0011 — Sources of F06: polygon stretched to its box

Spike SP-009, 2026-10-12. Answer to research request 0011, run by the Product Owner in claude.ai (Research mode); summed up here in the agent's words, every source marked verified re-read by the agent the same day.

What was asked

A regular polygon of n corners (3 to 12), its base flat at the bottom, symmetric about the vertical axis of its box, touching the four sides of a w × h box, its n corners convex and in the same order, its sides as even as possible.

Findings

No published construction

No book, paper or tool documents such a polygon. The rule has to be a derivation, each step citing a read source.

• Drawing tools stretch by a plain scaling, which makes the sides uneven: an affine map keeps lines and ratios on a line, not lengths nor angles (REF-MATHWORLD-AFFINE). Illustrator's live polygon scales non-uniformly and offers "Make Sides Equal", which returns to the regular polygon (REF-ADOBE-LIVE-SHAPES). Inkscape's polygon has no box-filling option (REF-INKSCAPE-POLYGON). Figma's polygon does not fill its box (Product Owner's test).
• PowerPoint's preset hexagon and octagon fill their box with insets measured from the shorter side, so their cut corners keep fixed angles and the straight sides absorb the stretch: a third rule, read only second-hand (REF-OOXML-ISSUE-465); the standard itself (REF-ECMA-376) is not read.
• An equilateral quadrilateral is a rhombus; a convex equilateral pentagon is fixed by two consecutive angles (REF-WIKI-EQUILATERAL-POLYGON).

Even sides alone are not enough

• The usual measure of evenness is the edge ratio, the longest side over the shortest, 1 when all are equal (REF-VTKM-EDGE-RATIO, from the Verdict library, REF-VERDICT-2007, not read). Other measures (variance, least squares) agree when equal sides exist.
• Minimizing it alone loses corners: a stretched square becomes a trapezoid; in a 300 × 50 box the pentagon's apex flattens; in 50 × 300 the hexagon's side corners vanish. Even when equal sides exist, corners can become almost flat: the equal-sided octagon of the λ family (below) in 100 × 50 has corners turning 13.6°, against 45° when regular.
• So a rule must also protect the corners, measured by their turning angle, the angle between the directions of the two sides meeting there (REF-CARROLL-2013); a regular polygon turns 2π/n at each corner.
• A closed polygon is fixed, up to its place, by its side lengths and turning angles (REF-CRANE-DISCRETE-CURVES): the shape can be built from directions of sides of length 1.
• Equal sides are unique only for n = 5 and 6; from 7 on, a family of equal-sided polygons fills the box, and a choice is needed. The research answer counted the octagon as unique too, assuming a central symmetry; the review of SP-009 found other equal-sided octagons filling a square (side directions 42°, 101.24°, 123.24°), so it is a family as well.

The λ family

• Every side has length 1, and takes the direction of the matching side of the regular polygon after a vertical scaling by λ: φⱼ = atan2(λ sin θⱼ, cos θⱼ), θⱼ = 2πj/n. For an odd n, the side reaching the apex takes the direction that closes the polygon. λ = 1 is the regular polygon; λ > 1 makes it taller.
• It gives the unique equal-sided polygon for n = 5 and 6, and one member of the family for the others (for the octagon, the one with a central symmetry).
• λ is the value for which the polygon's height over its width equals h / w, found by bisection on ln λ (REF-MATHWORLD-BISECTION: after k steps the error is at most 2^−k times the interval); closed forms for the hexagon and the octagon (φ₁ = atan r + asin(r / (2√(1 + r²))) for the hexagon, r = h/w).
• Bisection needs the aspect to grow with λ: checked on 601 values for every n from 5 to 12 by the research and on 6 001 values by the agent's tests, not proved.

Where equal sides stop being sound

With every corner kept turning at least half its regular turn (π/n), equal sides are possible for these ratios h / w:

n	Regular	Equal sides, corners kept
5	0.95106	0.63167 – 1.67401
6	0.86603	0.36603 – 1.27293
7	0.97493	0.62434 – 1.55632
8	1	0.61991 – 1.61313
9	0.98481	0.67233 – 1.59293
10	0.95106	0.54837 – 1.49813
11	0.98982	0.60000 – 1.60040
12	1	0.61699 – 1.62078

The triangle and the square have no freedom: plain stretch, an isosceles triangle and a rectangle.

The three rules

Rule	What it does	Hexagon 300 × 50 (edge ratio, smallest corner turn)
A — plain stretch	the regular polygon scaled on each axis to the box, as drawing tools do; simple, sides uneven	1.897, 18.4°
B — equal sides at any cost	equal sides whenever they exist; corners may almost vanish, impossible past a ratio	1, 14.2°
C — equal sides, then plain stretch	equal sides (λ family) while every corner turns at least π/n; past that limit, that limit polygon is scaled on each axis to the box	1.117, 14.7°

Rule C, the answer's recommendation: always touches the four sides, symmetric, base flat, n convex corners in order, unique and continuous as w and h change, depends only on h / w, closed forms or one bisection. Where the box is within the range above, its sides are exactly equal.

Test values (rule C, SVG frame, 5 decimals)

Drawing order of F02: clockwise on screen from the right end of the base.

Shape, box	Vertices	Sides
triangle 100 × 100	(100, 100), (0, 100), (50, 0)	100, 111.80340, 111.80340
square 300 × 50	(300, 50), (0, 50), (0, 0), (300, 0)	the rectangle
pentagon 100 × 100	(81.75623, 100), (18.24377, 100), (0, 39.16418), (50, 0), (100, 39.16418)	all 63.51246
pentagon 300 × 50	(228.85967, 50), (71.14033, 50), (0, 12.85950), (150, 0), (300, 12.85950)	157.71933, 80.25188, 150.55021
hexagon 100 × 100	(77.42919, 100), (22.57081, 100), (0, 50), (22.57081, 0), (77.42919, 0), (100, 50)	all 54.85838
hexagon 300 × 50	(204.90381, 50), (95.09619, 50), (0, 25), (95.09619, 0), (204.90381, 0), (300, 25)	109.80762, 98.32744
hexagon 50 × 300	(41.47297, 300), (8.52703, 300), (0, 150), (8.52703, 0), (41.47297, 0), (50, 150)	32.94593, 150.24217
octagon 100 × 100	the regular octagon	all 41.42136

Hand check, hexagon 100 × 100: φ₁ = 45° + asin(1/(2√2)) = 65.7048°; the unit half-width 1/2 + cos φ₁ = 0.91144; scale 50 / 0.91144 = 54.85838, the side. The agent's own value before the research was 54.858.

Decided by the Product Owner (2026-10-12)

1. Rule C: equal sides while every corner keeps enough turn, then a plain stretch of that limit polygon.
2. The corner guard is half of the regular turn (π/n, 30° for the hexagon): a design value, not a sourced one; one third and two thirds were offered.
3. The square stays a rectangle, and the triangle isosceles: neither has freedom (not the trapezoid that the edge ratio alone prefers).

Open, for the derivation (settled in DERIV-stretched-polygon, SP-009)

• Prove that the aspect of the λ family grows with λ for each n, and that the sound range contains λ = 1 and is one interval; or test it exhaustively.
• A source for the identity a sin φ − b cos φ = c ⇒ φ = atan(b/a) + asin(c / √(a² + b²)) (closed forms of the hexagon and octagon).
• Compute the limits λ_lo, λ_hi at run time or tabulate them with their derivation.
• Read ECMA-376 before citing the PowerPoint presets.

Research request

Request 0011-stretched-polygon-even-sides.md in research/requests/; the answer file was folded into this note and removed.