Unit II Summary: Euclid Book I Benchmarks Optimized¶
Comparative Operation Matrix for Problems XI–XXIV
1. Benchmark Table (Euclid Book I vs. Lemoine Géométrographie)¶
Prob. |
Construction |
Euclid |
Speculative (\(S, E\)) |
Lemoine (\(S, E\)) |
Saved |
|---|---|---|---|---|---|
XI |
Copy given angle at given point |
I.23 |
\(4R_1+2R_2+8C_1+C_2+4C_3\) (\(19, 13\)) |
\(2R_1+R_2+5C_1+3C_3\) (\(11, 7\)) |
\(-8\) |
XVI |
Triangle from three sides \(a, b, c\) |
I.22 |
\(6R_1+3R_2+9C_1+C_2+3C_3\) (\(22, 16\)) |
\(4R_1+3R_2+8C_1+C_2+3C_3\) (\(19, 13\)) |
\(-3\) |
XVII |
Parallel through external point \(A\) |
I.31 |
\(2R_1+R_2+5C_1+3C_3\) (\(11, 7\)) |
\(2R_1+R_2+3C_1+3C_3\) (\(9, 5\)) |
\(-2\) |
XVIII |
Perpendicular bisector & midpoint |
I.10 |
\(2R_1+R_2+3C_1+2C_3\) (\(8, 5\)) |
\(2R_1+R_2+2C_1+2C_3\) (\(7, 4\)) |
\(-1\) |
XVIII bis |
Symmetric reflection across line |
I.12 cor. |
\(4R_1+2R_2+5C_1+C_2+3C_3\) (\(15, 10\)) |
\(2C_1+2C_2+2C_3\) (\(6, 4\)) |
\(-9\) |
XIX |
Circle on diameter \(AB\) |
I.10+Post.3 |
\(2R_1+R_2+5C_1+3C_3\) (\(11, 7\)) |
\(2R_1+R_2+4C_1+3C_3\) (\(10, 6\)) |
\(-1\) |
XX.1 |
Perpendicular from external point |
I.12 |
\(2R_1+R_2+3C_1+C_2+3C_3\) (\(10, 6\)) |
\(2R_1+R_2+3C_1+C_2+2C_3\) (\(9, 6\)) |
\(-1\) |
XX.2 |
Perpendicular at point on line |
I.11 / III.31 |
\(2R_1+R_2+3C_1+3C_3\) (\(9, 5\)) |
\(4R_1+2R_2+C_1+C_3\) (\(8, 5\)) |
\(-1\) |
XXII |
Angle bisector |
I.9 |
\(2R_1+R_2+4C_1+3C_3\) (\(10, 6\)) |
\(2R_1+R_2+3C_1+3C_3\) (\(9, 5\)) |
\(-1\) |
2. Four Core Optimization Techniques Introduced in Unit II¶
Unconstrained Initial Carrier (:math:`R_2` vs. :math:`2R_1 + R_2`): When constructing a figure congruent to given magnitudes (Problem XVI), start from a free line \(R_2\) rather than placing two points first.
Aperture Persistence Across Operations: Retain an arbitrary sufficient radius across symmetric steps (Problems XVII, XVIII, XXII), eliminating redundant \(C_1\) measurements.
Eliminating Un-Needed Auxiliary Lines (Reflection Lemma XVIII *bis*): Reflect points across lines by two intersecting circles alone (\(0\) lines, \(S = 6\)), turning reflection into a zero-straightedge macro reused throughout Unit III.
Replacing Symmetric Compass Stepping by Thales Diameter Incidence (XX.2): Trade three circles and one line (\(S = 9\)) for one single circle and two lines (\(4R_1 + 2R_2 + C_1 + C_3, S = 8\)).