The tree of F2 = ⟨a, b⟩

The Cayley graph is drawn with a to the right and b upward; an edge at distance s from the viewpoint has length 2.5−s, so the whole tree always fits in the tilted square. Drag to slide the viewpoint along the tree. Click a vertex, on the canvas or in the list, to center it. Words use A = a−1, B = b−1.

Stage 1: choose A

Hold Shift and click to add the nearest vertex, or generate one of the examples below. Distances are word distances.

Example 1: a ball

A maximal R1-separated set of vertices in the ball of radius R2 about the vertex at the center of the view.

Example 2: k rays

A maximal R1-separated set in the R2-neighborhood of k geodesic rays of length L from the center vertex, spread as evenly as the tree allows and continuing straight.

Thin out and continue

Points are ordered by word length, then kept greedily if they are at least R′ from every point already kept. The kept set A′ replaces A.

Stage 2: halfspace scan

A halfspace at distance D from P is the component of the tree minus an edge whose far end is at distance D from P. For each point of A′, the view moves to it and scans these halfspaces in clockwise order from the upward one: all of them when there are at most 100, otherwise the ones that contain points. It keeps the halfspace with the most points of A′ and the best other one; two different halfspaces at the same distance are always disjoint.

Stage 3: sorting the red points

A″ is the set of red points, numbered by word length. Point 1 is the white anchor. Each point is sorted into an urn.

With radius r > 0, each halfspace H is replaced by H′ = H minus the r-neighborhood of its complement: the vertices of H at distance at least r from its nearest vertex. A direction toward a point closer than D continues straight past it.

Bad 0
Good 0
Undecided 0

Halfspaces of the selected point (click a point after the run):

Lineage lines (pink):

Points

Click a point to move the view to it.

    No points yet.