8 Typo in one coordinate of example.

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): $$P^0 = (\; (5,7),(6,9),(6,3),(6,2) \;),$$ $$P^1 = (\; (5,8),(6,6),(6,2),(5,4) \;),$$ $$P^2 = (\; (5,7),(6,4),(5,3),(5,6) \;),$$ $$P^3 = (\; (5,5),(5,3),(5,4),(5,6) \;),$$ $$P^4 = (\; (5,4),(5,3),(5,5) \;),$$ $$P^5 = (\; (5,3),(5,5) 5,3),(5,4) \;),$$ $$P^6 = (\; (5,3) \;).$$

7 More corrections, to figures.

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): $$P^0 = (\; (5,7),(6,9),(6,3),(6,2) \;),$$ $$P^1 = (\; (5,8),(6,6),(6,2),(5,4) \;),$$ $$P^2 = (\; (5,7),(6,4),(5,3),(5,6) \;),$$ $$P^3 = (\; (5,5),(5,3),(5,4),(5,6) \;),$$ $$P^4 = (\; (5,4),(5,3),(5,5) \;),$$ $$P^5 = (\; (5,3),(5,5) \;),$$ $$P^6 = (\; (5,3) \;).$$

6 Corrected, again, as per Douglas's remarks.

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): $$P^0 = (\; (5,7),(6,9),(6,3),(6,2) \;),$$ $$P^1 = (\; (5,4),(5,8),(6,2),(6,6) 5,8),(6,6),(6,2),(5,4) \;),$$ $$P^2 = (\; (5,5),(5,6),(6,4) 5,7),(6,4),(5,3),(5,6) \;),$$ $$P^3 = (\; (5,4),(5,5) 5,5),(5,3),(5,4),(5,6) \;),$$ $$P^4 = (\; (5,4) 5,4),(5,3),(5,5) \;),$$ $$P^5 = (\; (5,3),(5,5) \;),$$ $$P^6 = (\; (5,3) \;).$$

5 added 9 characters in body
4 added 2 characters in body; edited body
3 Corrected as per Barry Cipra's comments.
2 Moved image.
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