Heat a solid and it grows — each dimension by a fixed fraction per degree. The law is runnable: length stretches as dL = α·L·dT, area by twice that, volume by three times. Down the center, data flows: the bar and the heat go in, the engine computes the stretch, the proven result comes out. The blue team builds and defends it; the red team tries to break it.
source E. Grüneisen, “Theorie des festen Zustandes einatomiger Elemente,” Annalen der Physik 344(12), 257–306 (1912) — DOI 10.1002/andp.19123441202 AMBER (1912 German original; DOI stable, full text paywalled). Rendered, not quoted.
A solid's atoms sit in a slightly anharmonic well; heat widens their average spacing. To first order every length grows by the same fraction:
dL/L = α·dT. Because area is length² and volume length³, the same α drives all three:
linear coeff α · area coeff 2α · volume coeff 3α — the exact expansions are (1+αdT)² and (1+αdT)³; the 2 and 3 are their first-order slopes, verified live in the engine.
Typical α (10⁻⁶ / K): invar 1.2, steel 12, copper 17, brass 19, aluminium 23.
Heat swells a solid: dL = αL·dT, with 2α for area and 3α for volume — the same expansion strain that, when a bar is constrained, becomes a stress, and that bends a bimetallic strip.
Clamp the ends and the free strain αdT is fought by an elastic strain of equal size, so σ = E·αdT — thermal expansion feeding straight into the-stress-strain, kin sphere. Each sphere is the next one's premise.
The blue team's live check: re-run the law over a sweep of metals and temperatures and confirm dL = αL·dT to 1e−12, the 2α/3α slopes, and the bimetallic bend direction. If red tampers, this badge is where it shows.
Feed the engine three numbers: a metal (its coefficient α), an original length L₀, and a temperature change dT (positive heats, negative cools). Each of the four kinds of quantity responds by a different power of α:
| quantity | grows by | coeff |
|---|---|---|
| length L | α·dT | α |
| area A | ≈ 2α·dT | 2α |
| volume V | ≈ 3α·dT | 3α |
| angle / shape | 0 | — |
Shape and angles are invariant under uniform expansion — the solid grows but does not distort. That is what you feed the panel below.
Every value is computed from dL = αL·dT and the exact (1+αdT)n on the spot — never looked up.
What the machine produces, proven: the linear change strictly proportional to both L₀ and dT (double either, double the stretch); the area factor (1+αdT)² and volume factor (1+αdT)³ with first-order slopes exactly 2α and 3α; a bimetallic strip that bends toward the lower-α metal on heating and reverses on cooling.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
Worse: some materials shrink on heating (negative α — water 0–4°C, ZrW₂O₈, invar's near-zero). Anisotropic crystals expand differently along each axis, so a single scalar α is already an averaged fiction. The linear law is a small-strain, isotropic, first-order tangent — true near room temperature, a lie at the extremes.
“Area expands by α, same as length.” Cut. Area grows by 2α, volume by 3α — the powers of the dimension. The engine derives all three from one α.
“A hole in a heated plate shrinks.” Cut. A hole expands exactly as if it were filled metal — every dimension scales by (1+αdT). The cavity grows.
“The bimetal bends toward the metal that expands more.” Kept, corrected. The high-α side gets longer, so it sits on the outside of the curve — the strip bends toward the low-α metal.
The red team's move: swap the area coefficient 2α into the length change, so a heated bar expands twice as much as it should. The blue team's witness (window 7) is watching.
Feed the length change the area slope and dL = αL·dT breaks: the witness recomputes, finds the stretch doubled, disagrees with the exact law, and turns red. Nothing is faked; the attack is real and it is caught.