Bullet Hell Dodge & Covering Density

Move the mouse or drag a finger to fly. The dashed circle is the largest place on the screen where nothing can touch you right now, found by refinement rather than by sampling — watch it collapse as you raise the density. Everything runs in your browser; nothing is uploaded.

Covering the screen costs 13 to 16 times what covering its area suggests

The obvious estimate for "nowhere is safe" is the density at which the bullets' combined area equals the screen: A / πr². It is not close. Random discs land on top of one another, and the last few gaps are stubborn — closing them takes a logarithm's worth of extra bullets.

On a 600×600 screen, dodging with a combined radius of 14 pixels, the screen is not genuinely closed until about 7,508 bullets are on it — against 585 from the area. The multiple never strays far: 12.8× at one radius and 16.4× at another, because what separates the two columns is a logarithm rather than a constant.

A threshold is not a number until its criterion is stated, so here is this one: the density at which the mean largest empty circle, over four independent fields, falls to the radius. Change the criterion and you change the number.

The free area is exactly what the model says

Scatter n bullets of radius r over area A and the fraction of screen still safe is exp(−n·πr²/A). The measurement is not an approximation to it — it lands on it, worst case 0.0052 away across the whole range.

Which raises the obvious question: if the free fraction is an exponential, it is positive at every density, so is anywhere ever certainly covered? That is the wrong question to ask of the formula, and the reason the thresholds above are measured rather than derived. An exponential describes the average free area. What kills you is the largest single gap, and that is a different quantity with a different law.

The largest gap shrinks as √(A ln n / πn)

The largest empty circle in a field of n random points has a known scaling law, and the measured gap tracks it to a constant near 1.24 across a twenty-fold change in density. That ln n is the entire story of the table at the top of this page: it is what makes coverage cost so much more than area.

Measuring that gap with a grid is biased, not merely imprecise

The natural way to find the largest gap is to sample a grid and keep the best point. It cannot work, and the failure has a direction. A grid only finds a gap that happens to contain a sample point, so it can never report a gap larger than the true one — every estimate sits below the answer and climbs towards it as the grid tightens.

At 10,000 bullets even a 270×270 grid — 72,900 samples — is 9.5% short. A one-directional error is worse than a noisy one: it means a grid declares the screen covered earlier than it truly is, and a covering threshold built on grid samples comes out too low. So the numbers here come from a coarse pass that keeps eighteen candidates and refines each one locally with a shrinking step.

Sampling harder is not the fix — a finer grid can be worse

The obvious repair is to sample more finely. It does not reliably work, and this is the part that caught us out. "A finer grid cannot do worse" is only true when the finer sample set contains the coarser one — and samples at cell centres, (i + ½)·W/n, do not nest under doubling. A 60-grid reproduces 0 of a 30-grid's coordinates. A 90-grid reproduces all 30.

Quadrupling the work returned an answer 8.8% worse, because the doubled grid landed between the coarse grid's points and missed the one that had found the gap. Tripling recovered it exactly, as nesting requires. Across 120 trials on 40 fields, doubling lost 16 times (13% of them, once by 19.6%) and tripling lost 0. Nothing, at any resolution, ever beat refinement.

So the two halves of "a grid understates the gap" have completely different standing. Never overstating is geometry and holds always. Improving as you refine is a property of the sampling scheme, and only survives if the grids nest. The table further up steps 30 → 90 → 270 for exactly that reason.

Where the arithmetic gives up before the mathematics does

What this page does and does not answer

Everything above is the static question: at what density does the screen contain no safe point at all. Real survival is strictly harder, because a safe point also has to be reachable in the time you have and has to still be there when you arrive. The densities here are therefore an upper bound on what is survivable, not an estimate of it — the screen becomes undodgeable some distance before it becomes uncoverable.

The four measured columns are frozen — sweeping them live costs about twenty-eight seconds per build. Every derived quantity on this page is still computed at build time from those rows, and one row of each table is re-measured live by the same shipped function and compared: the free fraction at 200 bullets came back 0.7095 again just now.