Diamond Design  Editor's Notes
Summary
Ed. Note 1:
Tolkowsky shows a graphical way to determine the best crown angle and table angle
for a given pavilion angle. He tabulates the proportions
that best match just one pavilion angle (40° 45').
We can find formulas for the crown angle and table size,
so that a computer can calculate how changing the pavilion angle
changes the best crown angle and table size.
We will use the same pictures and analysis that Tolkowsky does.
(Corrected) Rounding Errors
Ed. Note 2:
On page 83 (near fig. 33),
Tolkowsky divides "all the light entering the diamond into three groups,
one of vertical rays and two of oblique rays, such that the light entering from
each group is the same." Using integrals, he shows that the angle of incidence
that divides the vertical rays from the oblique rays is:
i = 90°  arccos(1/3)
= 90°  70° 31' 44"
= 19° 28' 16"
(Tolkowsky rounds this off to 19½°.)
Ed. Note 3:
The corresponding angle of refraction is
r = arcsin(sin i / n)
= arcsin( 1/3 / 2.417 )
= arcsin( 0.1379 )
= 7° 55' 37"
(Tolkowsky rounds this off to 7° 52'.)
Ed. Note 4:
The critical angle is
arcsin(sin 90° / n)
= arcsin( 1 / 2.417)
= arcsin(0.4137)
= 24° 26' 23"
This 6 digit angle is absurdly precise,
because it is based on a number with only 4 digits (2.417).
The extra digits reduce the rounding errors
in further calculations in this article.
But all of the results should be rounded off.
(Tolkowsky rounds the critical angle off to 24° 26'.)
Note that this is valid for yellow sodium light.
Other colours have different indexes of refraction,
so they have different critical angles.
For example:
Index of Critical
Colour Refraction Angle Source Line Wavelength
Red 2.407 24° 33' Solar Bline 687 nm
Yellow 2.417 24° 26' Sodium Dline 589.3 nm
Violet 2.452 24° 4' Solar Gline 431 nm
The difference between the indexes of refraction (2.452  2.407) is the dispersion (0.044). (There is a small rounding error.)
The critical angles are calculated from the indexes of refraction.
All of the data besides the critical angles are taken from a GIA report:
Barak Green, Ilene Reinitz, Al Gilbertson, Mary Johnson, and James Shigley.
"Diamond Optics Part 2: Light Dispersion, Color Wavelengths, and Fire."
Gemological Institute of America, July 20, 2001.
Slightly different values are quoted by:
"Dana's System of Mineralogy."
7th Edition, Volume 1, page 148.
Ed. Note 5:
The average angle of incidence of the vertical rays is 0°,
just as Tolkowsky writes.
Ed. Note 6:
The average angle of incidence of the oblique rays is
i = 90°  arccos(2/3)
= 90°  48° 11' 23"
= 41° 48' 37"
(Tolkowsky rounds this off to 42°.)
Ed. Note 7:
The corresponding angle of refraction is
r = arcsin(sin i / n)
= arcsin( 2/3 / 2.417 )
= arcsin( 0.2758 )
= 16° 0' 40"
(Tolkowsky rounds this off to 16°.)
Ed. Note 8:
The limiting angle of refraction of the oblique rays is
r = 90°  (180°  (90°  24° 26' 23")  alpha)
= 90°  ( 90° + 24° 26' 23"  alpha)
= alpha  24° 26' 23"
For alpha = 40° 45', this gives r = 16° 18' 37".
(Tolkowsky rounds this off to 16° 19').
Ed. Note 9:
The corresponding angle of incidence is
i = arcsin(n * sin r)
For r = 16° 18' 37", this gives i = 42° 44' 56".
(Tolkowsky rounds this off to 42½°.)
Ed. Note 10:
Onethird of the light is incident from the right.
This light is incident from 19° 28' 16" to 90°.
[cos(90°  19° 28' 16")  cos(90°  90° )] / [cos(90° + 90°)  cos(90°  90°)]
= [sin( 19° 28' 16")  sin(
90° )] / [sin( 90°)  sin(
90°)]
= [1/3  1] / [1  1]
= [2/3] / [2]
= 1/3
Ed. Note 11:
Just part of the light incident from the right is effective.
This light is incident from 19° 28' 16"
to 42° 44' 56"
(actually an angle that depends on alpha).
[cos(90°  19° 28' 16")  cos(90°  42° 44' 56")] / [cos(90° + 90°)  cos(90°  90°)]
= [sin( 19° 28' 16") 
sin( 42° 44' 56")] / [sin( 90°) 
sin( 90°)]
= [1/3  0.6788] / [1  1]
= [0.3455] / [2]
= 0.1727
Ed. Note 12
So of the light incident from the right, only a fraction is effective:
0.1727 / (1/3)
= 0.5182
Actually, this fraction depends on alpha.
