Message Boards Message Boards


[GIF] Step Out (Recursive triangle morph)

Posted 1 year ago
1 Reply
5 Total Likes

Recursive triangle morph

Step Out

A follow-up to Advance. In Advance I recursively defined a nested sequence of squares by having the midpoint of each edge of the previous square be a vertex of the next square. Of course, there's no reason that the vertex at the next level has to be at the midpoint: we can put it at any fixed ratio of the distance along an edge and get a whirl.

The other part of Advance was to have each midpoint move radially outward until the convex hull of the original vertices and the scaled midpoints formed a new square. As @Sander Huisman pointed out in the comments, one could do the same for any $n$-gon and get a nice periodic motion.

We could just as well form the nested collection by putting the vertices at the next level at any fraction $w$ of the way along the edges of the previous level, and we could use that same point along the edge as the point to scale up to get new $n$-gons.

In general, this does not give a periodic motion, but for certain special values of $w$ it does. When $n=3$, the values of $w$ that give periodic motion seem to be $1/2$, $1/3$, and $\frac{1}{2}(1-\frac{1}{\sqrt{3}})$. The animation above is the one you get from $w=\frac{1}{2}(1-\frac{1}{\sqrt{3}})$.

Here's the code (notice that in the code I'm using $2w$ in the definition of WeightedMidpoints):

smootherstep[x_] := 6 x^5 - 15 x^4 + 10 x^3;

WeightedMidpoints[pts_, w_] := 
 Mean[({w #, (2 - w) RotateLeft[#]} &)[pts]]

 {n = 3, w = 1 - 1/Sqrt[3], depth = 100, 
  cols = RGBColor /@ {"#537780", "#FFFCCA"}, pts, t},
  pts = Nest[2 WeightedMidpoints[#, w] &, 2 Sqrt[2.] CirclePoints[n],  Floor[s]];
  t = smootherstep[Mod[s, 1]];
       {FaceForm[If[OddQ[i], cols[[1]], cols[[2]]]], Polygon[Riffle[#[[i]], (1 + t) WeightedMidpoints[#[[i]], w]]]},
       {i, 1, Length[#]}] &@NestList[WeightedMidpoints[#, w] &, pts, depth]},
   PlotRange -> .25, ImageSize -> 540],
  {s, 0, 4}]

enter image description here - Congratulations! This post is now Staff Pick! Thank you for your wonderful contributions. Please, keep them coming!

Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
or Discard

Group Abstract Group Abstract