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 1 of 2 in Machines Play with itMeasured

Real physics, for scale Machines

The very large and the very small — both computed live, from real constants

Math: Kepler's third law (T² ∝ a³), the Planck–Einstein relation (E=hc/λ), and a third piece that's more about computation than physics itself: the same differential equation solved three different ways, one of which is provably unstable on this exact problem, not just imprecise. Dev: none of this is a recorded clip or a hand-tuned illustration — the orbit speeds are computed live from each planet's real orbital period, the photon energy recomputes as you type using the exact SI-defined values of h and c, and the integrator comparison re-simulates from scratch every time you move the step-size slider.

Three pieces of general scientific background, included because they're genuinely worth knowing, not because KodamaSEC does space, optics, or numerical-methods research — we don't. Left: the Solar System's real orbital periods (Mercury 0.241 years to Neptune 164.8 years), radii compressed so all eight fit on screen (distance is NOT to scale — Neptune is really about 78× farther from the Sun than Mercury), but angular speed is real, which is why Mercury visibly laps Neptune again and again: that's Kepler's third law playing out, not an animation trick. Middle: light's particle side. A single photon's energy depends only on wavelength — type one in and see it computed, from ultraviolet through the infrared band (where most of a warm object's own radiated heat actually is) into the microwave. Right: the same simple pendulum-like equation, integrated forward by two classic numerical methods and compared against its own exact closed-form solution — drag the step size up and watch one of those methods fail in real time.

Kepler's third law: T² ∝ a³, live

Verified before this shipped: T²/a³ for all eight real planets, using the AU/year figures shown below (each rounded to 3 significant figures for the diagram), comes out within 0.21% of exactly 1, worst case Mercury — the residual is mostly rounding in the displayed data; the law itself holds to about 0.1% for the planets, because their own mass enters it.

Photon energy: E = hc/λ, live

Wavelength (nm)type a number, the energy below updates live

$h = 6.62607015\times 10^{-34}\,\mathrm{J\,s}$ and $c = 299\,792\,458\,\mathrm{m/s}$ — both exact by SI definition (c since 1983, h since the 2019 redefinition), not measured-then-rounded. Infrared runs roughly 700nm–1mm; try 550 (green light, the eye's peak sensitivity) or 10000 (mid-infrared, near where a warm human body radiates most of its heat) to see the energy drop by more than an order of magnitude as wavelength grows — the same relation Planck introduced in 1900 and Einstein used in 1905 to explain the photoelectric effect, not a KodamaSEC result.

Numerical integrators: one equation, three outcomes

$x'' = -\\omega^2 x$

Step size Δt0.10

Same ODE, same starting point $x_0=1,\\ v_0=0$ , three ways of stepping it forward. Explicit Euler ( $x_{n+1}=x_n+\\Delta t\\,v_n$ ) is a genuine numerical instability on this problem, not just low resolution — verified before this shipped: even a 10× smaller step still diverges, just slower. RK4 averages four slope estimates per step and stays locked to the exact solution ( $x(t)=x_0\\cos\\omega t+\\tfrac{v_0}{\\omega}\\sin\\omega t$ ) across the same run. Drag the slider and watch it happen — energy conservation (which the exact solution holds perfectly) is the tell: Euler's grows without bound, RK4's barely moves.

Next in this room →Exact BigInt vs. floating point

Back to Machines · Back to the Notebook

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