Friday, 21 January 2011

Optimal 2D Tesselation Shape

So, for the inaugural post, I have a question I honestly thought about while on the toilet.
Motivation: Let us say that I am a manufacturer of tubing. I make custom cross-sectional-area tubes for my clients and due to a special extracting machine, I am capable of making them in any (prismatic) shape I like!

The Process: (In 2D space) You are given an arbitrary list of N areas (An). You must select a shape S with area one. You will then be given N shapes strictly similar to S but scaled to An. You will place these into a plane such that there is no overlap. Let M be the smallest rectangle that entirely contains all such placed shapes.

The Problem Statement: Select shape S such that no matter what list A you are given, you can guarantee a "good" (small) area of M.

The Question: What is the shape? What kinds of things can you say about the area of M? Does this answer change if we instead make M a bounding circle?

Please comment your thoughts below!

Right now I'm thinking rectangles in the golden ratio....


  1. You worry me. Your sanity worries me.

  2. this is a real problem in mathmatics: