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Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of distinct points $P=P^0$ drawn from $\mathbb{N}^2$.
Define the midpoint of two points $a=(a_x,a_y)$ and $b=(b_x,b_y)$ to be
the point with coordinates
$$( \; \lfloor ( a_x + b_x ) / 2 \rfloor, \lfloor ( a_y + b_y ) / 2 \rfloor \; ). $$
Define $P^{k+1} =P^k \cup ($midpoints of $P^k )$ [strike that! instead:]
$P^{k+1} =($midpoints of $P^k )$
where by this notation I mean that the midpoints are interleaved between the
points of $P^k$, and then all duplicate points are removed to form $P^{k+1}$.
Thus, as $k$ increases, $|P_k|$ eventually reduces, I believe always down
to a single point $p^*$. Here are two examples, with $P^0$ the 20-point purple scribble,
and the last point marked in blue:
I would like to predict two aspects of this process, given $P^0$:
(1) The number of iterations to reach the final point $p^*$.
(2) The coordinates of $p^*$.
In the right example above, it took 39 iterations to reach $p^* = (20,7)$.
I had expected the number of iterations would be related to $\log_2 d_{max}$ where $d_{max}$ is
the largest coordinate difference between two adjacent points of $P^0$, but that is completely wrong
(in this example, $d_{max}=45$).
I am having difficulty analyzing this process.
Any ideas or literature pointers would be appreciated!
The same questions could be posed for points drawn from $\mathbb{N}^d$ for arbitrary $d$.
As the coordinate computations are independent, a key is understanding $d=1$.
Update. Prompted by Barry's question, I realize now (sorry!!!) I misdescribed
the process:
$P^{k+1} = ($midpoints of $P^k )$, not interleaved with $P^k$, but replacing $P^k$.
Here is a simple example, $n=4$ (now, I hope, corrected):
$$P^0 = (\; (5,7),(6,9),(6,3),(6,2) \;),$$
$$P^1 = (\; (5,8),(6,6),(6,2),(5,4) \;),$$
$$P^2 = (\; (5,7),(6,4),(5,3),(5,6) \;),$$
$$P^3 = (\; (5,5),(5,3),(5,4),(5,6) \;),$$
$$P^4 = (\; (5,4),(5,3),(5,5) \;),$$
$$P^5 = (\; (5,3),(5,5) 5,3),(5,4) \;),$$
$$P^6 = (\; (5,3) \;).$$
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7
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Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of distinct points $P=P^0$ drawn from $\mathbb{N}^2$.
Define the midpoint of two points $a=(a_x,a_y)$ and $b=(b_x,b_y)$ to be
the point with coordinates
$$( \; \lfloor ( a_x + b_x ) / 2 \rfloor, \lfloor ( a_y + b_y ) / 2 \rfloor \; ). $$
Define $P^{k+1} =P^k \cup ($midpoints of $P^k )$ [strike that! instead:]
$P^{k+1} =($midpoints of $P^k )$
where by this notation I mean that the midpoints are interleaved between the
points of $P^k$, and then all duplicate points are removed to form $P^{k+1}$.
Thus, as $k$ increases, $|P_k|$ eventually reduces, I believe always down
to a single point $p^*$. Here are two examples, with $P^0$ the 20-point purple scribble,
and the last point marked in blue:
I would like to predict two aspects of this process, given $P^0$:
(1) The number of iterations to reach the final point $p^*$.
(2) The coordinates of $p^*$.
In the right example above, it took 27 39 iterations to reach $p^* = (13,19)$.
20,7)$.
I had expected the number of iterations would be related to $\log_2 d_{max}$ where $d_{max}$ is
the largest coordinate difference between two adjacent points of $P^0$, but that is completely wrong
(in this example, $d_{max}=35$).
d_{max}=45$).
I am having difficulty analyzing this process.
Any ideas or literature pointers would be appreciated!
The same questions could be posed for points drawn from $\mathbb{N}^d$ for arbitrary $d$.
As the coordinate computations are independent, a key is understanding $d=1$.
Update. Prompted by Barry's question, I realize now (sorry!!!) I misdescribed
the process:
$P^{k+1} = ($midpoints of $P^k )$, not interleaved with $P^k$, but replacing $P^k$.
