So, for the inaugural post, I have a question I honestly thought about while on the toilet.

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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!

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PS:

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

You worry me. Your sanity worries me.

ReplyDeletethis is a real problem in mathmatics: http://www2.stetson.edu/~efriedma/squinsqu/

ReplyDelete