Figure out the lengths of edges
I try to recreate the following:
Here is the link.
http://tilings.math.uni-bielefeld.de/substitution/cyclotomic-trapezoids-9-fold/
This might help (formula is uncorrected formatted here cause I don't know how to do that):
each one the union of two triangles with edge lengths of the form
sin(k π/n).
So far I have figured out that there are 4 different lengths for the edges.
Here is a image of how P0 is split up:
a, b, c and d are edge lengths.
A, B, C etc. are corners.
The 2 long sides of shape P0 is a for example.
When splitting up a becomes a+a+c.
This is for all 4 edges:
a = a+a+c
b = b+d
c = a+c+d
d = b+c+d
They all depend on each other making it even harder.
Is there a way to figure the lengths out?
(more tags suggestion are welcome, I'm no math guru).
geometry
asked Nov 27 '18 at 16:05
clankill3r
1012
add a comment |
I try to recreate the following:
Here is the link.
http://tilings.math.uni-bielefeld.de/substitution/cyclotomic-trapezoids-9-fold/
This might help (formula is uncorrected formatted here cause I don't know how to do that):
each one the union of two triangles with edge lengths of the form
sin(k π/n).
So far I have figured out that there are 4 different lengths for the edges.
Here is a image of how P0 is split up:
a, b, c and d are edge lengths.
A, B, C etc. are corners.
The 2 long sides of shape P0 is a for example.
When splitting up a becomes a+a+c.
This is for all 4 edges:
a = a+a+c
b = b+d
c = a+c+d
d = b+c+d
They all depend on each other making it even harder.
Is there a way to figure the lengths out?
(more tags suggestion are welcome, I'm no math guru).
geometry
asked Nov 27 '18 at 16:05
clankill3r
1012
add a comment |
I try to recreate the following:
Here is the link.
http://tilings.math.uni-bielefeld.de/substitution/cyclotomic-trapezoids-9-fold/
This might help (formula is uncorrected formatted here cause I don't know how to do that):
each one the union of two triangles with edge lengths of the form
sin(k π/n).
So far I have figured out that there are 4 different lengths for the edges.
Here is a image of how P0 is split up:
a, b, c and d are edge lengths.
A, B, C etc. are corners.
The 2 long sides of shape P0 is a for example.
When splitting up a becomes a+a+c.
This is for all 4 edges:
a = a+a+c
b = b+d
c = a+c+d
d = b+c+d
They all depend on each other making it even harder.
Is there a way to figure the lengths out?
(more tags suggestion are welcome, I'm no math guru).
geometry
asked Nov 27 '18 at 16:05
clankill3r
1012
I try to recreate the following:
Here is the link.
http://tilings.math.uni-bielefeld.de/substitution/cyclotomic-trapezoids-9-fold/
This might help (formula is uncorrected formatted here cause I don't know how to do that):
each one the union of two triangles with edge lengths of the form
sin(k π/n).
So far I have figured out that there are 4 different lengths for the edges.
Here is a image of how P0 is split up:
a, b, c and d are edge lengths.
A, B, C etc. are corners.
The 2 long sides of shape P0 is a for example.
When splitting up a becomes a+a+c.
This is for all 4 edges:
a = a+a+c
b = b+d
c = a+c+d
d = b+c+d
They all depend on each other making it even harder.
Is there a way to figure the lengths out?
(more tags suggestion are welcome, I'm no math guru).
geometry
geometry
asked Nov 27 '18 at 16:05
clankill3r
1012
asked Nov 27 '18 at 16:05
clankill3r
1012
asked Nov 27 '18 at 16:05
clankill3r
1012
asked Nov 27 '18 at 16:05
clankill3r
1012
asked Nov 27 '18 at 16:05
clankill3r
1012
1012
add a comment |
add a comment |
2 Answers
2
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You might consider a limiting process. For that one just set up the substitution matrix
$$S=begin{pmatrix}
2 & 0 & 1 & 0\
0 & 1 & 0 & 1\
1 & 0 & 1 & 1\
0 & 1 & 1 & 1
end{pmatrix}$$
Then use for starting vector say $v=(1, 1, 1, 1)^T$ and apply $S$ to it over and over again, i.e. consider
$$lim_{ntoinfty}S^n v$$
- when being normalized on the way - that ought approximate the relative lengths.
--- rk
answered Nov 27 '18 at 20:05
Dr. Richard Klitzing
1,2966
I guess the Dr. in your name is well deserved.
– clankill3r
Nov 27 '18 at 21:27
add a comment |
The shorter answer already is contained within the cited website. There it states that the tiles have side length of the form $sin(pi k/n)$ and the title further states "9-fold", i.e. $n=9$.
--- rk
answered Nov 27 '18 at 20:12
Dr. Richard Klitzing
1,2966
Where does the k stand for?
