I first encountered Diamond Design in the Gemological Institute's archive, where the original 1919 pamphlet sits in a temperature-controlled drawer alongside Tolkowsky's hand-annotated correction sheets. What struck me was not the mathematics — Snell's law applied to a tetrahedral model is straightforward optics — but the audacity of the claim: that 34.5 degrees, the crown angle, is provably optimal for light return in a round brilliant diamond.
Pavilion Depth Is the Real Constraint
Most discussions of the Tolkowsky model fixate on the crown. But the pavilion — the lower portion below the girdle, where angle tolerances tighten to fractions of a degree — imposes harder physical limits. At a pavilion angle of 40.75 degrees, total internal reflection traps light for exactly one internal bounce before it exits through the crown. Steepen by half a degree and you gain fire at the cost of brilliance; shallow by the same margin and light leaks through the pavilion base into the mounting.
The ideal cut is not a single point. It is a narrow corridor through parameter space, and Tolkowsky was the first to map its walls.
Modern ray-tracing confirms what cutters in Antwerp and Surat have known empirically for decades: the Tolkowsky proportions describe a local optimum, not a universal one. Diamond rough varies; so must the cut. The 34.5-degree crown angle holds best for crystals with a refractive index of 2.417 and negligible birefringence — which is to say, for gem diamond and almost nothing else.