JavaScript is off, so the live parts of this page are not running. Everything written below is complete on its own and says what each computation does and what it produced — the empty rectangles are where it would have run in front of you. Nothing here needs an account, and nothing is hidden behind the scripts; they compute, they do not reveal.

Moving visuals on this page — measured at 0.26% brightness change per frame and zero flashes per second; your reduce motion setting stops them. What moves, and how it was measured

Mathematical plates that draw themselves — prime counts, orbits and lattices rendered live rather than shipped as pictures. It runs continuously while on screen, and it pauses on its own whenever it scrolls out of view.

We measured it rather than guessing: the largest frame-to-frame change in average brightness is 0.26%, and it produces zero flashes per second against the threshold of three set by WCAG 2.3.1. That is well under the limit, and it is not a promise about any individual: if moving images are a problem for you, you should not have to assess ours.

Everything animated on this site honours your system’s reduce motion setting — turn it on in your operating system or browser and these pages stop moving, with nothing else lost. All the mathematics is in the text and the still figures.

Entry 2 of 7 in Numbers & structure Run itMeasured

Accelerated Collatz map Numbers & structure

The accelerated Collatz map, stepped live

Math: the Collatz conjecture, one of the best-known open problems in number theory — still unproven for all positive integers, which is exactly why the histogram below is worth watching, not just decoration. Dev: every step runs on JavaScript BigInt, not floating point, so there is no precision loss no matter how large the numbers get.

$T(x) = \\dfrac{3x+1}{2^{v_2(3x+1)}}$

One accelerated step: multiply by 3, add 1, then divide out every factor of 2 at once — skipping the individual halving steps a plain Collatz map takes one at a time. $v_2(3x+1)$ is that factor count.

Current step

T(x) = …

A new starting value is drawn at random each time a trajectory reaches 1.

Don't want to trust the display above either? not yet run for this trajectory (starts at 27) — click to verify independently

$v_2(3x+1)$ histogram, this session (0 samples)

Bars: empirical frequency observed live. Line: the exact density of the valuation over odd numbers, $P(v_2=k) = 2^{-k}$ .

What that histogram is actually saying

It is easy to watch those bars settle onto the line and miss that they are an argument, not a decoration. Since $P(v_2=k)=2^{-k}$ , the mean number of twos divided out per accelerated step is exactly 2. Each step multiplies by three and then divides by $2^{v_2}$ , so on average it multiplies by

$\mathbb{E}\!\left[\log_2 \tfrac{T(x)}{x}\right] = \log_2 3 - 2 = -0.415\ldots$

— a factor of 3/4 per step, in geometric mean. Measured over the first 200,001 odd numbers, as log₂ 3 minus the mean valuation: −0.415047 against a predicted −0.415037 (the direct mean of log₂(T(x)/x), which keeps the +1, gives −0.415031). That is the main heuristic reason to believe the conjecture: the map is a downward drift dressed up as chaos, which is why the heuristic expects every orbit to fall to 1. It proves nothing on its own.

And it is not a proof, which is the part worth dwelling on. A negative average says nothing about any particular trajectory: 27 climbs to 9,232 before it comes down, and an average is exactly the wrong instrument for finding the one starting value that never does. The heuristic has been convincing since the 1970s and the conjecture has been open the whole time — those two facts are compatible, and holding both at once is the accurate position. Everything on this panel is a measurement; none of it is evidence that the conjecture is true.

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