The Number Seam | a staircase with nowhere to climb
Where the Motion Goes Still
Drag one control through two curves that rise by standing still. The first is made from ternary digits. The second is measured from a driven oscillator. Their resemblance is real, but it is not identity. The live computations below make the difference visible.
Layer one | one gesture, two staircases
Move through the stillness
On top, the chosen x is read in base three. A digit 1 means the point has entered a deleted middle third, so the function can never rise there. Below, the same position becomes the drive Omega of a lifted sine circle map. The map is iterated, not sketched.
The Cantor evaluator keeps at most twenty-four ternary digits. The circle map discards its transient before measuring rotation.
Cantor-Lebesgue function
F(x) = 0.50000000
deleted at the first ternary digit
Driven circle map
rho = computing
live orbit measurement
first ternary one, if present
zero on a resolved plateau
continued fractions
absolute distance from p/q
The upper curve is continuous, nondecreasing and locally constant outside the Cantor set. The retained set has length zero even though the function rises from zero to one. Its derivative is zero almost everywhere, not everywhere. The lower curve is a parameter response: each flat interval is a range of drives whose long-run rotation locks to one rational ratio.
Layer two | flagship depth computation
Turn the silhouette into a field
A staircase is only a cut. Each plateau opens into a two-dimensional Arnold tongue when forcing strength K is allowed to vary. Click the computed field to inspect the orbit that holds a tongue together. Then take the critical slice and ask how much parameter space remains outside the detected plateaus.
low denominator lockhigher denominator lockpainted K slice
Clicked orbit
Numerical witness
click a tongue
The product of map slopes over one candidate period estimates attraction. Magnitude below one is locally attracting.
recomputed on demand
transient plus measure
classification, not a theorem
where monotonicity can first fail
The residual set, measured rather than named
At the critical normalization K = 1, group adjacent samples with the same reduced rational label into plateaus. For each dyadic width r, sum the detected plateaus wider than r. The uncovered boxes are estimated by (1 - S(r))/r. Their log-log slope is the displayed dimension estimate.
No fit has been run in this session.
not run
slope of chosen window
not run
does not include systematics
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at selected cutoff
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grid-width sum
| samples | D | fit SE | plateaus | coverage | fit r range |
|---|---|---|---|---|---|
| Run one or all resolutions. | |||||
This can approach the historical estimate, but it cannot certify it. A sample is labeled rational when its measured rotation lies within half a net turn over the measurement window. That tolerance shrinks if the window grows. Near tongue boundaries, slow convergence moves classifications. Change the denominator ceiling or the fit window and the number moves. That movement is part of the result.
Layer three | the open edge
The decimal is not closed
The exact Hausdorff dimension of the non-mode-locked parameter set for this critical sine circle-map slice is not established by the sources opened for this page. Jensen, Bak and Bohr reported a numerical estimate of . Graczyk and Swiatek proved D >= 1/3 and D < 1 for the critical family they studied. A later Farey-Brocot calculation obtained for a related measure and explicitly framed the correspondence to the circle-map value as a conjectural universality argument. We could not find this as of 2026-08-01: a source that turns the sine-map decimal into an exact constant. The live fit above is therefore a convergence experiment against an old numerical estimate, not a proof or a new measurement standard.
The narrower open neighbor is universality. Which critical-map smoothness classes share the same residual dimension, and in which precise notion of dimension, remains a question that a single sine-map raster cannot answer.
The check: equations, transcriptions and free choices kept apart
recomputing the fixed anchors
| quantity | live result | route |
|---|
Every free choice and numerical uncertainty
- The Cantor control has a dyadic step of 1/8192. Its evaluator stops after 24 ternary digits. If no deleted interval is reached, it reports the midpoint of a binary interval of width 2^-24. Plateau values are exact within floating-point representation.
- The immediate circle-map readout fixes K = 0.9, initial lift x0 = 0.123456789, 256 discarded iterates, 768 measured iterates and denominator ceiling 32. The background slice uses 512 drive samples.
- The Arnold raster fixes 160 by 96 cells, 128 discarded and 256 measured iterates, denominator ceiling 24, a half-turn classification tolerance, and an attracting multiplier threshold of 0.98. These are detection choices. Dark cells can mean irrational response, unresolved locking, a weak boundary or a lock rejected by the stability threshold.
- The dimension sweep fixes K = 1, 512 discarded and 1536 measured iterates. A rational is accepted within 1/(2 x 1536). A plateau needs at least two adjacent samples. Width is measured between sample centers, so every boundary is uncertain by about one grid spacing.
- The default regression uses dyadic widths from 1/16 down to at least eight grid spacings. The fit-window control can discard additional fine scales. The printed standard error is only ordinary least-squares uncertainty conditional on those points. It excludes classification bias, correlated scales, finite transient error and the choice of denominator ceiling.
- All browser arithmetic uses IEEE 754 double precision. No interval arithmetic is claimed. Canvas pixels are a view of arrays whose numerical summaries are printed separately.
Source values that are transcribed, never called computed
What each staircase actually means
The Cantor-Lebesgue function is a specific singular cumulative function. Its complement of plateaus is the middle-third Cantor set, with dimension computed from exact self-similarity. The circle-map staircase is a response curve of a nonlinear dynamical system. Its residual set, mechanism and dimension differ. A monotone graph itself has graph dimension one, so neither 0.630929... nor the historical 0.87 is the graph dimension.
Complete does not mean that every point is inside a plateau. It means the plateau interiors have full Lebesgue measure. The residual set can still be uncountable. Arnold tongues are the two-dimensional locking regions; a staircase is one fixed-K cut through them.