# The Finger Trap

> An in-browser laboratory and escape game built on the biaxial braid — the geometry of the
> finger trap, the cable grip and the orthopaedic traction sleeve. Pulling a braided tube
> lengthens and narrows it; this page reproduces that from exact helix kinematics and then
> computes, in closed form, exactly how hard you would have to pull to get out.

Site: https://finger-trap.skillsafe.ai/
Cost: free. No account, no server call, no model. Everything runs in the browser.

## What it answers

The claim under test is the one everybody repeats: "the harder you pull, the tighter it
grips, so pulling can never free you."

- The first half is TRUE and exactly so. The inward line load is W = W0 + T tan^2(theta)/R,
  linear in the braid tension T. Doubling the pull doubles the squeeze and no more.
- The second half is FALSE. Because the squeeze is exactly proportional, the applied force
  cancels out of the escape question. The holding force is finite:
  Fmax = W0 R cot^2(theta) (e^Lambda - 1), with Lambda = mu ell tan^2(theta) / R.
- The sharpest form: set the snugness W0 to zero — friction and braid intact, no
  interference — and Fmax is exactly zero at every braid angle and every friction
  coefficient. The braid multiplies snugness; it cannot create it. A trap loose on a thin
  finger cannot be made to grip by pulling, however hard.
- Push or pull? Fmax / Prelease = e^Lambda - 1 exactly, so the folk advice ("push the ends
  together") is right precisely when Lambda > ln 2, and wrong below it.
- The locking boundary: the trap holds iff mu (ell/R) tan^2(theta) > ln(1 + F* tan^2(theta)
  / (W0 R)). If mu*ell/R exceeds the normalised pull it holds at every braid angle;
  otherwise there is a critical braid angle below which you pull straight out.

## How it is verified

Every headline number is produced twice by routes that share no code.

- Kinematics L = b cos(theta), D = b sin(theta)/(n pi) are checked against a polyline helix
  whose length is a chord sum and whose radius is found by bisection. Second-order
  convergence table included.
- The widest-tube angle arctan(sqrt 2) = 54.735610317 degrees is RECOVERED by a
  golden-section maximiser handed only the numerically measured volume and never told the
  answer. The same maximiser on lateral area returns 45 degrees — a different angle.
- The squeeze law W = T tan^2(theta)/R is checked against discrete polygon statics on a
  closed helix, with no curvature formula used.
- The holding force is checked against a dynamic Coulomb simulator — springs, masses,
  friction impulses, a stick test — that contains no exponential and no differential
  equation.
- The critical braid angle is located analytically as a transcendental root and,
  independently, by bisecting the simulator's own hold/escape verdict.

## Pages

- / — Pull (kinematics lab), Escape (six traps, four of them beatable by pulling),
  Locking map, Findings (every measured number), Help.
- /CREDITS.txt — full provenance, every constant tagged DOCUMENTED / qualified / DERIVED /
  MEASURED / RECONSTRUCTED, with a URL per source.
- /LICENSE.txt — MIT. The finger trap itself is a public-domain folk object.

## Provenance and honesty

The finger trap has no inventor and no trademark is claimed. A one-ended German version is
documented from 1870; the device is not of Chinese origin. A 2024 refereed paper on the
geometry and mechanics of the finger trap exists (Lu et al., Extreme Mechanics Letters 71,
102200); only its abstract could be read here, and the two claims taken from it are marked
as such. No physical trap was measured, so all six traps' dimensions are RECONSTRUCTED and
labelled. The model has no strand-strength limit, no strand stretch, a rigid finger and no
twist, and Help says so.