The Fibonacci 60-Ring
Last digits of the Fibonacci sequence (seeds 0, 1, mod 10), one digit every 6°: position 0 at the left (9 o'clock), counted clockwise, so 15 sits at the top, 30 at the right and 45 at the bottom. Mirror pairs that sum to ten are the chords of Neal's Net: horizontal at odd positions, vertical at even, gaps only at the four cardinal 0s. The same mirror law draws the net's twin, twenty-eight equal-digit chords crossing the net at right angles, and the two grids cut the ring into fourteen inscribed rectangles: corners n, 30−n, 30+n, 60−n, digits d, d, 10−d, 10−d (V67).
The ring, position 0 → 59
Top row positions 0–29, bottom row 30–59. Every column is an antipodal pair: top + bottom = 10 — the half-turn law, visible at a glance (the two 0-columns are the cardinals). Generated live as Fibonacci mod 10 — not stored.
Digit laws on this ring
Rectangle repair — one struck digit, named by its three corners (V68)
A corrupted corner is the unique corner that disagrees with the other three while those three still agree. Its twin restores it — no total consulted. The paper's example: rectangle 7, 23, 37, 53 reads 3, 7, 7, 3; strike 23 to 2.
strike position to digitAngle input & checksummed folds
Quantize: position = θ ÷ 6 (mod 60). Folds are jumps on the ring; every jump lands on a digit that must stand in a known relation to the starting digit. Signs come from the jump direction — never from lookup. The three reductions n+30, 60−n, 30−n are the other three corners of the position's rectangle, and every jump is a move inside the position's octad: the landing digit is the start digit times 1, 3, 7 or 9 — one test for every move (V70, V71).
Fold ledger (this angle)
COS — the two-ring ruler, read between the centres
Two unit rings slide apart until their crossings pin to marks 2k steps apart (6k° = θ). The centers then stand exactly 2·cos 6k° ring-radii apart — half the measured distance is the cosine — and the two crossings stand 2·sin 6k° apart, half of that the sine: one setting reads both, and their ratio is the tangent (V73). With the join axis through positions 0 and 30 (the classic side-by-side vesica of the worked examples) the crossings sit at k and 60−k on the sliding ring and the still ring's marks under them are 30−k and 30+k: the four corners of one rectangle, digits d, d, 10−d, 10−d, so every setting certifies four digits and the fourteen settings are the fourteen rectangles (V69). With the axis through 15 and 45 the crossings sit at 15−k and 15+k under the same parity test — equal for odd k, sum to ten for even k. Only these two cardinal axes certify every setting; the other eight 18° orientations certify only k = 5 and k = 10 (V69).
SIN — the same setting, read across the crossings
The sine is read at the cosine's own setting. Pin the crossings 2k steps apart (6k° = θ) and the two crossing points stand exactly 2·sin 6k° ring-radii apart — half the crossing-to-crossing distance is the sine. One setting gives both projections: the centers read the cosine, the crossings read the sine, and their ratio e ÷ d is the tangent (the companion paper Two Sliding Rings; V73). The quarter-turn stamp d(15−n) ≡ 7·d(n) for odd n, ≡ 3·d(n) for even n (mod 10) certifies the link sin 6k° = cos 6(15−k)°, the octad move of a reflection followed by a quarter turn (V70). Both readings appear in the ledger.
TAN — the gnomon at the east cardinal
A vertical line stands at the east cardinal 0 (position 30). The ray from the center through position 30−n, the position’s partner across the up-down line, extended to that line, marks the length tan 6n°. Certificates: level pairs (n, n+30 — equal tangent, digits sum to ten), reciprocal pairs (n, 15−n — product exactly 1, the ×7 stamp: the octad move of a reflection followed by a quarter turn, V70), and the triple-product row tan θ·tan(60°−θ)·tan(60°+θ) = tan 3θ at the 53 valid rows.
The zoom ladder
Every step opens into a finer 60-division ring: level L has 60·60L positions of 6⁄60L degrees each, and the digits tile with period 60. Because 10 divides 60, any angle typed with finitely many decimal digits of degrees lands exactly on a rung.
The rungs — the top ring is level 0
| Level | Step size | Positions | Angles on this ring |
|---|---|---|---|
| 0 | 6° | 60 | 0°, 6°, 12°, … 354° — the 6° grid |
| 1 | 6′ = 0.1° | 3,600 | every multiple of 0.1°: 0.1°, 0.2°, 7.3°, 359.9° … |
| 2 | 6″ ≈ 0.00167° | 216,000 | every multiple of 1/600 of a degree |
| L | 6/60L ° | 60·60L | every multiple of that step |
The step is always six of the next unit down (6° → 6′ → 6″). The rungs nest — every coarser angle also sits on every finer ring — and the app reports the minimal rung: the shallowest level where the angle is a whole position.
