[math-fun] Many cheerful facts about the Hurwitz quaternions
The 24 quaternions G = {±1, ±i, ±j, ±k, (±1 ±i ±j ±k)/2} form a group under multiplication isomorphic to the binary tetrahedral group 2T, which can be defined as the inverse image of the rotation group T of the regular tetrahedron under the double covering map π : S^3 —> SO(3) from the unit sphere S^3 (the group of unit quaternions) to the rotation group SO(3) of 3-space: 2T = π^(-1)(T), where the map π is defined via π(u) = (q |—> u q u^(-1)) for q = xi + yj + zk in the 3-space of pure quaternions. The subring of the quaternions generated by the 24 elements of 2T is the Hurwitz "integral" quaternions J: J = <2T>. As a lattice in R^4, J is just J = Z^4 ∪ (1/2,1/2,1/2,1/2) + Z^4, the union of the integer points and the so-called half-integer points. This lattice is called the D_4 lattice, aka the 4D checkerboard lattice, and is believed to be the unique lattice in R^4 whose points form the centers of the densest possible sphere-packing in R^4. J forms a maximal order of the quaternions (https://en.wikipedia.org/wiki/Order_(ring_theory)). The points of R^4 closer to the origin than any other lattice point form the interior points of the regular polytope called the 24-cell, which is the convex hull of the points of 2T. All of R^4 is tiled by these "Voronoi polytope" 24-cells, one for each lattice point of D^4. It's also true that the Voronoi 24-cell of the origin is surrounded by 24 other 24-cells, and much like a rosette of 7 hexagons in the plane can tessellate the plane, 4-space can be tessellated by these clusters of 25 24-cells, translates of the original cluster by quaternions in the left ideal (2+i)J of the Hurwitz quaternions. This means that if R^4 is quotiented out by the lattice (2+i)J, the result is a certain 4-dimensional torus R^4/(2+i)J that is itself tessellated by 25 24-cells ... each one sharing a 3-dimensional face (an octahedron) with each other one. Which I find to be very cool! —Dan Fred Lunnon wrote: ----- Hurwitz quaternions work in 3-space --- see for example my write-up for when this algorithm last appeared on math-fun, https://arxiv.org/abs/1202.3198 I never needed to consider 4-space, since nobody has managed to discover a Heronian pentatope! On 10/22/20, Henry Baker <hbaker1@pipeline.com> wrote:
Hurwitz quaternions might work for 4D. .....
Something I noticed about the D_4 lattice is that the union of the lattice together with its deep holes is geometrically similar to the original D_4 lattice, scaled by 1/sqrt(2). This means that if you take the 4-dimensional torus R^4 / D_4 and start greedily placing points to be as far away as possible from the nearest point, you'll end up progressively filling ever-finer copies of the D_4 lattice. (This also works on the more conventional torus R^4 / Z^4, because after the first two points are emplaced they'll form D_4 / Z^4.)
Sent: Thursday, October 22, 2020 at 6:36 PM From: "Dan Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Subject: [math-fun] Many cheerful facts about the Hurwitz quaternions
The 24 quaternions
G = {±1, ±i, ±j, ±k, (±1 ±i ±j ±k)/2}
form a group under multiplication isomorphic to the binary tetrahedral group 2T, which can be defined as the inverse image of the rotation group T of the regular tetrahedron under the double covering map
π : S^3 —> SO(3)
from the unit sphere S^3 (the group of unit quaternions) to the rotation group SO(3) of 3-space:
2T = π^(-1)(T),
where the map π is defined via
π(u) = (q |—> u q u^(-1))
for q = xi + yj + zk in the 3-space of pure quaternions.
The subring of the quaternions generated by the 24 elements of 2T is the Hurwitz "integral" quaternions J:
J = <2T>.
As a lattice in R^4, J is just
J = Z^4 ∪ (1/2,1/2,1/2,1/2) + Z^4,
the union of the integer points and the so-called half-integer points.
This lattice is called the D_4 lattice, aka the 4D checkerboard lattice, and is believed to be the unique lattice in R^4 whose points form the centers of the densest possible sphere-packing in R^4.
J forms a maximal order of the quaternions (https://en.wikipedia.org/wiki/Order_(ring_theory)).
The points of R^4 closer to the origin than any other lattice point form the interior points of the regular polytope called the 24-cell, which is the convex hull of the points of 2T. All of R^4 is tiled by these "Voronoi polytope" 24-cells, one for each lattice point of D^4.
It's also true that the Voronoi 24-cell of the origin is surrounded by 24 other 24-cells, and much like a rosette of 7 hexagons in the plane can tessellate the plane, 4-space can be tessellated by these clusters of 25 24-cells, translates of the original cluster by quaternions in the left ideal (2+i)J of the Hurwitz quaternions.
This means that if R^4 is quotiented out by the lattice (2+i)J, the result is a certain 4-dimensional torus R^4/(2+i)J that is itself tessellated by 25 24-cells ... each one sharing a 3-dimensional face (an octahedron) with each other one.
Which I find to be very cool!
—Dan
Fred Lunnon wrote: ----- Hurwitz quaternions work in 3-space --- see for example my write-up for when this algorithm last appeared on math-fun, https://arxiv.org/abs/1202.3198
I never needed to consider 4-space, since nobody has managed to discover a Heronian pentatope!
On 10/22/20, Henry Baker <hbaker1@pipeline.com> wrote:
Hurwitz quaternions might work for 4D. .....
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