent, along the Y axis, and a reflected copy. However, the third tile required dissecting each square into the L-shaped fringes from the sum-of-odd-numbers-is-a-square proof, flattening the fringes into odd-length stripes, and sorting and reassembling the stripes into a stack similar to the Empire State Building. This violates the tiling problem conditions. ccc a ccc ccc b ccc ccc ccc c ccc cc c bb bbb bb c c c bb ccc bb a ccccc a Rich --------- Quoting Allan Wechsler <acwacw@gmail.com>:
I think we can still get every positive N. A notched rectangle can tile the plane, regardless of the dimensions of the rectangle or the notch. I can make a notched rectangle with N distinct squares, for any N.
On Mon, May 29, 2017 at 5:10 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Yes ? when I said N distinct sizes I meant a net of N squares, of distinct sizes, per fundamental domain.
Or else N squares of distinct symmetry classes (regardless of whether some or all of the squares are the same size).
?Dan ---- * By a symmetry class is meant a maximal set of squares that can be taken to each other by a symmetry of the entire pattern. Even though more than one of those squares could belong to the same fundamental domain.
On May 29, 2017, at 1:55 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Dan, do you mean one square of each size (per fundamental domain)? Because if you can have any number, the answer is "all positive integers". Start with a single square, and subdivide it as needed.
On Mon, May 29, 2017 at 4:36 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Which suggests the question: For which N do there exist periodic tessellations of the plane using tiles that are squares of N distinct sizes?
(Or else N distinct symmetry classes.)
?Dan
On May 29, 2017, at 1:12 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Nicely spotted --- thanks! WFL
On 5/29/17, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote: Wikipedia has a good article on these tilings;
Wikipedia.org/wiki/Pythagorean_tiling There are 4 names given: Pythagorean tiling Two squares tessellation Hopscotch pattern Pinwheel pattern (not to be confused with pinwheel tiling)
The only other thing about these tilings I am aware of that is not mentioned in the article is that these tilings can tile a torus. See Geoffrey Morley's article www.squaring.net/sq/st/st.html
To make the tiles really dance ... Danny Caligari; https://www.google.com.au/amp/s/lamington.wordpress.com/ 2013/01/13/kenyons-squarespirals/amp/
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participants (5)
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Allan Wechsler -
Dan Asimov -
Fred Lunnon -
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Stuart Anderson