References
Claim Register, Fun Facts, and Sources
This page keeps the evidence model explicit: claims, facts, and source trail all stay inspectable.
Claims
| ID | Status | Statement |
|---|---|---|
| verified-euclid-cf | Verified | Running the Euclidean algorithm on positive integers yields the finite continued fraction of a/b. |
| verified-bezout | Verified | Extended Euclid produces coefficients x and y with a*x + b*y = gcd(a, b). |
| verified-mod-inverse | Verified | If gcd(a, b) = 1, the x coefficient from extended Euclid gives the inverse of a modulo b. |
| verified-rectangle | Verified | Rectangle-to-squares dissection follows the same quotient sequence as the Euclidean algorithm. |
| verified-golden-special | Verified | The logarithmic-spiral story belongs to the special golden-rectangle case, not to every Euclidean rectangle dissection. |
| verified-gear-design | Verified | Continued fractions are a valid way to design good rational gear ratios; the resulting gear train then realizes the chosen fixed ratio. |
| verified-visible-coprime | Verified | Visible lattice points from the origin correspond to coprime integer pairs. |
| verified-rhythms | Verified | Euclidean rhythm constructions distribute pulses as evenly as possible across a cycle and connect directly to the Euclidean algorithm. |
| analogy-phase | Analogy Only | A single division step can be visualized as q full turns plus a residual phase r/b, but that is an analogy rather than a full autonomous machine. |
| analogy-automatic-gears | Analogy Only | Automatic compound-gear or sliding-gear Euclid machines remain a conceptual analogy unless sensing, control, and reconfiguration are added beyond a simple gear train. |
| false-center-shift-remainder | False | Changing the center distance of involute gears does not make the shift itself equal the integer remainder. |
| false-generic-spiral | False | Generic inward-spiral or logarithmic-spiral claims for arbitrary gear or rectangle models are not valid in general. |
| false-historical-gear-euclid | False | Historical compound gear trains did not themselves implement Euclid as an exact iterative algorithm on arbitrary inputs. |
Fun Facts
| Title | Body | Source |
|---|---|---|
| Ancient, Still Useful | Euclid's algorithm is one of the oldest named algorithms that is still used directly in modern software. | euclid_book_vii |
| Lamé's 1844 Result | Gabriel Lamé's Euclid analysis is a classic early complexity result: Fibonacci pairs force the slowest standard run. | mathworld_euclidean_algorithm |
| A Clockmaker's Tree | Achille Brocot was a clockmaker. The Stern-Brocot tree has real gear-ratio history, not just abstract number theory. | mathworld_stern_brocot |
| Not That Ford | Ford circles are named after Lester Ford, the mathematician, not Henry Ford. | mathworld_ford |
| Coprime Means Visible | A lattice point is visible from the origin exactly when its coordinates share no common factor. | mathworld_orchard |
| Math That You Can Hear | Euclidean rhythm patterns show up in multiple traditional musical families because they spread beats as evenly as possible. | toussaint_rhythms |
| Special, Not Generic | The golden spiral story depends on self-similarity. Arbitrary Euclidean rectangle cuts do not automatically inherit it. | golden_rectangle |
| Fast Math Rendering | Typeset equations help when the same integer process is shown as division, continued fractions, and recurrences. | bell |
Sources
| Title | Note | Link |
|---|---|---|
| Jordan Bell, The Euclidean algorithm and finite continued fractions | Primary reference for Euclid, continued fractions, and convergents. | https://jordanbell.info/LaTeX/euclideanalgorithm/euclideanalgorithm.pdf |
| Clark University, Euclid's Elements, Book VII | Readable translation of Euclid's arithmetic algorithm setting. | http://aleph0.clarku.edu/~djoyce/elements/bookVII/bookVII.html |
| Cambridge DANotes, Gear Meshing | Mechanical grounding for what actual involute gears do and do not compute. | https://www-mdp.eng.cam.ac.uk/web/library/enginfo/textbooks_dvd_only/DAN/gears/meshing/meshing.html |
| The Mathematical Gazette, Golden rectangles and the logarithmic spiral | Reference for the special golden-rectangle/logarithmic-spiral case. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/8B3A00A26C1E9FF5CB8A4D9340D87EBD/S0008439500004603a.pdf/div-classtitleGolden-rectangles-and-the-logarithmic-spiraldiv.pdf |
| G. S. Chirikjian, The Golden Spiral | Clarifies the special self-similar spiral construction. | https://link.springer.com/chapter/10.1007/978-3-662-68931-8_8 |
| GlobalSpec, Graphical method of using continued fractions to find the best gear ratio | Historical engineering direction: Euclid and continued fractions help design gear trains. | https://www.globalspec.com/reference/68680/203279/graphical-method-of-using-continued-fractions-to-find-the-best-gear-ratio |
| Godfried Toussaint, The Euclidean Algorithm Generates Traditional Musical Rhythms | Foundational source for Euclidean rhythms. | https://archive.bridgesmathart.org/2005/bridges2005-47.html |
| Wolfram MathWorld, Euclidean Algorithm | Compact reference for Lamé, Fibonacci worst cases, and algorithm variants. | https://mathworld.wolfram.com/EuclideanAlgorithm.html |
| Wolfram MathWorld, Farey Sequence | Reference for Farey neighbors and rational ordering. | https://mathworld.wolfram.com/FareySequence.html |
| Wolfram MathWorld, Ford Circle | Reference for Ford-circle geometry. | https://mathworld.wolfram.com/FordCircle.html |
| Wolfram MathWorld, Euclid's Orchard | Reference for visible lattice points and orchard geometry. | https://mathworld.wolfram.com/EuclidsOrchard.html |
| Wolfram MathWorld, Stern-Brocot Tree | Reference for the mediant tree and gear-ratio history. | https://mathworld.wolfram.com/Stern-BrocotTree.html |
| Wolfram MathWorld, Modular Inverse | Reference for modular inversion and coprime gating. | https://mathworld.wolfram.com/ModularInverse.html |
| Wolfram MathWorld, Periodic Continued Fraction | Reference for periodic continued fractions of quadratic surds. | https://mathworld.wolfram.com/PeriodicContinuedFraction.html |
| Wolfram MathWorld, Lagrange's Continued Fraction Theorem | Reference for eventual periodicity of quadratic surds. | https://mathworld.wolfram.com/LagrangesContinuedFractionTheorem.html |