Editor's note 17
shows the formula used to calculate the values in Table III:
Table III
alpha fraction
39° 0.4114
40° 0.4725
40° 20' 0.4928
40° 45' 0.5182
41° 0.5334
42° 0.5938
(Tolkowsky rounds the fraction off to 0.493 .)
Derivation of Formulas
Ed. Note 13:
The crown angle is a weighted average of two angles shown in
fig. 35.
CrownAngle = ( FirstAngle * FirstWeight
+ SecondAngle * SecondWeight)
/ (FirstWeight + SecondWeight)
Ed. Note 14:
The first angle is between R'S' and the vertical.
FirstAngle = 180°  4 * alpha.
(Tolkowsky rounds this to 17°.)
Ed. Note 15:
The weight given to the first angle is based on the area of the pavilion
directly underneath the table, because these rays entered the table
more or less vertically. Tolkowsky calculates the weight using TC * TC.
FirstWeight = TC * TC
TC = PM / cos alpha
= (DE / 2) * t / cos alpha
where t is the table ratio.
Note that the FirstWeight depends on both
the pavilion angle and the table ratio.
Ed. Note 16:
The second angle is between Q_{1}R_{1} and the vertical.
It is also the angle between Q_{2}E and the vertical,
as well as the angle between Q_{3}R_{3} and the vertical.
SecondAngle = 2 * alpha  12° 05'.
(Tolkowsky rounds this to 69½°.)
Editor's note 17 shows that the angle being subtracted (12° 05')
depends on alpha.
Ed. Note 17:
The weight given to the second angle is based on the area of the pavilion
that reflects oblique rays to the crown facets, multiplied by the effective
fraction of the oblique rays. Tolkowsky writes that (SC * SC  Q_{2}C * Q_{2}C)
is proportional to the area.
SecondWeight = (SC * SC  Q_{2}C * Q_{2}C) * EffectiveFraction
The fraction was found in editor's note 12,
using this formula:
EffectiveAngle = arcsin(2.417 * sin (alpha  24° 26' 23"))
EffectiveFraction = [cos(90°  19° 28' 16")  cos(90°  EffectiveAngle)] / (2) / (1/3)
= [sin( 19° 28' 16")  sin( EffectiveAngle)] / (2) / (1/3)
= [1/3  sin(EffectiveAngle)] * 3 / (2)
In editor's note 16, the angle being subtracted depends on alpha.
The angle being subtracted is between the ray P_{2}Q_{2} and the vertical.
This ray is incident at:
AverageEffectiveAngle = arcsin(( (1/3) + sin EffectiveAngle ) / 2)
This ray is refracted to follow P_{2}Q_{2}:
AverageRefractedAngle = arcsin((sin AverageEffectiveAngle) / 2.417)
= arcsin(( (1/3) + sin EffectiveAngle ) / 2 / 2.417)
For alpha = 40° 45', this gives angles of 30° 24' and 12° 5' respectively.
(Tolkowsky rounds them off to 30° 15' and 12° 0').
Ed. Note 18:
Fig. 38 is a simple version of fig. 35.
SC = ST + TC
Let point F be where the ray PT crosses the girdle DE.
Let point G be the point on the ray PT at the same height as Q.
Let point H be the point on the ray PT at the same height as S.
(QG and SH are perpendicular to PT).
Then,
TC = PM / cos alpha
SH = AP * (tan SPT) / (tan SPT + 1 / tan alpha)
SC = TC + SH / cos alpha
So we need to know AP and PM to find SC and TC.
Finding AP and PM
Ed. Note 19:
Look at triangle PQR. Find PM in terms of QG.
PG = (QG + 2 * PM) * tan QRP
QG = PG * tan QPT
PG = QG / tan QPT
(QG + 2 * PM) * tan QRP = QG / tan QPT
2 * PM * tan QRP = QG * (1 / tan QPT  tan QRP)
PM = QG * (1 / tan QPT  tan QRP) / 2 / tan QRP
Let
g = (1 / tan QPT  tan QRP) / 2 / tan QRP
PM = QG * g
Ed. Note 20:
We know that ray PQ is reflected to QR,
so we can find angle QRP.
QRP = RQC  alpha
RQC = AQP
AQP = 90°  alpha + QPT
QRP = 90°  alpha + QPT  alpha
= 90°  2 * alpha + QPT
Ed. Note 21:
Triangle PQT is split by QG. Find QG in terms of PT.
PT = PG + GT
GT = QG * tan alpha
PG = QG / tan QPT
PT = QG * tan alpha + QG / tan QPT
PT = QG * [tan alpha + (1 / tan QPT)]
QG = PT / [tan alpha + (1 / tan QPT)]
Ed. Note 22:
We also know PT in terms of AP.
