construction in ten plates

construction presentation · 2026-08-17

the aligned
distinct ring.

one ray, repeated ten times. one ring, solved from its center and its air. a construction of whole angles, simple fractions, and φ.

tip circle · 27 = 3³symmetry · D10exact svg · download
selected aligned distinct-ring sol mark
selected geometry embedded exactly · this deck explains its construction01 / 10

02 · source grammar

one path, one circle, nine rotations.

1 × pathone mirrored ray
+ 9 rotations36° apart
+ 1 circlethe complete mark
<path id="ray" />the only authored contour
rotate(36° × n)n = 1…9 around one center
circle r = 13/2stroke = √5 − 1/2
identity by repetition · no independently drawn copies02 / 10

03 · construction field · closed-form targets

five concentric radii describe the selected silhouette.

five-radius construction field the selected mark overlaid with the tip circle, ray base, ring outer edge, ring centerline, and ring inner edge. tip · 27/2 base · 8 + √5/2 outer · 25/4 + √5/2 ring · 13/2 inner · 27/4 − √5/2
tip circle27/2
ray base8 + √5/2
ring outer edge25/4 + √5/2
ring centerline13/2
ring inner edge27/4 − √5/2
one center.

all five boundaries share the same origin. only the ray sides depart from a circle, and they do it symmetrically.

closed forms are construction targets · the selected source stores them to finite precision03 / 10

04 · ray anatomy · source-precision target

the anchors stay fixed. the sides lean inward.

inward ray curve compared with a straight-sided ray the selected quadratic ray overlays a straight-sided reference with the same tip and base anchors. tip anchor base anchor quadratic control straight reference selected curve
same tip · same base · same 33° arconly the side contour changes
visible bow target27/200

one half of one percent of the 27-unit span, and one percent of the tip radius by design.

2 : 1

the control target is 27/100; the visible midpoint moves half as far.

inward, not soft.

the adjustment is small enough to preserve the silhouette and large enough to keep the rays from reading as petals.

quadratic Bézier geometry · optical correction expressed as a ratio04 / 10

05 · ten-ray cadence · source-precision target

each 36° base sector is eleven parts ray, one part air.

thirty-six degree sector divided into thirty-three degrees of ray and three degrees of air the selected mark is faded while the intended sweep of one ray and its neighboring air gaps are highlighted. 33° ray 1½° 1½° 11/12 ray air = 1/24 + 1/24
base-circle cadence33° : 3° = 11 : 1
one sector
36° = 33° + 3°

across ten repeats, the mark becomes 330° of ray rhythm held apart by 30° of air.

11/12

ray on the base circle

1/12

air on the base circle

intended cadence around the full base circle · one source ray repeated05 / 10

06 · the ring, determined · closed-form target

the stroke is what remains after the center and the air are fixed.

inner edge27/4 − √5/2
ring center13/2
outer edge25/4 + √5/2
ray base8 + √5/2
tip27/2
stroke = √5 − 1/2
air = 7/4
chosenr = 13/2

the distinct ring centerline

chosenair = 7/4

one quarter of the seven-unit center-to-tip span

determinedw = √5 − 1/2

the only stroke that satisfies both choices

two choices determine the third quantity06 / 10

07 · target cancellation

one irrational unit moves through the radii, then disappears.

radial cutaway showing the ray base and ring outer edge the ray base and ring outer edge both include the same square-root-of-five-over-two term. 7/4 air base = 8 + √5/2 outer = 25/4 + √5/2
same irrational offsettwo neighboring radii

in the closed-form construction, both boundaries carry the same √5/2 offset. the visible air between them does not.

8 + √5/2
− (25/4 + √5/2)
= 8 − 25/4
= 7/4
φ²

ray base minus ring center

7

tip minus ring center

¼ × 7

the air between ray and ring

the irrational terms shape the field; their difference becomes a simple fraction07 / 10

08 · the golden skeleton

ten points contain two pentagons, so φ belongs to the silhouette.

decagon and two pentagons inside the selected mark the ten ray tips make a regular decagon. alternating tips form two regular pentagons. 1 φ
10

tips at 36° intervals make one regular decagon.

2 × 5

alternating tips reveal two regular pentagons.

φ

inside either pentagon, diagonal ÷ side is the golden ratio.

2 sin 18° = 1/φ
decagon side = 27/(2φ)
pentagon diagonal / side = φ
φ is carried by the tip lattice before it appears in any formula08 / 10

09 · integer colour

the two hues are made from small whole-number vectors.

#FFCC33
51 × (5, 4, 1)

gold · hue 45°

green sits at 3/4 of the red-to-blue span.
3/4 → 45°
#E8913A
29 × (8, 5, 2)

orange · hue 30°

green sits at 1/2 of the red-to-blue span.
1/2 → 30°
45° − 30° = 15° of warm separation.
exact sRGB bytes · exact hue angles · no sampled colour09 / 10

10 · the result

the math is the balance

a single ray repeated ten times. a ring resolved from its center and its air. a golden lattice held by the silhouette. every relationship survives scale because it belongs to the construction.

16
22
32
48
64

geometry selected · 2026-08-15
download sha-256 ·
0b2d956e62d94bb9fd2f116319c44352
06895a43717d78297da9b1ba51777e43
presentation built · 2026-08-17 · selected geometry embedded exactly

selected aligned distinct-ring sol mark
selected geometry · static, self-contained, and drawn from one source10 / 10