A couple of "Apollonian gems":
1. Pencils of lines and circles turn out to be points of intersection of 2D subspaces of $\mathbb M^{1,3}$ with the unit sphere (actually, hyperboloid).
2. The Barning matrices that form a semigroup the orbit of which through [3,4,5] recovers all irreducible Pythagorean triples may be viewed as derived from certain three inversions in the Apollonian window.
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