PT = AP * tan alpha
Ed. Note 23:
Find AP in terms of QG.
QG = AP * tan alpha / [tan alpha + (1 / tan QPT)]
AP = QG * [tan alpha + (1 / tan QPT)] / tan alpha
Let
f = [tan alpha + (1 / tan QPT)] / tan alpha
f = 1 + (1 / tan QPT / tan alpha)
AP = QG * f
Ed. Note 24:
We now know both AP and PM in terms of QG, but QG is still unknown.
Use triangle ADP to find AP in terms of DE and PM.
In fig. 38, let D' be the point along AP directly above point D.
Then DD' is perpendicular to AP.
AP = AD' + D'P
DD' = D'P * tan beta
DD' = AD' * tan alpha
AD' * tan alpha = D'P * tan beta
AD' = D'P * tan beta / tan alpha
AP = AD' + D'P
= D'P * tan beta / tan alpha + D'P
= D'P * (1 + tan beta / tan alpha)
Let
h = 1 + tan beta / tan alpha
AP = D'P * h
D'P = DE / 2  PM
AP = (DE / 2  PM) * h
Ed. Note 25:
We can now combine the following equations to solve for QG:
AP = (DE / 2  PM) * h
AP = QG * f
PM = QG * g
QG * f = (DE / 2  PM) * h
QG * f = (DE / 2  QG * g) * h
QG * (f + g * h) = DE / 2 * h
QG = DE / 2 * h / (f + g * h)
Ed. Note 26:
We can now solve for AP and PM in terms of DE.
This also gives the table ratio of a diamond with a knife edge girdle.
PM = DE / 2 * h / (f + g * h) * g
PM = DE / 2 * g * h / (f + g * h)
Let
t = g * h / (f + g * h)
PM = (DE / 2) * t
AP = QG * f
PM = QG * g
QG = PM / g
AP = PM * f / g
Ed. Note 27:
Fig. 39 is a simple version of fig. 35.
We will use ray Q_{2}E to find Q_{2}C.
Let point J be where the girdle DE crosses the axis MC.
Let point K be where ray Q_{2}E crosses the axis MC.
Let point L be the point on the axis MC at the same height as Q_{2}.
(Q_{2}L is perpendicular to MC).
Then,
JC = JE * tan alpha
JK = JE * tan Q_{2}ED
KL = Q_{2}L * tan Q_{2}ED
LC = Q_{2}L * tan alpha
JC = JK + KL + LC
LC + KL = JC  JK
Q_{2}L * (tan alpha + tan Q_{2}ED) = JE * (tan alpha  tan Q_{2}ED)
Q_{2}L = JE * (tan alpha  tan Q_{2}ED) / (tan alpha + tan Q_{2}ED)
Q_{2}C = Q_{2}L / cos alpha
= JE / cos alpha * (tan alpha  tan Q_{2}ED) / (tan alpha + tan Q_{2}ED)
JE = (DE / 2)
Q_{2}C = (DE / 2) / cos alpha * (tan alpha  tan Q_{2}ED) / (tan alpha + tan Q_{2}ED)
Ed. Note 28:
We can use line Q_{2}KE to solve for angle Q_{2}ED.
Angle Q_{2}ED is between Q_{2}E and the horizontal.
The SecondAngle is between Q_{2}E and the vertical.
The AverageRefractedAngle is SPT
in figs. 35 and 38.
Q_{2}ED = 90°  SecondAngle
SecondAngle = 2 * alpha  AverageRefractedAngle
Q_{2}ED = 90°  2 * alpha + AverageRefractedAngle
Culet
Ed. Note 29:
If the diamond has a culet,
part of the pavilion will be chopped off. How much is chopped off?
The culet is hard to measure accurately,
so there is no standard definition of how to measure it.
If we assume that the "culet" is:
culet = maximum diameter of the culet facet
If the culet is an octagon,
culet * cos(22° 30') = minimum diameter of the culet facet
Each side of the octagon crosses a pavilion main facet.
The midpoint of each side is on the centerline of the main facet.
The "culet height" is:
CuletHeight = (culet / 2) * cos(22° 30') * tan alpha
For alpha = 40° 45',
CuletHeight = (0.4) * culet
As of 2001, most diamond measuring machines measure the culet
very inaccurately. For example, "pointed" or "no" culets
are usually reported as 0.3% to 0.7% culets.
It seems that these diamond measuring machines
do NOT measure the culet itself.
Instead, they measure other parameters, and do some math.
Usually, the numbers do NOT add up.
The machines attribute the error to the culet.
Fisheye diamonds
Ed. Note 30:
The formulas derived above are summarized in
editor's note 36.
Unfortunately, not all of the proportions that satisfy these formulas
result in ideal diamonds. Two problems are fisheye diamonds
and nailhead diamonds.
