Diamnod

Tolkowsky focused on diamonds with a pavilion angle of 40° 45'. He wrote that this angle "gives the most vivid fire and the greatest brilliancy, and that although a greater angle would give better reflection, this would not compensate for the loss due to the corresponding reduction in dispersion."

Tolkowsky also developed a geometric model of the crown, on pages 80-94 of Diamond Design. To keep his model simple, Tolkowsky used a knife-edge girdle. This geometric model let him take a given pavilion angle and calculate the best crown angle and table size for that pavilion angle. Tolkowsky gave only one example: He calculated that a diamond with a pavilion angle of 40° 45' looks best with a crown angle of 34° 30', and a table size of 53%. This example includes some rounding errors. This point is starred (*) in Figure 3.

Tolkowsky's geometric model of the crown can be used to calculate the crown angles and table sizes that correspond to other pavilion angles. But it is hard to draw geometric models very accurately. Even fig. 35 in Diamond Design has inaccuracies.

To get around this problem, the editor's notes convert Tolkowsky's geometric model of the crown into formulas. Tolkowsky's assumptions about the crown (including the knife-edge girdle) have been kept, and his logic has been followed step-by-step. The editor's notes do NOT have his rounding errors, however. The bold line in Figure 3 shows points that satisfy these formulas.

As Tolkowsky writes,

"The calculations have been made as simple as possible, so as not to be beyond the range of readers with a knowledge of elementary geometry, algebra, and trigonometry. Where, however, it was found that the accuracy of the results would be impaired without the introduction of more advanced mathematics, these have been used, and graphical methods have been explained as an alternative."

Despite being so simple, the conclusions are surprisingly accurate. Figure 3 broadly agrees with actual diamonds, and with 3-D computerized simulations of diamonds. The simulations agree that a larger pavilion angle can be offset by a smaller crown angle. (The simulations suggest slightly smaller pavilion angles. The difference is about the same as Tolkowsky's rounding error.)

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For reference is a diagram of a diamond showing the basic parts.