F.40 Regular Polygons and Nested Radicals

Archimedean Edge-Doubling Recursion, Golden Decagon/Pentagon Radicals, and the Pentadecagon

For a regular N-gon inscribed in a unit circle (R = 1), doubling the number of sides maps edge length E_{N/2} to E_N = sqrt(2 - sqrt(4 - E_{N/2}^2)), generating closed-form nested quadratic radicals for the 2^k, 3*2^k, 5*2^k, and 15*2^k polygon families.

Interactive GEOMETOR construction for F.40 (Regular Polygons and Nested Radicals). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point A : Point ===== c_unit = circle(O, A) l_oa = line(O, A) perp = perpendicular(O, l_oa) B = intersection(perp, c_unit) M = midpoint(O, B) c_five = circle(M, A)

Mathematical Synthesis

Spanning 18 folios (SeeligNotebooks), Seelig’s treatise on regular polygons derives the Archimedean edge-doubling recursion \(E_N = \sqrt{2 - \sqrt{4 - E_{N/2}^2}}\) (for circumradius \(R=1\), where \(E_N = 2\sin(\pi/N)\) and apothem \(r_N = \frac{1}{2}\sqrt{4 - E_N^2}\)) and compiles complete nested-radical tables across four constructible families: the square/octagon chain (\(E_4 = \sqrt{2}, E_8 = \sqrt{2-\sqrt{2}}, E_{16}, E_{32}, E_{64}\)), the triangle/hexagon/dodecagon chain (\(E_3 = \sqrt{3}, E_6 = 1, E_{12} = \sqrt{2-\sqrt{3}}, E_{24}\)), and the golden ratio pentagon/decagon chain (\(E_5 = \sqrt{3-\phi} = \frac{1}{2}\sqrt{10-2\sqrt{5}} \approx 1.17557, E_{10} = \sqrt{2-\phi} = \phi - 1 \approx 0.61803, E_{20} = \sqrt{2-\sqrt{2+\phi}}\)).

Dated April 20, 2015, the second half of the dossier investigates the regular 15-gon (Euclid Elements IV.16, \(E_{15} = 2\sin 12^\circ \approx 0.41582\)), citing James Stafford’s exact radical table for multiples of \(\pi/30 = 6^\circ\), alongside an angle-chasing exploration of the regular enneagon (\(N=9\), requiring a cubic trisection) and two auxiliary Cartesian cevian sheets.

Key Algebraic Identities

\[E_N = 2\sin\frac{\pi}{N} = \sqrt{2 - \sqrt{4 - E_{N/2}^2}}, \qquad r_N = \cos\frac{\pi}{N} = \frac{1}{2}\sqrt{4 - E_N^2}\]

\[E_{10} = \phi - 1 = \sqrt{2 - \phi}, \qquad E_5 = \sqrt{3 - \phi}, \qquad E_{15} = 2\sin 12^\circ = \frac{1}{4}\left(\sqrt{10 + 2\sqrt{5}} + \sqrt{3} - \sqrt{15}\right)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.40 Regular Polygons and Nested Radicals — Folios (Folios 0239–0256, 18 scanned manuscript pages in resources/seelig/40_polygons)
Euclid’s Elements	IV.11, IV.15, IV.16, XIII.9, XIII.10 (EuclidElements)
GEOMETOR Elements (G-Index)	G.9 figure, G.10 circle
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Polygon Table, Golden Triangle (Spreadsheets)

Computational Workbook Verification

The following worksheets from seelig’s Triangles and Circles.ods workbook verify the numerical and symbolic formulas for this topic on the canonical \((a, b, c) = (5, 6, 7)\) triangle:

Worksheet: Golden Triangle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
F11	F11	0.09549	=F13^2
F12	F12	0.19098	=1-[.$F$36]/2
F13	F13	0.30902	=1/(2*[.F$36])
F14	F14	0.32492	=TAN(RADIANS(18))
F15	|IM|	0.36327	=F35/F42
F16	|GM|	0.51295	=F35/3
F17	F17	0.52786	=(F36-0.5)/(F36+0.5)
F18	F18	0.58779	=F31/2

Cell Excerpt

The Golden Triangle	—	—	—	—	—
Isosceles triangle with equal sides	—	—	—	—	—
Unity base length, vertex angle 36°,	—	—	—	—	—
Fred W. Seelig	—	—	—	—	—
Mar. 22, 2015	—	—	Created	—	—
Mar. 31, 2015	—	—	Last Modified	—	—

Constructive Dependency Graph

Prerequisites: F.23 Golden Triangles

Required for: F.45 Squares, Subdivided Lattices, and the Japanese Theorem