Little Laboratory — Interactive Scientific Explorations

The Little Laboratory is a small space on this site where equations become something to explore. It exists because a model one has only read is not yet understood; the moment of understanding usually comes from moving a parameter and seeing what changes. Each of its three instruments takes one standard model — a textbook result, not a research finding — writes it down exactly, and puts its parameters under the visitor’s hand.

§The approach

Every experiment is built the same way. The mathematics lives in a small, dependency-free TypeScript module that evaluates the model in closed form; a page component draws it as inline SVG and exposes its parameters as sliders and buttons; readouts print the quantities the model defines; and notes beneath the instrument state the equations, the assumptions and what to look for. A shared clock advances time for the animated experiments, re-rendering about thirty times a second, pausing while the tab is hidden, and never starting on its own for a visitor who prefers reduced motion. The sliders are native range inputs that announce their values, so the instruments work from the keyboard.

What a visitor can investigate is bounded by the model, deliberately. Nothing is measured, fitted or simulated beyond the equation on the page, and the notes say so. Two of the experiments are linked to research entries elsewhere on the site where the same mathematics or the same idea appears, with a sentence saying what the connection is and what it is not.

§The experiments

§I. Wave Superposition

Two sine waves travel along one string, $y_1 = A_1 \sin(k_1 x - \omega_1 t)$ and $y_2 = A_2 \sin(k_2 x - \omega_2 t + \Delta\varphi)$, and the string shows their sum. Both move at one speed $v$, so $\omega = v k$ — the non-dispersive case, an assumption of the instrument rather than a law. Sliders set the two amplitudes, the two wave numbers, the relative phase and the speed; the clock can be paused and reset; four presets jump to the instructive cases; and a toggle shades, at each point and instant, whether the two displacements pull the same way or opposite ways. When the wave numbers are equal the readout gives the resultant amplitude $\sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\Delta\varphi}$ and names the interference constructive, destructive or partial; when they differ it says, correctly, that there is no single amplitude and the sum beats. Open Wave Superposition.

§II. Gradient Descent

A point on the quadratic bowl $f(x,y) = \tfrac{1}{2}(a x^2 + b y^2)$ is moved against the gradient, $\mathbf{p}_{n+1} = \mathbf{p}_n - \eta\,\nabla f(\mathbf{p}_n)$ with $\nabla f = (a x, b y)$. The curvatures $a$ and $b$, the learning rate $\eta$ and the starting point are adjustable — the start can also be placed by clicking on the surface — and the path is stepped through one iteration at a time or played. The whole trajectory is computed in advance, at most two hundred steps, stopping when the gradient falls below $10^{-6}$ or the point has left any sensible frame. Because the coordinates decouple, each step multiplies $x$ by $(1 - \eta a)$ and $y$ by $(1 - \eta b)$; the instrument reads the regime off those factors and says whether the rate is converging, oscillating or diverging, with the bound $\eta < 2/\max(a, b)$ printed beside it. Level curves of $f$ are drawn as the ellipses they are. Open Gradient Descent.

§III. Reaction Kinetics

An idealised first-order reaction $\mathrm{A} \to \text{products}$: $d[\mathrm{A}]/dt = -k[\mathrm{A}]$, which integrates to $[\mathrm{A}](t) = [\mathrm{A}]_0\,e^{-kt}$ with half-life $t_{1/2} = \ln 2 / k$. Sliders set the initial concentration, the rate constant and the elapsed time, which can also be played; the plot’s window spans about six half-lives, dashed markers show the successive halvings, equally spaced, and a panel of one hundred dots keeps a tally of the fraction remaining — a tally, the caption says, not a simulation. The readouts give the concentration, the fraction remaining, the half-life and the instantaneous rate $k[\mathrm{A}]$; with $k = 0$ nothing reacts and the instrument says there is no half-life. No numerical integration is involved: the exponential is evaluated directly. Open Reaction Kinetics.

§Where it stands

Three instruments, open and working. The laboratory is meant to grow slowly, one well-understood model at a time, and only when there is something true and small enough to put under a slider. It is a way of thinking, not a research output: nothing here is new, and that is the point.