Here is a simple example, $n=4$ (now, I hope, corrected):
$$P^0 = (\; (5,7),(6,9),(6,3),(6,2) \;),$$
$$P^1 = (\; (5,8),(6,6),(6,2),(5,4) \;),$$
$$P^2 = (\; (5,7),(6,4),(5,3),(5,6) \;),$$
$$P^3 = (\; (5,5),(5,3),(5,4),(5,6) \;),$$
$$P^4 = (\; (5,4),(5,3),(5,5) \;),$$
$$P^5 = (\; (5,3),(5,5) \;),$$
$$P^6 = (\; (5,3) \;).$$
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6
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Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of distinct points $P=P^0$ drawn from $\mathbb{N}^2$.
Define the midpoint of two points $a=(a_x,a_y)$ and $b=(b_x,b_y)$ to be
the point with coordinates
$$( \; \lfloor ( a_x + b_x ) / 2 \rfloor, \lfloor ( a_y + b_y ) / 2 \rfloor \; ). $$
Define $P^{k+1} =P^k \cup ($midpoints of $P^k )$ [strike that! instead:]
$P^{k+1} =($midpoints of $P^k )$
where by this notation I mean that the midpoints are interleaved between the
points of $P^k$, and then all duplicate points are removed to form $P^{k+1}$.
Thus, as $k$ increases, $|P_k|$ eventually reduces, I believe always down
to a single point $p^*$. Here are two examples, with $P^0$ the 20-point purple scribble,
and the last point marked in blue:
I would like to predict two aspects of this process, given $P^0$:
(1) The number of iterations to reach the final point $p^*$.
(2) The coordinates of $p^*$.
In the right example above, it took 27 iterations to reach $p^* = (13,19)$.
I had expected the number of iterations would be related to $\log_2 d_{max}$ where $d_{max}$ is
the largest coordinate difference between two adjacent points of $P^0$, but that is completely wrong
(in this example, $d_{max}=35$).
I am having difficulty analyzing this process.
Any ideas or literature pointers would be appreciated!
The same questions could be posed for points drawn from $\mathbb{N}^d$ for arbitrary $d$.
As the coordinate computations are independent, a key is understanding $d=1$.
Update. Prompted by Barry's question, I realize now (sorry!!!) I misdescribed
the process:
$P^{k+1} = ($midpoints of $P^k )$, not interleaved with $P^k$, but replacing $P^k$.
Here is a simple example, $n=4$:
n=4$ (now, I hope, corrected):
$$P^0 = (\; (5,7),(6,9),(6,3),(6,2) \;),$$
$$P^1 = (\; (5,4),(5,8),(6,2),(6,6) 5,8),(6,6),(6,2),(5,4) \;),$$
$$P^2 = (\; (5,5),(5,6),(6,4) 5,7),(6,4),(5,3),(5,6) \;),$$
$$P^3 = (\; (5,4),(5,5) 5,5),(5,3),(5,4),(5,6) \;),$$
$$P^4 = (\; (5,4) 5,4),(5,3),(5,5) \;),$$
$$P^5 = (\; (5,3),(5,5) \;),$$
$$P^6 = (\; (5,3) \;).$$
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5
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Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of distinct points $P=P^0$ drawn from $\mathbb{N}^2$.
Define the midpoint of two points $a=(a_x,a_y)$ and $b=(b_x,b_y)$ to be
the point with coordinates
$$( \; \lfloor ( a_x + b_x ) / 2 \rfloor, \lfloor ( a_y + b_y ) / 2 \rfloor \; ). $$
Define $P^{k+1} =P^k \cup ($midpoints of $P^k )$ [strike that! instead:]
$P^{k+1} =($midpoints of $P^k )$
where by this notation I mean that the midpoints are interleaved between the
points of $P^k$, and then all duplicate points are removed to form $P^{k+1}$.
Thus, as $k$ increases, $|P_k|$ eventually reduces, I believe always down
to a single point $p^*$. Here are two examples, with $P^0$ the 20-point purple scribble,
and the last point marked in blue:
I would like to predict two aspects of this process, given $P^0$:
(1) The number of iterations to reach the final point $p^*$.
(2) The coordinates of $p^*$.
In the right example above, it took 27 iterations to reach $p^* = (13,19)$.
