A Proof About Where Symmetries Can’t Exist

In a major mathematical achievement, a small team of researchers has proven Zimmer’s conjecture.

more…

Tau.Neutrino said:

A Proof About Where Symmetries Can’t Exist

In a major mathematical achievement, a small team of researchers has proven Zimmer’s conjecture.

more…

TATE says it was proved in 2017.

The Rev Dodgson said:

Tau.Neutrino said:

A Proof About Where Symmetries Can’t Exist

In a major mathematical achievement, a small team of researchers has proven Zimmer’s conjecture.

more…

TATE says it was proved in 2017.

ok seems an old article

Still interesting.

Tau.Neutrino said:

The Rev Dodgson said:

Tau.Neutrino said:

A Proof About Where Symmetries Can’t Exist

In a major mathematical achievement, a small team of researchers has proven Zimmer’s conjecture.

more…

TATE says it was proved in 2017.

ok seems an old article

Still interesting.

Sure.

If you like that sort of thing ;)

The Rev Dodgson said:

Tau.Neutrino said:

The Rev Dodgson said:

TATE says it was proved in 2017.

ok seems an old article

Still interesting.

Sure.

If you like that sort of thing ;)

I do like that sort of thing :-)

> The conjecture states that there can exist symmetries (specifically higher-rank lattices) in a higher dimension that cannot exist in lower dimensions.

(Warning: tongue in cheek, but I do have a point).

Well, that’s obvious. Symmetries exist in 4-D that don’t exist in 3-D.

Take the lattice of the cube vs the hypercube for example. It is impossible to rotate the cubic lattice to lay it over itself such that the corners of the rotated lattice line up with the lattice of the unrotated cube.

But in 4-D we have the relationship that the distance between the point (0,0,0,0) and (1,1,1,1), ie. the corners of a hypercube lattice, is exactly the same as the distance between the point (0,0,0,0) and (2,0,0,0) because 1^2+1^2+1^2+1^2 = 2^2+0^0+0^2+0^2.

And following that through, the hypercube lattice can be rotated and placed on top of itself such that the corners of the rotated lattice do line up with the lattice of the unrotated hypercube. The hypercube lattice can sit on top of itself, sharing corner nodes, in three different rotational states.

This is an additional symmetry that is exists in 4-D but not in 3-D. I’ve played around with this a lot. I could tell you things about 24-cells and cross polytopes that would make your head spin.