Hilbert's 3rd Problem

[byosys]

I've been asked to explain one of Hilbert's problems as part of my math class. Before everyone flames me, I want to say that this is NOT for a grade; its strictly enrichment and something that I want to do.

I've read through this Wikipedia Link, but I still don't quite undertand what Dehn invariant means. Could someone please explain this and Hilbert's 3rd problem in general to me?

Thanks.

 

[CycloWizard]

It's hard to know where to start without knowing your math background, but I'll take a stab at this. An invariant is some value that describes the characteristics of a system. The example I'm most familiar with is the stress tensor, which has three invariants. The simplest is the trace of the tensor (the sum of the elements on the main diagonal of the tensor). The value of this invariant tells you something about the tensor. Let's say that if this invariant is zero, then the fluid is incompressible. If it's non-zero, then the fluid is not incompressible.

In the case of Dehn's invariant, the invariant characterizes the geometry of the system by considering the lengths and angles of the various segments. This invariant value allows you to compare otherwise incomparable objects in a specific manner by comparing a specific set of characteristics. For example, it will specifically let you look at a cube and a sphere in the same sense, despite their disparate gemoetries, and arrive at the same conclusion: that both are scissors-congruent.

Hopefully I didn't confuse the issue too much.

 

[byosys]

Thanks Cyclo. I got the basic concept, which was all I could hope to grasp at this stage. I think the real problem is that I just finished BC Calc this year, and from what my teacher said, for me to understand this problem I would need another 2-3 years of math.

Thanks!