I had expected the number of iterations would be related to $\log_2 d_{max}$ where $d_{max}$ is
the largest coordinate difference between two adjacent points of $P^0$, but that is completely wrong
(in this example, $d_{max}=35$).
I am having difficulty analyzing this process.
Any ideas or literature pointers would be appreciated!
The same questions could be posed for points drawn from $\mathbb{N}^d$ for arbitrary $d$.
As the coordinate computations are independent, a key is understanding $d=1$.
Update. Prompted by Barry's question, I realize now (sorry!!!) I misdescribed
the process:
$P^{k+1} = ($midpoints of $P^k )$, not interleaved with $P^k$, but replacing $P^k$.
Here is a simple example, $n=4$:
$$P^0 = (\; (5,7),(6,9),(6,3),(6,2) \;),$$
$$P^1 = (\; (5,4),(5,8),(6,2),(6,6) \;),$$
$$P^2 = (\; (5,5),(5,6),(6,4) \;),$$
$$P^3 = (\; (5,4),(5,5) \;),$$
$$P^4 = (\; (5,4) \;).$$
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4
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Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of points $P=P^0$ drawn from $\mathbb{N}^2$.
Define the midpoint of two points $a=(a_x,a_y)$ and $b=(b_x,b_y)$ to be
the point with coordinates
$$( \; \lfloor ( a_x + b_x ) / 2 \rfloor, \lfloor ( a_y + b_y ) / 2 \rfloor \; ). $$
Define $P^{k+1} =P^k \cup ($midpoints of $P^k )$ [strike that! instead:]
$P^{k+1} =($midpoints of $P^k )$
where by this notation I mean that the midpoints are interleaved between the
points of $P^k$, and then all duplicate points are removed to form $P^{k+1}$.
Thus, as $k$ increases, $|P_k|$ eventually reduces, I believe always down
to a single point $p^*$. Here are two examples, with $P^0$ the 20-point purple scribble,
and the last point marked in blue:
I would like to predict two aspects of this process, given $P^0$:
(1) The number of iterations to reach the final point $p^*$.
(2) The coordinates of $p^*$.
In the right example above, it took 27 iterations to reach $p^* = (13,19)$.
I had expected the number of iterations would be related to $\log_2 d_{max}$ where $d_{max}$ is
the largest coordinate difference between two adjacent points of $P^0$, but that is completely wrong
(in this example, $d_{max}=35$).
I am having difficulty analyzing this process.
Any ideas or literature pointers would be appreciated!
The same questions could be posed for points drawn from $\mathbb{N}^d$ for arbitrary $d$.
As the coordinate computations are independent, a key is understanding $d=1$.
Update. Prompted by Barry's question, I realize now (sorry!!!) I misdescribed
the process:
$P^{k+1} = ($midpoints of $P^k )$, not interleaved with $P^k$, but replacing $P^k$.
Here is a simple example, $n=4$:
$$P^0 = (\; (5,7),(6,9),(6,3),(6,2) \;),$$
$$P^1 = (\; (5,4),(5,8),(6,2),(6,6) \;),$$
$$P^2 = (\; (5,5),(5,6),(6,4) \;),$$
$$P^3 = (\; (5,4),(5,5) \;),$$
$$P^4 = (\; (5,4) \;).$$
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3
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Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of points $P=P^0$ drawn from $\mathbb{N}^2$.
Define the midpoint of two points $a=(a_x,a_y)$ and $b=(b_x,b_y)$ to be
the point with coordinates
$$( \; \lfloor ( a_x + b_x ) / 2 \rfloor, \lfloor ( a_y + b_y ) / 2 \rfloor \; ). $$
Define $P^{k+1} =P^k \cup ($midpoints of $P^k )$,
$ [strike that! instead:]
$P^{k+1} =($midpoints of $P^k )$
where by this notation I mean that the midpoints are interleaved between the
points of $P^k$, and then all duplicate points are removed to form $P^{k+1}$.
Thus, as $k$ increases, $|P_k|$ eventually reduces, I believe always down
to a single point $p^*$. Here are two examples, with $P^0$ the 20-point purple scribble,
and the last point marked in blue:
I would like to predict two aspects of this process, given $P^0$:
(1) The number of iterations to reach the final point $p^*$.
