100 Degree Theorem: Analytic Data Boxes
These cyan boxes contain words (or, in a few cases, just
polygons) which are dealt with analytically in Rich's paper.
Here is a description of these various boxes:
-
P1 is a neighborhood of the (degenerate)
right triangle whose small angle is 0.
This polygon is covered by an infinite
union of orbit tiles. The 20 data boxes
contain the starts of the two infinite
series of tiles which cover P1.
-
P2 is the set of acute triangles whose big
angle is at least 569/1024 radians.
This number just slightly exceeds 100
degrees. We could have made P2 be the
region consisting in triangles whose
big angle is at least 100 degrees,
but we take this slightly smaller region
because its vertices have dyadic rational coordinates.
- P3 is the set of triangles whose second
angle is bigger than the first angle.
By symmetry we do not need to consider
this region of parameter space.
- P4 is a neighborhood of the right triangle
whose small angle is Pi/4. This region
is covered by the orbit tiles associated
to the 5 data boxes shown. We cannot
use just numerical methods to verify
that that P is contained in O(W) for
the relevant polygon and word, because
P shares some of its boundary with O(w).
We use the same trick as for the polygons P7,...,P29:
We ignore some of the defining functions and
then verify by hand that these ignored
defining functions cut out the right shape.
Rich's paper has the details of this.
- P5 is a neighborhood of the right triangle
whose small angle is Pi/5. This region
is covered by the orbit tiles associated
to the 2 data boxes shown. We cannot use
just numerical methods to verify that
that P is contained in O(W) for the
relevant polygon and word, because
P shares some of its boundary with O(w).
We use the same trick as for the polygons
P7,...,P29: We ignore some of the defining
functions and then verify by hand that these
ignored defining functions cut out the
right shape. Rich's paper has the details of this.
- P6 is a neighborhood of the right triangle
whose small angle is Pi/6. This polygon
is covered by an infinite union of orbit
tiles. The 20 data boxes contain the starts
of the two infinite series of tiles which
cover P6.