Tolkowsky, 1919, and what the arithmetic left out

In 1919 Marcel Tolkowsky, a member of an Antwerp diamond family reading engineering in London, published a short book called Diamond Design. In it he calculated, from the refractive index of diamond and the geometry of a round brilliant, the proportions that would return the most light to an observer looking straight down at the stone. He arrived at a pavilion angle of 40.75 degrees, a crown angle of 34.5 degrees, and a table 53 per cent of the girdle diameter.

A century later those three numbers are still the reference against which round brilliants are described, and they are the reference stone rendered on the front page of this site.

What he actually did

Tolkowsky traced a single ray. He followed it into the stone through the crown, down to a pavilion facet, across to the opposite pavilion facet, and back out, and he found the angles at which that ray both stayed inside the stone at each bounce — total internal reflection, requiring more than 24.4 degrees from the normal — and left the crown steeply enough to reach the eye rather than skimming away.

He then balanced brilliance against fire: a shallower crown lets more light out through the table with less dispersion, a steeper crown spreads more of the exit through the crown facets, where the different wavelengths separate. His table of 53 per cent is the point at which he judged the trade-off best made.

It is remarkable work and it is worth reading, because he is explicit about his simplifications in a way that the people quoting him usually are not.

What he left out, by his own account

The calculation is two-dimensional. It follows rays in a single plane through the middle of the stone, and a round brilliant is emphatically not a two-dimensional object: it has fifty-eight facets in three dimensions, and light entering off-axis takes paths his plane never contains.

It considers one ray, not a distribution. Real lighting is a room — windows, lamps, ceilings, and the observer’s own head blocking a cone of it. A stone optimised for a single ray from directly above is not necessarily optimal under a hemisphere of diffuse light with a hole in the middle of it.

It does not model scintillation at all: the pattern of flashes as the stone or the observer moves, which is the thing a person actually notices across a room. Nor does it model obstruction, contrast, or the way the eye reads a pattern of light and dark rather than a total quantity.

What came afterwards

Eppler’s work in Germany in the 1930s arrived at slightly different numbers — a 56 per cent table, a 33 degree crown — and was for decades the European standard while Tolkowsky’s was the American one. Both were widely described as ‘the ideal cut’, which should have been the first clue.

When the Gemological Institute of America finally ran the problem properly, over three-dimensional models and thousands of proportion combinations under realistic lighting, published in 2004, the answer was not a point. It was a region: a large family of combinations of table, crown and pavilion that all perform at the top grade, trading brightness against fire against scintillation in ways that are matters of preference. Tolkowsky’s numbers sit comfortably inside that region. So do a great many combinations he would not have recognised.

Why we still quote him

Because a reference has to be a single stone, and his is the one the whole trade knows. When this site shows you two stones side by side and tells you one of them is cut correctly, the correct one is Tolkowsky’s — 40.75, 34.5, 53 — and the comparison is honest as long as we say what the reference is. A stone that returns four per cent less light than his is not a bad stone. A stone that returns thirty per cent less is, and the difference between those two statements is the entire reason we built the demonstration rather than writing another paragraph about brilliance.