# Paper Boat

> Fold a paper boat from a sheet of A4, launch it down a rain gutter, and watch its metacentric
> height decide whether it survives. Free, entirely client-side, no account, no data
> collection, no model calls.

https://paper-boat.skillsafe.ai/

## What it actually does

The page computes the hydrostatics of a prismatic hull exactly, in the browser, by two independent
routes and checks them against each other:

- A SIMULATION that knows no stability formula. It rotates the hull section, bisects for the
  waterline that displaces the right area, and takes the centroid of whatever is under water. The
  metacentric height falls out of it as the initial slope of the righting lever.
- A CLOSED FORM that integrates the section curve symbolically and applies BM = I/V, GM = KB + BM - KG.

Over eight hulls the two agree to 4.616e-8 relative. The small-angle roll period T = 2*pi*k/sqrt(g*GM),
integrated from the simulated righting curve, agrees with the closed form to 2.780e-6 - and the same
comparison against a metacentric height that is wrong by 7.223e-4 percent still fails, so it is a
control rather than a restatement.

## The finding

A sheet of paper has a fixed girth, so every millimetre of beam is taken off the sides.

- A wider fold IS stiffer. Over the whole range in which these boats float upright, GM rises with
  beam, the roll period falls, and the deck-edge immersion angle falls. All three strictly monotone.
  Going from 35% of the girth in beam to 85% multiplies peak roll acceleration by 7.72.
- The widest fold is NOT the safest. The safest puts 0.690607424 of the girth into beam. At 0.97 the
  boat survives 135 times less gust energy. Folds within 1% of the best run from 0.660880704 to 0.717352377,
  bisected to 1e-9 and re-verified at each boundary.
- The objective is bimodal: a very narrow, very deep fold near 0.015 is a second local optimum,
  stable through a deep centre of buoyancy rather than a wide waterplane. Measured: beta=0.015 E=2.1131e-3 (37.5% of the global optimum)
- The failure mode is SWAMPING, not capsize. For a box section the free surface of shipped water is
  as wide as the waterplane, so i/V and BM are the same number: water aboard cancels the entire
  metacentric radius and leaves KB - KG. Measured over drifting voyages: {"landed":73,"capsized":14,"swamped":39} over 126 drifting runs
- A 12 g-laden A4 boat carries GM at 624% of its beam. Naval architects design ships for 3-5%.

## Three things the literature got adjusted

- The textbook small-angle range (0-15 degrees) does not transfer. For a paper boat the wall-sided
  law dies at 6.4623 degrees, where the WINDWARD BILGE lifts clear of the water - confirmed to equal
  atan(2d/B) by an independent geometric route (the closed form and the simulation differ by 3.083e-8). The boat floats too high for the usual limit to bind.
- The closed-form landmark B = W/3 that the thin-draught small-angle algebra produces has NO regime
  of its own: its two assumptions together demand D much less than B, i.e. beta much greater than
  1/3, contradicting its own answer. At beta = 1/3 the deck edge immerses at 83.5 degrees, where
  tan(theta)/theta = 6.009.
- The IMO A.562(14) empirical ship formula T = 2*C*B/sqrt(GM) overestimates a paper boat's roll
  period by a factor of 10.370, and the whole error lives in the radius of gyration that C stands in for.

## Contents

- / - the app. One page, no framework, no CDN, no third-party anything.
- /CREDITS.txt - the sources, what differs from a real paper boat, and the trade-mark record.
- /LICENSE.txt - MIT.

## Trade marks

TMview returned 65 marks matching "paper boat" on 2026-09-20. PAPER BOAT is a live registered word
mark in class 9 (software), class 41 (entertainment) and class 28 (games) in several jurisdictions.
No clearance is claimed and none was sought. See CREDITS.txt for the verbatim extract.

## Not a service

There is no API. Nothing is sent anywhere. The only persistence is your last fold and your best
distance, in your own browser's local storage.