Unit V: Axial Reflection & Chapter III — The Theory of Revolution — Notes

Historical, Mathematical & Cross-Repository Commentary (pp. 60–68)

Historical, Mathematical & GEOMETOR Commentary

1. Petersen’s Theory of Revolution as Complex Multiplication

Petersen’s Chapter III (pp. 65–68) is one of the earliest and cleanest elementary expositions of planar spiral similarity. Placing the pivot point \(O\) at \(0 \in \mathbb{C}\), Petersen’s revolution \((O, v, f)\) is multiplication by the complex constant \(\lambda = f e^{i v}\):

\[z_A = f e^{i v} z_a.\]

Look at how transparently this explains § 5 (Miquel’s Pivot Point, p. 68) and Problem 349 (p. 67):

  • Any non-translation direct similarity \(z \mapsto \lambda z + \mu\) (\(\lambda \neq 1\)) has a unique fixed point \(z_O = \mu / (1 - \lambda)\)—Petersen’s pivot point, constructed synthetically via the Miquel circles \((PAa) \cap (PBb)\).

  • In Problem 349 (equilateral triangle with vertices on three parallel lines \(L_1, L_2, L_3\)), multiplication around \(P \in L_1\) by \(e^{i\pi/3} = \frac{1 + i\sqrt{3}}{2}\) (\(v = 60^\circ, f = 1\)) maps \(Q \in L_2\) to \(R \in e^{i\pi/3}(L_2) \cap L_3\), solving the problem in a single line-line intersection!

2. Porismatic Parity in Problem 340 (Minimum-Perimeter Inscribed Polygons)

In Problem 340 (p. 62), unfolding an inscribed \(n\)-gon across the \(n\) sides of a host \(n\)-gon composes \(n\) axial reflections:

  • When \(n\) is odd, the product of \(n\) reflections is an orientation-reversing glide/reflection, yielding a unique determinate fixed-direction cycle (for \(n = 3\) in an acute triangle, Fagnano’s 1775 orthic triangle, pedal of orthocenter X(4)).

  • When \(n\) is even, the product of \(n\) reflections is an orientation-preserving rotation or translation: if a non-trivial rotation, no parallel periodic path exists (0 solutions); if a pure translation (e.g. a cyclic quadrilateral whose opposite angles match, or a billiard rectangle), infinitely many equal-perimeter inscribed polygons exist!

3. Exact geometor.model Realizations: equilateral_on_three_parallels & hero_reflection_shortest_path

  • equilateral_on_three_parallels() (Problem 349, p. 67): On \(y_1 = 0, y_2 = 1, y_3 = 3\) with pivot \(P = (0, 0)\), rotating \(L_2\) by \(+60^\circ\) via Euclid I.1 circles produces \(L'_2: \sqrt{3}\,x - y + 2 = 0\), meeting \(L_3\) at \(R = (\frac{\sqrt{3}}{3}, 3)\) and rotating back to \(Q = (\frac{5\sqrt{3}}{3}, 1)\) in \(\mathbb{Q}(\sqrt{3})\). All three side quadrances equal \(\frac{28}{3}\), and all interior spreads equal \(\frac{3}{4}\) (\(\sin^2 60^\circ\)).

  • hero_reflection_shortest_path() (Problem 338, p. 61): Reflecting \(A = (2, 6)\) across \(y = 0\) to \(A_1 = (2, -6)\) and joining \(A_1\) to \(B = (14, 10)\) meets the mirror at \(X = (\frac{13}{2}, 0)\) with equal incidence/reflection spreads \(\frac{16}{25}\) and minimum unfolded path quadrance \(Q(A_1, B) = 12^2 + 16^2 = 400 = 20^2\).