The Dehn Invariant - Numberphile
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as an accessible explanation of the Dehn invariant and Hilbert's third problem, and as a concrete motivating example for what the tensor product does.
Recommended by 2 people across 2 different months, 2019 to 2023.
What it is
A Numberphile video in which Daniel Litt explains the Dehn invariant, the tool Max Dehn used to solve Hilbert's third problem by showing that polyhedra of equal volume cannot always be cut into congruent pieces. It walks through computing the invariant using a tensor product of edge lengths and dihedral angles.
Cited in 1 Wikipedia article, including Dehn invariant.
What people said
Recommends Mark Bennet · 2022-11 · Mathematics Stack Exchange · on “Proving volume formulae”
It is also easier to use congruence in two dimensions than in three - for which Dehn proved (Hilbert's third problem) that two polyhedra of the same volume need not be dissectible into a collection of congruent pieces. This meant that a new idea had to be introduced to extend the notion of measure from two dimensions to three. See this Numberphile video for more information. [this video]
Recommends Clemens Bartholdy · 2023-01 · Mathematics Stack Exchange · on “Geometric intuition of tensor product”
For a totally different angle to the subject, we can get a taste of what the tensor product means by considering Dehn's invariant. I'll try to shortly explain what it is, if we have a polyhedron with edge lengths $l_i$ and dihedral angles [LINK] $\theta_i$ , we can define the Dehn's invariant through the quotient: $$ R \otimes_{\mathbb{Z} } ( R / 2 \pi \mathbb{Z})$$ Now, the point is, if we have a polyhedron, then upon cutting it into two pieces, the Dehn's invariant is fixed! For a beautiful exposition of this topic, see this Numberphile video [this video]
Cites Christian Albina · 2019-07 · Mathematics Stack Exchange · on “What does the tensor symbol "$\otimes$" (or whatever this symbol is) do?”
I encountered the symbol " $\otimes$ " in a Numberphile video about the Dehn Invariant [this video] , and I have no clue what it means. What, in layman terms, does this symbol do? I'm not all that experienced in maths and often fail to understand the jargon used in the descriptions of what the tensor symbol does and what a "tensor product" is.
Cites Clemens Bartholdy · 2023-01 · Mathematics Stack Exchange · on “Computing the Dehn Invariant for a cube (Numberphile)”
In this video [this video] by Numberphile, the Dehn invariant is computed by noting that there are 12 edges all having dihedral angle $\frac{\pi}{2}$ We compute the invariant using linearity in first slow $$ 1 \otimes \frac{\pi}{2} + 1 \otimes \frac{\pi}{2} ... \text{12 times} = 12 \otimes \frac{\pi}{2 } \tag{1}$$ But, suppose we had added things in the second slot, and then use the fact that the second slot is mod $2\pi$ : $$ 1 \otimes \frac{\pi}{2} + 1 \otimes \frac{\pi}{2} ... \text{12 times} = 1 \otimes (12 \frac{\pi}{2}) = 1 \otimes 6 \pi = 1 \otimes 0 \tag{2}$$ How would I show (1)=(2)?