Pythagoras tree bounding size
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The Pythagoras tree is a fractal generated by squares. For each square, two new smaller squares are constructed and connected by their corners to the original square. The angle of the triangle formed can be varied.
The problem is finding the size of the bounding box (to a certain precision) for this fractal. The graph is essentially a rooted tree. Each node represents a square with a certain size and orientation. The tree is infinitely deep. For a certain precision, the problem is easily solved with a computer using branch and bound. I want to know if there are any estimates or hard limits on the size. Any bounds on the area of the fractal are also appreciated.
The isosceles right triangle case gives a nice bounding box of $6 times 4$, which can be calculated easily with a geometric series. Other cases are more difficult for me (the maximal tree paths follow a zigzag pattern - for a while).
limits trees fractals
edited Dec 24 '18 at 9:33
Glorfindel
3,41381930
asked May 28 '16 at 2:10
qwrqwr
6,69242755
$endgroup$
|
show 2 more comments
$begingroup$
The Pythagoras tree is a fractal generated by squares. For each square, two new smaller squares are constructed and connected by their corners to the original square. The angle of the triangle formed can be varied.
The problem is finding the size of the bounding box (to a certain precision) for this fractal. The graph is essentially a rooted tree. Each node represents a square with a certain size and orientation. The tree is infinitely deep. For a certain precision, the problem is easily solved with a computer using branch and bound. I want to know if there are any estimates or hard limits on the size. Any bounds on the area of the fractal are also appreciated.
The isosceles right triangle case gives a nice bounding box of $6 times 4$, which can be calculated easily with a geometric series. Other cases are more difficult for me (the maximal tree paths follow a zigzag pattern - for a while).
limits trees fractals
edited Dec 24 '18 at 9:33
Glorfindel
3,41381930
asked May 28 '16 at 2:10
qwrqwr
6,69242755
$endgroup$
1
$begingroup$
It seems there's no upper bound on the height of such boxes, considering the sequence of fractals where the right triangle becomes closer and closer to degenerate.
$endgroup$
– Greg Martin
May 28 '16 at 22:07
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@GregMartin I mean for a certain case of triangle (ex. 30-60-90)
$endgroup$
– qwr
May 28 '16 at 22:44
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You asked several questions; I answered one of them.
$endgroup$
– Greg Martin
May 29 '16 at 0:16
1
$begingroup$
Are you doing projecteuler.net/problem=395 ? In order not to ruin the fun, I will only give you a hint: at the $i$-th iteration, although $2^i$ squares are added, there are only $i + 1$ different shapes and positions.
$endgroup$
– WhatsUp
May 30 '16 at 15:10
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Not just the width and the height, also the position of the first square within the bounding box would be nice to know.
$endgroup$
– Paul
May 31 '16 at 18:28
|
show 2 more comments
$begingroup$
The Pythagoras tree is a fractal generated by squares. For each square, two new smaller squares are constructed and connected by their corners to the original square. The angle of the triangle formed can be varied.
The problem is finding the size of the bounding box (to a certain precision) for this fractal. The graph is essentially a rooted tree. Each node represents a square with a certain size and orientation. The tree is infinitely deep. For a certain precision, the problem is easily solved with a computer using branch and bound. I want to know if there are any estimates or hard limits on the size. Any bounds on the area of the fractal are also appreciated.
The isosceles right triangle case gives a nice bounding box of $6 times 4$, which can be calculated easily with a geometric series. Other cases are more difficult for me (the maximal tree paths follow a zigzag pattern - for a while).
limits trees fractals
edited Dec 24 '18 at 9:33
Glorfindel
3,41381930
asked May 28 '16 at 2:10
qwrqwr
6,69242755
$endgroup$
The Pythagoras tree is a fractal generated by squares. For each square, two new smaller squares are constructed and connected by their corners to the original square. The angle of the triangle formed can be varied.
The problem is finding the size of the bounding box (to a certain precision) for this fractal. The graph is essentially a rooted tree. Each node represents a square with a certain size and orientation. The tree is infinitely deep. For a certain precision, the problem is easily solved with a computer using branch and bound. I want to know if there are any estimates or hard limits on the size. Any bounds on the area of the fractal are also appreciated.
The isosceles right triangle case gives a nice bounding box of $6 times 4$, which can be calculated easily with a geometric series. Other cases are more difficult for me (the maximal tree paths follow a zigzag pattern - for a while).
limits trees fractals
limits trees fractals
edited Dec 24 '18 at 9:33
Glorfindel
3,41381930
asked May 28 '16 at 2:10
qwrqwr
6,69242755
edited Dec 24 '18 at 9:33
Glorfindel
3,41381930
asked May 28 '16 at 2:10
qwrqwr
6,69242755
edited Dec 24 '18 at 9:33
Glorfindel
3,41381930
edited Dec 24 '18 at 9:33
Glorfindel
3,41381930
edited Dec 24 '18 at 9:33
Glorfindel
3,41381930
3,41381930
asked May 28 '16 at 2:10
qwrqwr
6,69242755
asked May 28 '16 at 2:10
qwrqwr
6,69242755
asked May 28 '16 at 2:10
qwrqwr
6,69242755
6,69242755
1
$begingroup$
It seems there's no upper bound on the height of such boxes, considering the sequence of fractals where the right triangle becomes closer and closer to degenerate.
