Tolkowsky ideal

In 1919 a young mathematician, Marcel Tolkowsky, wrote a Masters thesis on the proportions for round brilliant cut diamonds. This became the basis for what is now known as the “Ideal Cut".

It is said he asked passers by, in the streets of London, to select the most appealing diamond from a small group. From these casual observer opinions, together with those of the diamond cutters in his family's Belgian business, he confirmed the proportions of the best looking diamonds. He then applied math and physics to prove why.

The proportions he chose produce a beautiful diamond. But it has taken more than 80 years for the industry to realize that his theorem predicted a range of proportions, not just the single set of parameters shown on the image; he was aware of a trade off between maximum brilliance and maximum fire that results from varying combinations of crown and pavilion angles.

You can read Tolkowsky's book here and play with his mathematic model thanks to Mathematician Jasper Paulsen.

Adamas Gemological Laboratory

What Tolkowsky Didn't Say

©1996 Martin D. Haske found at Adamas Gemological Laboratory

Have you ever heard the rumor story, started at one end of a line and listened to at the other end; every one adds their own spin and the original story, oft mistaken for fact, winds up quite differently; and fact and fiction become inter-twinned. Take the theoretical American Brilliant Cut Diamond, for example.

Every one in the jewelry industry has seen the Standard American Cut Brilliant picture (shown below), attributed to Tolkowsky, with a 53% Table, 34.5 degree crown angle and 43.1% pavilion. Added to that is a girdle thickness of 0.7% to 1.7% giving to those accepting what they see and read as "ideal", the perception that this is perfect cut or "ideal cut" or a "standard".
Ignoring, for the moment, any inaccuracy in the calculations Marcel Tolkowsky performed without the aid of a computer, Marcel Tolkowsky DID NOT SPECIFY GIRDLE THICKNESS. The origin of the 0.7% to 1.7% "ideal" girdle thickness, cannot be attributed to Tolkowsky, at least in the Mathematics portion, Part III (Chapter III) , of his original work, "Diamond Design"!

Tolkowsky's Ideal cut, 34.5 degree crown, 53% Table, 43.1 % pavilion depth, added up to 59.3%, period. The industry interpretation, however, took a different bent. Lost in the woodwork, was a 1979 statement in the GIA Diamonds Course that the ideal cut diamond "would be 61 to 62%", recognizing the fact that the 0.7% to 1.7% would give a "rather thin girdle". Indeed, mathematically, for the approximately (standard) 66 % pavilion girdle facet length, the 59.3% Tolkowsky brilliant, as defined, would have sixteen (16) knife edges forming the girdle ! Tolkowsky's picture of the "ideal" had knife edges.

If we look at the mathematics, the depth of the scallops have to be a MINIMUM 1.7%, between the WIDEST part of the girdle scallops, and in that condition, they meet at a POINT. Hardly an "ideal" cut!

If we change the pavilion girdle break facet length to approximately 80%, which appears the norm for today, this 1.7% limitation is reduced to approximately 1.5%.

Communication may be lacking in the industry, and everyone has their own interpretation, but the FACTS are, that the GIA, at least in their Diamond Grading Course(s), to the limits of my research, has never specified Total Depth Percentage!.

GIA has defined a medium to slightly thick girdle to be a requirement for a Class 1 make. If one looks at the published GIA girdle thickness charts, curve fits through the inherent discontinuities, one finds that the implied recommended TOTAL DEPTH PERCENTAGE RANGE, for a 34.5 degree crown, 53% Table, and 43.1% pavilion depth to be at least 61.0% Total depth. IN FACT, for a theoretical one (1) carat Tolkowsky round brilliant, 6.53 mm in diameter, the acceptable range (interpolated) that GIA recommends, is 61.5 to 62.7 PERCENT TOTAL DEPTH. Industry publications suggest 60 to 61% TOTAL DEPTH for a CLASS 1 Cut, consistent with the third edition of the Diamond Dictionary which says that the Tolkowsky Ideal cut was 60-61% Total Depth.

With all those deep pockets in the industry, most never did their homework (if any at all), or READ, and understood, the complicated (or at least background) issues. They probably tried, and knew they couldn't cut a "Tolkowsky Ideal", because, with a girdle that didn't have a KNIFE EDGE, it wouldn't add up to what was presented. So the "ideal", as defined by Tolkowsky, was apparently "modified" by the industry, and took form as spread tables, reducing the crown height, so that the 53-60 table/60-61 depth "ideal" relationship, so oft quoted, could be realized!

A simple mathematical proof is:

Consider the girdle break facets, 16 in number, each subtending an arc (alpha) of 22.5 degrees (360/16). The chord of this arc (2a), and the length (b) from the center of the circular diamond outline are related by the trigonometric relationship

Cos(alpha/2)=b/r

Cos(22.5/2)=Cos(11.25)=0.98078=b/r

Therefore b ~ 0.98*r ~ 0.49*diameter ~ 0.49*D

And c = r - b ~ 0.50*D - 0.49*D ~ 0.01*D

The ungula cut by the girdle break facets form the familiar scallop on the girdle edge, the depth of which (h) is related to c, and the angle at which the girdle break facet (beta) cut the girdle by the trigonometric relationship

Tan(beta) = h / c

or h = c*Tan(beta)

The crown girdle break facets are at an angle GREATER than the crown mains (34.5 degrees) or approximately 39.45 degrees (for a 50% star facet length) and the pavilion break facets are at angle greater than the pavilion mains (40.76 degrees , zero culet, 43.1% pavilion depth) or approximately 41.4 degrees (for an 66% pavilion break facet length).

For the crown scallop depth (h1)

h1 = c*Tan (39.45) > c*Tan(34.5)

h1 = 0.01*D*0.82287 ~ 0.00822*D > 0.01*D*0.68728

For the pavilion scallop depth (h2)

h2 = c*Tan(41.4) > c*Tan(40.76)

h2 = 0.01*D*0.88162 ~ 0.0088*D > 0.01*D*0.86195

If we allow the crown scallop and the pavilion scallop to meet (at a knife edge) , the girdle thickness would be the sum of h1 and h2

h1+h2 ~ 0.0088*D + 0.0082*D ~ 0.017*D or 1.7%D

Any girdle depth less than 1.7%, for the Tolkowsky crown and pavilion, would result in not only knife edges at the girdle, but flattened girdle edges and not a true round, surely not an Ideal cut from the industries point of view.

The Ideal Tolkowsky cut therefore, has to have a MINIMUM girdle thickness (defined at the widest part of the scallops) of at LEAST 1.7%, and for a 16.2% crown and 43.1% pavilion, at LEAST 61% depth (16.2+1.7+43.1).