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I am trying to create subdivision tilings inspired by the work of Brian Rushton (eg. page 77 of this paper). The challenge is to subdivide tiles according to a certain rule, and then apply this rule recursively to the result.

For example, a square might be subdivided by this rule which splits it into 5 more quadrilaterals. Each of these must then be subdivided again by the same rule, but since the tiles to subdivide are no longer square, a mapping function must be applied to find the vertex positions of the child tiles. What is the best mapping function to use? Here is an illustration of what I mean.

I think the definition of best is:

  • Vertices on the edges of the un-mapped tile end up on the edge of the mapped tile.
  • All vertices stay within the parent tile.
  • The appearance of the child tiles are distorted as little as possible.

I am developing a web app that makes this process interactive which is here, and you can try out the two functions that I have explored so far, but they both have issues. If you would like to experiment or contribute the source code is on github here!

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Have you looked at Cannon, Floyd & Parry's subdivide and tilepack programs ? https://personal.math.vt.edu/floyd/research/software/subdiv.html Designed to create input files for Ken Stephenson's CirclePack software http://www.circlepack.com/software.html

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  • $\begingroup$ Thanks for the suggestion, I think their software is the best way to do it :) I am wondering how to use it in my app, maybe I will need to read through the source code for that. I think it is also worth noting that due to the circle-packing approach, only convex tiles can be created with this method $\endgroup$ Commented Oct 9, 2023 at 22:19
  • $\begingroup$ I think you can define any shape you want as the Riemann Mapping Theorem applies $\endgroup$ Commented Oct 11, 2023 at 2:03

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