The Heronian Triangle 13–14–15 as an Arithmetic, Crystallographic, and CM–Elliptic Node

We show how the Heronian triangle 13–14–15 sits at a common node
between four worlds: (i) the family of consecutive triangles
(a−1,a,a+1) with integer area and inradius, governed by Pell equations
in the real quadratic field Q(√3); (ii) an exact periodic “corona crystal”
in the Euclidean plane, whose wallpaper group is p6m and whose reflection
subgroup is the Euclidean Coxeter triangle group Δ(2,3,6); (iii) the cyclotomic
field Q(ζ12), which contains both Q(√−3) and the biquadratic field
Q(√2,√3) arising from Pell-type equations; and (iv) a CM elliptic curve
with j-invariant 0, on which the corona crystal defines a (2,3,6)-torsion
configuration with coordinates in cyclotomic subfields of Q(ζ12).

The real quadratic field Q(√3) appears as the intersection K1 ∩ K2 of
the biquadratic field K1 = Q(√2,√3) and the cyclotomic field
K2 = Q(ζ12), and plays the role of arithmetic node connecting Pell
arithmetic, planar crystallography, CM elliptic curves, and cyclotomic
Galois symmetry around the triangle 13–14–15.