I meant "all directions" to mean taking unit steps in each of the three coordinate directions: +-x, +-y, or +-z. With the important proviso that the group is completely non-commutative: Different (non-backtracking) paths always get to different places, the graph has no loops, there's only one way to reach a point. 2+i, 2+j, 2+k generate a non-commutative 3-generator free group. Dividing by sqrt5 maps everything down to unit length, but there are still no relations. AKAIK, there's no relationship to the Hurwitz quaternions: They are all divisors of 1, i.e. units. 2+-i &c have norm 5, and don't divide 1. If you make them unit length by dividing by sqrt5, then the coordinates are irrational. All the Hurwitz coords are multiples of 1/2. Rich --- Quoting Henry Baker <hbaker1@pipeline.com>:

Hi Rich:

I'm not SUre that I understand your dimensions and trees.

"All directions": is this in 3-space or 4-space?

Is there any relation to the 24 Hurwitz quaternions?

At 04:47 PM 3/15/2018, rcs@xmission.com wrote:

...

It's a different covering structure, but the products of 2+-i, 2+-j, 2+-k generate a dense covering of all directions. (Divide the generators by sqrt5 to get units.) Except for the cancellations of conjugates, like (2+i)(2-i) = 5, every product is different. You can view the pre-space as a tree with node degree 5, except the root is degree 6. The post-space is points on the unit quaternion sphere. I assume the covering is near uniform, but haven't seen the confirming theorem.

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