function construction - How to generate a series of bisection points of a given line segment?

I want to get the coordinates of the bisection points for a given line segment to a specified resolution. For example, when the line is given by {0,0} and {1,0}, the output coordinates should be {0.5,0} for the first iteration, and then {0.25,0} and {0.75,0} for the second iteration, and so on, until the distances between those points are less than some resolution, say 1/16. Is there a convenient way?

Answer

Here's a cute method based on the way Mathematica reduces fractions. Suppose you want to divide the line into 16ths:

n = 16;
x = Reverse @ GatherBy[Range[n - 1]/n, Denominator]

(* {{1/2}, {1/4, 3/4}, {1/8, 3/8, 5/8, 7/8}, 
  {1/16, 3/16, 5/16, 7/16, 9/16, 11/16, 13/16, 15/16}} *)

The actual coordinates can be obtained with:

line = {{0, 0}, {1, 0}};

Map[{1 - #, #}.line &, x, {-1}]

(* {{{1/2, 0}}, {{1/4, 0}, {3/4, 0}}, {{1/8, 0}, {3/8, 0}, {5/8, 0}, {7/8, 0}}, ... *)

Comments

front end - keyboard shortcut to invoke Insert new matrix

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plotting - How to draw lines between specified dots on ListPlot?

I would like to create a plot where I have unconnected dots and some connected. So far, I have figured out how to draw the dots. My code is the following: ListPlot[{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4,13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full] I have thought using ListLinePlot command, but I don't know how to specify to the command to draw only selected lines between the dots. Do have any suggestions/hints on how to do that? Thank you. Answer One possibility would be to use Epilog with Line : ListPlot[ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4, 13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full, Epilog -> { Line[ ...

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