(2) The coordinates of $p^*$.
In the right example above, it took 27 iterations to reach $p^* = (13,19)$.
I had expected the number of iterations would be related to $\log_2 d_{max}$ where $d_{max}$ is
the largest coordinate difference between two adjacent points of $P^0$, but that is completely wrong
(in this example, $d_{max}=35$).
I am having difficulty analyzing this process.
Any ideas or literature pointers would be appreciated!
The same questions could be posed for points drawn from $\mathbb{N}^d$ for arbitrary $d$.
As the coordinate computations are independent, a key is understanding $d=1$.
Update. Prompted by Barry's question, I realize now (sorry!!!) I misdescribed
the process:
$P^{k+1} = ($midpoints of $P^k )$, not interleaved with $P^k$, but replacing $P^k$.
Here is a simple example, $n=4$:
$$P^0 = (\; (5,7),(6,9),(6,3),(6,2) \;),$$
$$P^1 = (\; (5,4),(5,8),(6,2),(6,6) \;),$$
$$P^2 = (\; (5,5),(5,6),(6,4) \;),$$
$$P^3 = (\; (5,4),(5,5) \;),$$
$$P^4 = (\; (5,4) \;).$$
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2
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Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of points $P=P^0$ drawn from $\mathbb{N}^2$.
Define the midpoint of two points $a=(a_x,a_y)$ and $b=(b_x,b_y)$ to be
the point with coordinates
$$( \; \lfloor ( a_x + b_x ) / 2 \rfloor, \lfloor ( a_y + b_y ) / 2 \rfloor \; ). $$
Define $P^{k+1} = P^k \cup ($midpoints of $P^k )$,
where by this notation I mean that the midpoints are interleaved between the
points of $P^k$, and then all duplicate points are removed to form $P^{k+1}$.
Thus, as $k$ increases, $|P_k|$ eventually reduces, I believe always down
to a single point $p^*$. Here are two examples, with $P^0$ the 20-point purple scribble,
and the last point marked in blue:
I would like to predict two aspects of this process, given $P^0$:
(1) The number of iterations to reach the final point $p^*$.
(2) The coordinates of $p^*$.
In the right example above, it took 27 iterations to reach $p^* = (13,19)$.
I had expected the number of iterations would be related to $\log_2 d_{max}$ where $d_{max}$ is
the largest coordinate difference between two adjacent points of $P^0$, but that is completely wrong
(in this example, $d_{max}=35$).
I am having difficulty analyzing this process.
Any ideas or literature pointers would be appreciated!
The same questions could be posed for points drawn from $\mathbb{N}^d$ for arbitrary $d$.
As the coordinate computations are independent, a key is understanding $d=1$.
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1
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Midpoint lattice polygons
Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of points $P=P^0$ drawn from $\mathbb{N}^2$.
Define the midpoint of two points $a=(a_x,a_y)$ and $b=(b_x,b_y)$ to be
the point with coordinates
$$( \; \lfloor ( a_x + b_x ) / 2 \rfloor, \lfloor ( a_y + b_y ) / 2 \rfloor \; ). $$
Define $P^{k+1} = P^k \cup ($midpoints of $P^k )$,
where by this notation I mean that the midpoints are interleaved between the
points of $P^k$, and then all duplicate points are removed to form $P^{k+1}$.
Thus, as $k$ increases, $|P_k|$ eventually reduces, I believe always down
to a single point $p^*$. Here are two examples, with $P^0$ the 20-point purple scribble,
and the last point marked in blue:
I would like to predict two aspects of this process, given $P^0$:
(1) The number of iterations to reach the final point $p^*$.
(2) The coordinates of $p^*$.
In the right example above, it took 27 iterations to reach $p^* = (13,19)$.
I had expected the number of iterations would be related to $\log_2 d_{max}$ where $d_{max}$ is
the largest coordinate difference between two adjacent points of $P^0$, but that is completely wrong
(in this example, $d_{max}=35$).
I am having difficulty analyzing this process.
Any ideas or literature pointers would be appreciated!
The same questions could be posed for points drawn from $\mathbb{N}^d$ for arbitrary $d$.
As the coordinate computations are independent, a key is understanding $d=1$.
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