– clankill3r
Nov 27 '18 at 21:26
add a comment |
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2 Answers
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active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
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active
oldest
votes
You might consider a limiting process. For that one just set up the substitution matrix
$$S=begin{pmatrix}
2 & 0 & 1 & 0\
0 & 1 & 0 & 1\
1 & 0 & 1 & 1\
0 & 1 & 1 & 1
end{pmatrix}$$
Then use for starting vector say $v=(1, 1, 1, 1)^T$ and apply $S$ to it over and over again, i.e. consider
$$lim_{ntoinfty}S^n v$$
- when being normalized on the way - that ought approximate the relative lengths.
--- rk
answered Nov 27 '18 at 20:05
Dr. Richard Klitzing
1,2966
I guess the Dr. in your name is well deserved.
– clankill3r
Nov 27 '18 at 21:27
add a comment |
You might consider a limiting process. For that one just set up the substitution matrix
$$S=begin{pmatrix}
2 & 0 & 1 & 0\
0 & 1 & 0 & 1\
1 & 0 & 1 & 1\
0 & 1 & 1 & 1
end{pmatrix}$$
Then use for starting vector say $v=(1, 1, 1, 1)^T$ and apply $S$ to it over and over again, i.e. consider
$$lim_{ntoinfty}S^n v$$
- when being normalized on the way - that ought approximate the relative lengths.
--- rk
answered Nov 27 '18 at 20:05
Dr. Richard Klitzing
1,2966
I guess the Dr. in your name is well deserved.
– clankill3r
Nov 27 '18 at 21:27
add a comment |
You might consider a limiting process. For that one just set up the substitution matrix
$$S=begin{pmatrix}
2 & 0 & 1 & 0\
0 & 1 & 0 & 1\
1 & 0 & 1 & 1\
0 & 1 & 1 & 1
end{pmatrix}$$
Then use for starting vector say $v=(1, 1, 1, 1)^T$ and apply $S$ to it over and over again, i.e. consider
$$lim_{ntoinfty}S^n v$$
- when being normalized on the way - that ought approximate the relative lengths.
--- rk
answered Nov 27 '18 at 20:05
Dr. Richard Klitzing
1,2966
You might consider a limiting process. For that one just set up the substitution matrix
$$S=begin{pmatrix}
2 & 0 & 1 & 0\
0 & 1 & 0 & 1\
1 & 0 & 1 & 1\
0 & 1 & 1 & 1
end{pmatrix}$$
Then use for starting vector say $v=(1, 1, 1, 1)^T$ and apply $S$ to it over and over again, i.e. consider
$$lim_{ntoinfty}S^n v$$
- when being normalized on the way - that ought approximate the relative lengths.
--- rk
answered Nov 27 '18 at 20:05
Dr. Richard Klitzing
1,2966
answered Nov 27 '18 at 20:05
Dr. Richard Klitzing
1,2966
answered Nov 27 '18 at 20:05
Dr. Richard Klitzing
1,2966
answered Nov 27 '18 at 20:05
Dr. Richard Klitzing
1,2966
1,2966
I guess the Dr. in your name is well deserved.
– clankill3r
Nov 27 '18 at 21:27
add a comment |
I guess the Dr. in your name is well deserved.
– clankill3r
Nov 27 '18 at 21:27
I guess the Dr. in your name is well deserved.
– clankill3r
Nov 27 '18 at 21:27
I guess the Dr. in your name is well deserved.
– clankill3r
Nov 27 '18 at 21:27
add a comment |
The shorter answer already is contained within the cited website. There it states that the tiles have side length of the form $sin(pi k/n)$ and the title further states "9-fold", i.e. $n=9$.
--- rk
answered Nov 27 '18 at 20:12
Dr. Richard Klitzing
1,2966
Where does the k stand for?
– clankill3r
Nov 27 '18 at 21:26
add a comment |
The shorter answer already is contained within the cited website. There it states that the tiles have side length of the form $sin(pi k/n)$ and the title further states "9-fold", i.e. $n=9$.
--- rk
answered Nov 27 '18 at 20:12
Dr. Richard Klitzing
1,2966
Where does the k stand for?
– clankill3r
Nov 27 '18 at 21:26
add a comment |
The shorter answer already is contained within the cited website. There it states that the tiles have side length of the form $sin(pi k/n)$ and the title further states "9-fold", i.e. $n=9$.
--- rk
answered Nov 27 '18 at 20:12
Dr. Richard Klitzing
1,2966
The shorter answer already is contained within the cited website. There it states that the tiles have side length of the form $sin(pi k/n)$ and the title further states "9-fold", i.e. $n=9$.
--- rk
answered Nov 27 '18 at 20:12
Dr. Richard Klitzing
1,2966
answered Nov 27 '18 at 20:12
Dr. Richard Klitzing
1,2966
answered Nov 27 '18 at 20:12
Dr. Richard Klitzing
1,2966
answered Nov 27 '18 at 20:12
Dr. Richard Klitzing
1,2966
1,2966
Where does the k stand for?
– clankill3r
Nov 27 '18 at 21:26
add a comment |
Where does the k stand for?
– clankill3r
Nov 27 '18 at 21:26
Where does the k stand for?
– clankill3r
Nov 27 '18 at 21:26
Where does the k stand for?
– clankill3r
Nov 27 '18 at 21:26
add a comment |
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