Teboho Tsotetsi
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Alien machinery

That's what it looks like to me.  But it's an image created by Greg Egan, the science fiction author.   And there's a story behind it.

Egan and I figured out a bunch of stuff about the McGee graph, a highly symmetrical graph with 24 vertices and 36 edges.   I wrote an article about it on Visual Insight, my blog for beautiful math pictures.

Later I got an email from Ed Pegg, Jr saying he'd worked out a unit-distance embedding of the McGee graph: a way of drawing it in the plane so that any two vertices connected by an edge are distance 1 apart.  He wanted to know if this was rigid or flexible.  In other words, he wanted to know whether you can change its shape slightly while it remains a unit-distance embedding.

Egan thought about it a lot and did a lot of computations and discovered that this unit-distance embedding is flexible.  And here it is, flexing!

For Pegg and Egan's work, go here:

http://math.stackexchange.com/questions/1484002/is-unit-mcgee-rigid

What's the practical use of all this?  Mainly, it's a practice problem in structural rigidity: the study of whether a structure is flexible or rigid.  This is important in engineering:

https://en.wikipedia.org/wiki/Structural_rigidity

A structure is infinitesimally flexible if, roughly, we can bend it a teeny weeny bit.  As the name suggests, infinitesimal rigidity can be determined by using calculus to take the derivative of all the edge lengths as a function of all the vertex positions and then using linear algebra to see in which directions this derivative is zero.  This is easy in principle, though complicated when you have 24 vertices and 36 edges.

Puzzle 1: with a minimum of explicit computation, prove that any unit-distance embedding of the McGee graph is infinitesimally flexible.

Infinitesimal flexibility is a necessary but not sufficient condition for true flexibility.

Puzzle 2: find a unit-distance embedding of a graph that is infinitesimally flexible but not flexible.

So, Egan had to do more work to show Pegg's unit-distance embedding of the McGee graph was actually flexible.  There is probably a high-powered theoretical way to do this, and it's probably not even very complicated, but I don't know it.   Do you?

For my Visual Insight post on the McGee graph, go here:

http://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/

By the way, I don't like the phrase 'unit-distance embedding' - we're not really embedding the McGee graph in the plane, because we're letting the edges cross.   The word 'immersion' would be better.

#geometry
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