Inverse relationship between Crown and Pavilion Angles

After studying diamond cut as a student in 1984, and while buying and selling diamonds in my business, I observed that small variations in pavilion angles could be compensated for by a larger inverse variation in crown angle. Over time I formulated an approximation of 2C to 1P and I used this discovery secretly and commercially to buy a larger range of beautiful diamonds than the commonly applied tolerances based on Tolkowsky’s design criteria.
Diamonds in Figures 1 and 2 show a wide range of proportions that look and perform with similar characteristics. Knowledge of this inverse relationship between crown and pavilion angles makes it possible to cut diamonds more efficiently from the normal distribution of economically viable diamond rough. When buying diamonds with an Ideal-Scope I would find diamonds with proportions like these that had a beautiful appearance.

Figure 1. Even though the crown (top) of the blue profile is 6.5 steeper a compensating or inverse reduction in pavilion angle of just 1.2 results in similar ray paths. The red profile diamond weighs less and will be shown to have a higher light return, diamonds with shallow crowns and deeper pavilions are described here as a brilliant ideal cut (BIC). The steeper crown on the blue profile improves fire or dispersion, while the reduced pavilion angle optimizes light return. This steep / shallow combination is called a firey1 ideal cut (FIC). Figure 2 shows images of diamonds with these proportions that would be considered poor and extreme under existing cut grade standards.

It appears there is also a less critical relationship between crown and pavilion angles and table sizes. This would warrant further study.

Figure 2. These virtual diamonds and GilbertsonScope images were generated using DiamCalc by design to have similar visual properties (DiamCalc and GilbertsonScope will be discussed later). Stone 1 has a 6.5 shallower crown and a 1.2 deeper pavilion than stone 3 corresponding to an inverse ratio2 of 5.4:1 between crown and pavilion angles. Different table sizes were used in an effort to match the size of the central table reflections. The scores in the table below are from the HCA system proposed here. The strong similarities between these three stones are evident in the complex GilbertsonScope images. The shallow crown diamond on the left has more saturated colors that indicate a higher light return.

Table. Proportions of diamonds in Figure 2
Stone 1 (AGS 7) 2 (AGS 0) 3 (AGS 2)
Crown 30 34.5 36.5
Pavilion 41.55 40.75 40.35
Table 53.3% 57% 60%
HCA Score 1.2 BIC 0.7 1.1 FIC

1 ‘Firey’ is a word created by the author to better communicate fire or dispersion than the correct word - ‘fiery’.
2 After I purchased DiamCalc and had personal communications with its designers and the scientists from Moscow State University (MSU) I reviewed my earlier approximations of 2:1 and today I believe a relationship of around 5.4:1 optimises many crown :pavilion proportion combinations.