Easy to state and hard to solve problems make mathematics interesting. Packing problems are one such type. In fact there is a very nice Wikipedia article on this topic:

Packing problems involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible.

I came across this problem a couple of years ago, while watching the following TED Talk by Eduardo Sáenz de Cabezón:

In this talk, the object of attraction is the Weaire-Phelan structure made from six 14-hedrons and two dodechedrons. This object is believed to be the solution of Kelvin’s problem:

How you can chop 3d space into cells of equal volume with the minimum surface area per cell?

The packing problem for higher dimensions was in news last year, since a problem about “Densest Packing Problem in Dimensions 8 and 24” was solved by a young mathematician (Maryna Viazovska). The mathematics involved in the solution is very advanced but we can start gaining knowledge from this book:

A classic reference in this field by two well known geniuses.

Recently, while reading Matt Parker’s book, I discovered a wonderful website called Packomania by Eckard Specht (Otto-von-Guericke-Universität Magdeburg) containing data about packing problems in 2D and 3D. (also checkout his Math4u.de website, it’s a good reference for elementary triangle geometry and inequalities problems.)

Screen-shot of Dr. Eckard Specht’s homepage, his online problem collection (with solutions; each problem in GIF, PS and PDF formats) and Packomania.

No arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements.

The book by Stephen Wicker is meant chiefly for EE/CS majors. Whereas for mathematicians like you, there is a masterpiece “introduction to Coding Theory” by van Lint. It is quite dense !!!

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Packing problems of some type are used in error control codes (integral part of digital communications systems) .

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Can you please share a link where I can read more about it?

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If u can, refer to the book “Error Control Systems” by Stephen Wicker. It is a core concept in error control codes.

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The book by Stephen Wicker is meant chiefly for EE/CS majors. Whereas for mathematicians like you, there is a masterpiece “introduction to Coding Theory” by van Lint. It is quite dense !!!

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Thanks!

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