Given a segment
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
of the length 1, define the set
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
of points in the
following way: it contains two points
![A,B,](/media/m/2/1/4/21452c07a542ecf62097510d7c00f7a9.png)
and also all points obtained from
![A,B](/media/m/7/1/7/7174f8a9f33236ee137c01b144237389.png)
by iterating the following rule: With every pair of points
![X,Y](/media/m/9/c/0/9c05fd567d1ce9c07fdc0f1f286c474a.png)
the set
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
contains also the point
![Z](/media/m/7/9/4/794ff2bd637e30ea27e50e57eecd0b76.png)
of the segment
![XY](/media/m/1/c/e/1ce2b6bc5783d5ee7b3276a845f41d6e.png)
for which
%V0
Given a segment $AB$ of the length 1, define the set $M$ of points in the
following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$