The digits tile with period 60, so every step opens into a complete 60-ring carrying the identical net — the net survives verbatim at every rung (V10: zero violations at level 1). Hover the zoomed ring to match pairs at fine resolution.
The wheels — the precision panel (companion paper The Sixty-Ring as a Calculator)
Carried in the companion paper The Sixty-Ring as a Calculator. Arbitrary precision without series: the half-angle wheel descends from any anchor, Heron's wheel closes on any square root quadratically, and the π demo runs Archimedes' ladder from the vesica anchor sin 30° = ½. The convergence moduli are the system's own numbers: Heron doubles its digits per turn; the polygon error quarters per doubling.
Half-angle wheel cos(θ/2) = √((1+cos θ)/2)
| rung | angle | value | err vs Math |
|---|
Heron's root wheel x → (x + a/x)/2
a = x₀ =| turn | x | error | digits |
|---|
The π demo — Archimedes' polygon ladder from sin 30° = ½
| k | polygon | n·sin(180°/n) | error vs π | ratio (≈4 per doubling) |
|---|
The number π never closes — only its digits flow, to any depth. This is the system's boundary, shared by every computational system ever built.
The three polygon checks (Section 5)
Draw the triangle (corners every 20 steps), the square (every 15), or the pentagon (every 12) from any start. Each shape carries its own digit check from a different engine of the ring — and the same three shapes are the ones that force the modulus in the first place.
The check, read live
Why sixty — the surd engine (companion paper The Sixty-Ring Beyond the Net)
A ring that hosts the triangle, the square and the pentagon at once must have a position count divisible by 3, 4 and 5 — and the smallest such count is 60. Read the rulers at the polygons’ lock angles and the three construction surds fall out.
The locks, computed live
The four sister nets — one net, four slants (companion paper The Sixty-Ring Beyond the Net, V36)
Lay the 24-cycle of digital roots (add a number’s digits until one digit remains) from a different starting pair and the whole sum-to-nine net tilts — by exact 15° clicks. The tilt is a fingerprint of where the ring was started.
Families found by enumeration
The Verify page — Appendix A: the core set V1–V26, V28, V29, V38, V40, V57–V71, V73, with V27, V30, V36, V46, plus acceptance
Exact claims by exhaustive enumeration; value identities to machine precision. If any check fails, the implementation — not the mathematics — has a bug. Acceptance: all checks green, and sin/cos/tan of every grid angle agree with the platform's Math functions to 10⁻¹².
The boundary — what this system does NOT claim
Stated before anyone has to ask. Each of these is a boundary the paper itself draws, not a concession extracted by a critic:
| check | claim | result |
|---|
What is this thing? (the plain-language tour)
A companion to the paper Neal's Net: A Self-Verifying Trigonometric System on the Fibonacci 60-Ring by Neal Strassner — written here for anyone from middle school up.
The one-minute version
Take the Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… (each number is the two before it, added). Now keep only the last digit of each: 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9… Here's the surprise: that list of last digits repeats forever every 60 steps. Sixty — the same number as minutes on a clock. So put the 60 digits around a circle, one every 6°, like a clock face: this page starts at the left (9 o'clock) and counts clockwise.
Then comes the discovery this whole page is named after: pick almost any digit on that circle and look at its mirror partner across the ring — the two digits always add up to 10. Draw a line between every pair and you get a woven grid of perfectly horizontal and vertical lines: Neal's Net. The only gaps are four zeros sitting exactly at the compass points (0°, 90°, 180°, 270°).
Why "net" and not just "pretty pattern"?
Look at the barcode on any cereal box: its last digit is a check digit — a built-in test. If a scanner misreads one number, the check digit won't match, and the machine knows something went wrong. Neal's Net does the same job for this circle: if any single digit were wrong, its partner across the ring wouldn't sum to 10 anymore, and the net would visibly break at that exact spot. The pattern checks itself.
The big idea of the paper: since angle math can be done on this circle, every step of angle math can carry that built-in test. Your calculator gives you an answer and asks you to trust it. This system gives you an answer and a receipt.
How each page works
1 · Ring — the object itself. Hover any digit to light up its partner and watch them add to 10. The "trace the net" button walks through all 28 pairs one by one. Turn on the twin grid and the same hover lights a whole rectangle: four corners reading d, d, 10−d, 10−d — and the repair card shows that one struck corner is named by the other three and restored from its twin.
2 · Angles & Folds — angle math as clock math. The angle 216° becomes "position 36" (just divide by 6). Adding angles is adding positions; flipping an angle is a jump across the ring. After every jump, the digits at takeoff and landing must match by a known rule — that's the receipt. Get the jump wrong, the digits won't match, and you see a red ✗.
