### Spherical Triangles with Vertices in the Binary Tetrahedral Group

Up to (possibly orientation-reversing) isometry, a spherical triangle is determined by the lengths of its edges. Using as edges the possible lengths between elements of the binary tetrahedral group, the possible faces (spherical triangles with vertices in the group) that can occur are:

A = ppp
B = pgg
C = pvg (has a non-congruent mirror image)
D = pvv
E = ggg
F = gvv

using as before the notation: g = pi/3, p = pi/2, v = 2*pi/3.

NOTATION: Edge lengths are encoded as before:
p = pi/2, g = pi/3, v = 2*pi/3.
Face angles and areas are from the following set:
x = arccos(1/3), y = arccos(-1/3), z = arccos(1/sqrt(3)),
w = arccos(-1/sqrt(3)), and pi/2.

Face: A
Edge lengths: p,p,p
Face angles between labelled edges: --p--(pi/2)--p--(pi/2)--p--(pi/2)--
Area: pi/2

Face: B
Edge lengths: p,g,g
Face angles between labelled edges: --p--(w)--g--(y)--g--(w)--
Area: y-x

Face: C
Edge lengths: p,v,g
Face angles between labelled edges: --p--(z)--v--(x)--g--(w)--
Area: x

Face: D
Edge lengths: p,v,v
Face angles between labelled edges: --p--(w)--v--(y)--v--(w)--
Area: pi

Face: E
Edge lengths: g,g,g
Face angles between labelled edges: --g--(x)--g--(x)--g--(x)--
Area: 2x-y

Face: F
Edge lengths: g,v,v
Face angles between labelled edges: --g--(y)--v--(x)--v--(y)--
Area: y

Angles are given in radians; areas scaled so that a great sphere has area 4pi. These computations were made using the Spherical Law of Cosines. Note that w = x + z, y = 2* z, 2*w + y = 2*pi,
y + x = pi, etc. These triangles admit the following

DECOMPOSITIONS:

D = B + 2*C
C + C = C + C
B + B = B + B
C = B + E
D + D + D + D = S^2

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