Mean Light Angle

Mean Light Angle = Bezel Angle

Now all the data are available to calculate the mean angle of light approaching the bezel (also the bezel angle)...

B = (B_CI_CA_C + B_RI_RA_R) / ( I_CA_C + I_RA_R)

= (B_CI_CA_C + B_R(0.521 I_C)A_R) / (I_CA_C + (0.521 I_C)A_R)

= (B_CA_C + B_R(0.521)A_R) / (A_C + (0.521)A_R)

= (17^o(0.125T²) + 69.4^o(0.521)(0.125T²)) / (0.125T² + (0.521)(0.125T²))

= (17^o + 69.4^o(0.521)) / (1+0.521))

= 35.0^o

Tolkowsky got 34.5^o, using measurements from a graphical layout he made by trial-and-error; a copy of this layout is furnished with the book. He developed no mathematical formulas for T/W*, Y/T, or Z/T but measured them from the layout. It is probable that the analytical answer of B = 35.0^o is more correct than Tolkowsky's graphical one of 34.5^o, but no practical faceter would argue the point.

Final Angles And Proportions

Figure 5

It was determined here that the pavilion angle should be

P=40.75^o (same as Tolkowsky)

The corresponding bezel angle has been calculated at

B = 35.0^o (Tolkowsky got 34.5^o)

The table width, vs. the flat crown, was shown to be

T/W* = 0.392

The final girdle width is therefore ...

W/W* = (tanP + (T/W*)tanB) / (tanP + tanB)
= (0.861 + (0.392)(0.700)) / (0.861 + 0.700) = 0.727

The final table:girdle ratio is therefore ...

T / W = (T/W*) / (W/W*)= 0.392 / 0.727 = 0.539 ............ (Tolkowsky got 0.530)

The total depth (H) is ...

H / W = (H/W*) / (W/W*) =
= ((tanP)/2) / 0.727= (0.861/2) / 0.727 = 0.593 ............ (same as Tolkowsky))

The pavilion depth (HP) is ...

HP / W = (tanP)/2= 0.861/2 = 0.431 ............ (same as Tolkowsky))

The crown depth (Hc) is ...

Hc / W = ((1 - T/W)(tanB)/2)= 0.461(0.700)/2 = 0.162 ............ (same as Tolkowsky))

The results of this analysis, following Tolkowsky's logic and general method, but improving on his mathematics, yields virtually identical results.

Girdle Facets

Tolkowsky goes on to discuss the rays of the group oblique from the right which are reflected off the second side of the pavilion also before passing through the bezel. Because these rays hit high on the pavilion and pass through the edge of the bezel, he claims that they are influenced more by the pavilion and crown girdle facets than by the mains.

With the pavilion girdle facet 2^o steeper than the pavilion main (P' = 42.75^o) the light coming across the stone at q₂ = 69.4^o becomes, according to Eq. (1b) ...

q3' = 180^o - 2P' - q2 = 180^o - 2 (42.75^o) - 69.4^o = 25.1^o

By making the crown girdle facets at B' = 42^o, he points out, these rays will approach the facets at B'-q3' = 17^o to give the optimum combination of brilliance and dispersion discussed in the design of the pavilion. In actuality, the crown girdle facets are generally cut about 5^o steeper than the crown main facets, whereby B' = B+5^o = 35^o+5^o = 40^o.

Star Facets

Tolkowsky then mentions the star facets, pointing out that they don't do much except make the design look better and add to the distribution of light, giving more 'sparkle' to the gem.

Comparison To Actual Cuts

Tolkowsky shows the dimensions of five diamonds which had been taken from production because of their exceptional appearance and because they were large enough to provide accurate proportion measurements. He averages these data and shows a close correlation to his theoretical findings.