3 · COS — two identical rings overlap like two hula hoops (the lens shape in the middle is the ancient "vesica piscis"). Slide them until their crossing points land on marked dots, then measure the distance between the two centers. Half of that distance IS the cosine — and the two crossing points stand apart by twice the sine, so the same setting hands you the sine as well (tab 4 reads it that way). You didn't look the value up — you made it, and the digits at the crossings certify the setting was right.
4 · SIN — the same two rings, read the other way. At the setting that gave the cosine, the two crossing points stand apart by twice the sine, so half that distance is the sine: one setting gives both, centers for cosine, crossings for sine. The quarter-turn digit stamp certifies the link sin 36° = cos 54°.
5 · TAN — a sundial. Stand a pole at the edge of the circle and shine a ray from the center; the length of the "shadow" on the pole is the tangent. It carries three different receipts, including pairs of tangents that must multiply to exactly 1.
6 · Zoom ladder — "but the circle only has marks every 6°!" True — and a clock only has marks every hour, until you look at the minute hand. Every step of this ring opens into a complete smaller ring of 60 finer steps (6° → 6 arcminutes → 6 arcseconds, forever), and — this is the remarkable part — every smaller ring carries the exact same digits and the exact same net. The checking never stops working, no matter how deep you zoom.
7 · Wheels (companion paper The Sixty-Ring as a Calculator) — how people computed before electronics: take a simple step and repeat it. Heron's wheel squeezes out square roots (the correct digits double every turn). The π wheel is Archimedes' actual 2,200-year-old method: start with a hexagon, keep doubling its sides, and watch 3.000 → 3.106 → 3.139 → 3.14159… pour out.
8 · Polygon checks — the triangle, the square and the pentagon inscribed on the ring, each with its own digit test (Section 5). 9 · Why 60 and 10 · Sister nets — two readings carried in the companion paper The Sixty-Ring Beyond the Net: why the count must be sixty, and the sum-to-nine nets of the digital-root ring.
11 · Verify — the report card. Every claim of the paper's Appendix A core set (V1–V26, V28, V29, V38, V40, V57–V71, V73, with V27, V30, V36, V46) is re-tested live on your computer the moment you press the button, plus a final exam: the system's homemade sine/cosine/tangent values must agree with your device's built-in math to 12 decimal places.
"How do I know it's really doing what it says?"
Three answers, from quickest to deepest:
1. Every answer shows its work. Open the Ledger panel (right side) and click "▸ show the work" under any entry — you'll see the actual computation, line by line, with the real numbers: the division, the jumps, the digit tests, the measurement, the halving. Nothing happens off-stage.
2. The Verify page re-proves the paper on your machine. It doesn't display stored results — it re-runs every enumeration and identity from scratch, in front of you, every time. If this app's code had a bug, those lights would go red.
3. The whole program is one open file. No internet connection, no hidden libraries, no server. Right-click this page → "View Page Source" and you are reading every line of code that exists. The ring digits aren't even stored — they're regenerated as Fibonacci-mod-10 each time the page loads, and the net immediately checks them.
Who made this
The net — the pairing-to-ten structure on the Fibonacci last-digit ring — was independently identified by Neal Strassner around 2015 (github.com/NealStrassner). The paper develops it into a complete, self-checking system for trigonometry, and this app implements the paper's Appendix B so anyone can drive it. The underlying facts have a documented lineage: the 60-cycle itself is classical (Lagrange recorded it in 1774; Wall formalized it in 1960); the circle arrangement and the opposite-digits-sum-to-ten observation circulate widely in recreational literature from at least the early 2010s (Jain 108, Lucien Khan, and others all claim the wheel — a priority contest this work does not adjudicate); and the antipodal and quarter-turn congruences were proven rigorously by Guyer, Mbirika & Scott (Rocky Mountain J. Math., 2024). Many independent readings have been laid over these same sixty digits — numerological, harmonic, horological, theorem-proving — and not one of them ever drew the net: the parity-alternating grid of perfectly horizontal and vertical sum-to-ten chords, and its reading as a checksum, appear in no prior source found. The net, and the verification layer built on it — angle arithmetic that checks itself — are this work's contribution, beginning exactly where the prior literature stops.
The code itself was written by Claude (Anthropic's AI) working under Neal's direction — the same division of labor the paper's acknowledgments describe: the mathematics, the discoveries, and the design decisions are Neal's; the implementation and the machine-checking are the AI's. Fittingly for this project, you don't have to take that on faith either: the Verify tab re-tests everything the code claims.
License: CC BY 4.0 — you may share and adapt this file, but you must credit Neal Strassner and link back.