$endgroup$
– Greg Martin
May 28 '16 at 22:07
$begingroup$
@GregMartin I mean for a certain case of triangle (ex. 30-60-90)
$endgroup$
– qwr
May 28 '16 at 22:44
$begingroup$
You asked several questions; I answered one of them.
$endgroup$
– Greg Martin
May 29 '16 at 0:16
1
$begingroup$
Are you doing projecteuler.net/problem=395 ? In order not to ruin the fun, I will only give you a hint: at the $i$-th iteration, although $2^i$ squares are added, there are only $i + 1$ different shapes and positions.
$endgroup$
– WhatsUp
May 30 '16 at 15:10
$begingroup$
Not just the width and the height, also the position of the first square within the bounding box would be nice to know.
$endgroup$
– Paul
May 31 '16 at 18:28
|
show 2 more comments
1
$begingroup$
It seems there's no upper bound on the height of such boxes, considering the sequence of fractals where the right triangle becomes closer and closer to degenerate.
$endgroup$
– Greg Martin
May 28 '16 at 22:07
$begingroup$
@GregMartin I mean for a certain case of triangle (ex. 30-60-90)
$endgroup$
– qwr
May 28 '16 at 22:44
$begingroup$
You asked several questions; I answered one of them.
$endgroup$
– Greg Martin
May 29 '16 at 0:16
1
$begingroup$
Are you doing projecteuler.net/problem=395 ? In order not to ruin the fun, I will only give you a hint: at the $i$-th iteration, although $2^i$ squares are added, there are only $i + 1$ different shapes and positions.
$endgroup$
– WhatsUp
May 30 '16 at 15:10
$begingroup$
Not just the width and the height, also the position of the first square within the bounding box would be nice to know.
$endgroup$
– Paul
May 31 '16 at 18:28
1
1
$begingroup$
It seems there's no upper bound on the height of such boxes, considering the sequence of fractals where the right triangle becomes closer and closer to degenerate.
$endgroup$
– Greg Martin
May 28 '16 at 22:07
$begingroup$
It seems there's no upper bound on the height of such boxes, considering the sequence of fractals where the right triangle becomes closer and closer to degenerate.
$endgroup$
– Greg Martin
May 28 '16 at 22:07
$begingroup$
@GregMartin I mean for a certain case of triangle (ex. 30-60-90)
$endgroup$
– qwr
May 28 '16 at 22:44
$begingroup$
@GregMartin I mean for a certain case of triangle (ex. 30-60-90)
$endgroup$
– qwr
May 28 '16 at 22:44
$begingroup$
You asked several questions; I answered one of them.
$endgroup$
– Greg Martin
May 29 '16 at 0:16
$begingroup$
You asked several questions; I answered one of them.
$endgroup$
– Greg Martin
May 29 '16 at 0:16
1
1
$begingroup$
Are you doing projecteuler.net/problem=395 ? In order not to ruin the fun, I will only give you a hint: at the $i$-th iteration, although $2^i$ squares are added, there are only $i + 1$ different shapes and positions.
$endgroup$
– WhatsUp
May 30 '16 at 15:10
$begingroup$
Are you doing projecteuler.net/problem=395 ? In order not to ruin the fun, I will only give you a hint: at the $i$-th iteration, although $2^i$ squares are added, there are only $i + 1$ different shapes and positions.
$endgroup$
– WhatsUp
May 30 '16 at 15:10
$begingroup$
Not just the width and the height, also the position of the first square within the bounding box would be nice to know.
$endgroup$
– Paul
May 31 '16 at 18:28
$begingroup$
Not just the width and the height, also the position of the first square within the bounding box would be nice to know.
$endgroup$
– Paul
May 31 '16 at 18:28
|
show 2 more comments
0
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1
$begingroup$
It seems there's no upper bound on the height of such boxes, considering the sequence of fractals where the right triangle becomes closer and closer to degenerate.
$endgroup$
– Greg Martin
May 28 '16 at 22:07
$begingroup$
@GregMartin I mean for a certain case of triangle (ex. 30-60-90)
$endgroup$
– qwr
May 28 '16 at 22:44
$begingroup$
You asked several questions; I answered one of them.
$endgroup$
– Greg Martin
May 29 '16 at 0:16
1
$begingroup$
Are you doing projecteuler.net/problem=395 ? In order not to ruin the fun, I will only give you a hint: at the $i$-th iteration, although $2^i$ squares are added, there are only $i + 1$ different shapes and positions.
$endgroup$
– WhatsUp
May 30 '16 at 15:10
$begingroup$
Not just the width and the height, also the position of the first square within the bounding box would be nice to know.
$endgroup$
– Paul
May 31 '16 at 18:28