Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Wednesday, May 21, 2008

M Theory Lesson 191

As one moves up in dimension, it quickly becomes difficult to draw all intersections. The tetrahedron comes from four sets, with single (orange), double (blue), triple (pink) and quadruple (green) intersections. This gives an Euler characteristic of $\chi = 4 - 6 + 4 - 1 = 1$ for the ball in three dimensional space. Observe that by alternating signs we lose the information that there are 15 (= 4 + 6 + 4 + 1) pieces of Venn diagram. An invariant that combines both pieces of information is the Pauli circulant

$A$ $B$
$B$ $A$

for $A = V + F$ and $B = E + I$ ($I$ meaning 3d pieces) in this three dimensional example. Recall that the eigenvalues of the Pauli matrix are $A - B$ and $A + B$, the first being $\chi$ and the second the subset counter. This works in all dimensions.


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