Editor's notes 3032 are based on the explanation of fisheyes
in the PriceScope fisheye tutorial.
Fisheye diamonds occur when the pavilion angle
is small enough to break one of Tolkowsky's assumptions.
Tolkowsky assumes that the rays that enter the table vertically
are doubly reflected before exiting the crown.
If the reflection of PT ends up higher than the girdle,
then a reflection of the girdle will appear in the table.
If the girdle is unfinished, this reflection looks like a gigantic inclusion.
Ed. Note 31:
We can use the reflection of PT to guess whether a diamond will have a fisheye.
The reflection of PT has an inclination of:
= 90°  2 * alpha
The reflection's slope is:
= tan(90°  2 * alpha)
Point T is beneath the girdle, by this much:
= diameter / 2 * (1  t) * tan alpha
Point T is to the left of point E, by this much:
= diameter / 2 * (1 + t)
So we have fish eye if:
diameter / 2 * (1 + t) * tan(90°  2 * alpha)
> diameter / 2 * (1  t) * tan alpha
(1 + t) * tan(90°  2 * alpha)
>
(1  t) * tan alpha
Of course, we need to adjust for the girdle thickness.
Also, if the diamond is tilted, we need to use a different ray
instead of PT. But we can guess whether the diamond will have a fisheye,
even if we don't make these adjustments.
Ed. Note 32:
A diamond with:
alpha = 38° 15',
beta = 39° 14',
t = 0.532
satisfies the equations in editor's note 36,
but is on the verge of being a fisheye.
These two solutions have been removed from
Table IV (in editor's note 37):
Pavilion Crown Table
Angle Angle Ratio Total Height Ratio
(alpha) (beta) (t)
37° 40° 56' 0.528 0.581  culet height fisheye
38° 39° 35' 0.531
0.584  culet height fisheye
Nailhead diamonds
Ed. Note 33:
A nailhead occurs when the person looking at the diamond
sees the reflection of their own head.
It is caused by pavilion and lower girdle facet angles close to 45°.
Michael Cowing has published a picture of a nailhead diamond with:
alpha = 42° 36',
beta = 30°,
t = 0.69
The diamond is pictured in Figure 9, and described in Table I, of:
Michael Cowing. "Diamond brilliance: theories, measurement, and judgement."
Journal of Gemmology. Volume 27, Number 4, October 2000, pages 209227.
Copyright by the Gemmological Association and Gem Testing Laboratory of Great Britain.
Ed. Note 34:
A diamond with:
alpha = 42° 36',
beta = 29° 30',
t = 0.537
satisfies Tolkowsky's model. This diamond is very similar
(in terms of crown angle and pavilion angle) to the nailhead diamond
in Cowing's article.
These two solutions have been removed from
Table IV:
Pavilion Crown Table
Angle Angle Ratio Total Height Ratio
(alpha) (beta) (t)
43° 27° 40' 0.535 0.588  culet height nailhead
44° 20° 30' 0.52
0.572  culet height nailhead
Also, this solution (still in
Table IV) is questionable:
42° 31° 41' 0.539 0.593  culet height
Girdle Thickness
Ed. Note 35:
Tolkowsky mentions the girdle. He even says that the diamond is circular.
But he does not include the thickness of the girdle in the overall depth of the diamond.
To finish the diamond design, we need to estimate the thickness of the girdle.
I have written an article titled,
"Diamond Girdles  How Adding a Girdle Changes Marcel Tolkowsky's Diamond Design."
The article uses the same logic as Tolkowsky, and most of the same formulas,
but adds in the girdle. For any given pavilion angle,
the girdle increases the optimum table ratio,
and reduces the optimum crown angle.
Formulas for Best Proportions
Ed. Note 36:
These formulas are the same as Tolkowsky's geometric model.
They were derived in the editor's notes (above).
These formulas start with a pavilion angle (alpha)
and a guess at the crown angle (beta).
Then we need to loop through the formulas a few times
to find the best crown angle and table ratio
for the pavilion angle.
Then we will need to make sure that the diamond really is pretty.
Editor's notes 3034 explain that some combinations
of proportions that satisfy this model make
fisheye or
nailhead diamonds,
which are NOT ideal. And of course, the only way to judge
if the diamond really is beautiful is to look at it in appropriate light.
Caution: These formulas use the same assumptions and logic as Tolkowsky 
including a knifeedge girdle.
Start.
We choose alpha.
We start with a guess for beta (say, 35°).
Step 1. We look at the girdle:
DE = diameter of a knifeedge diamond. (1 mm is easiest.)
Step 2. We find out what fraction of the oblique rays are effective, and their average angle:
CriticalAngle = arcsin(1 / 2.417) = 24° 26' 23"
EffectiveAngle = arcsin(sin (alpha  24° 26' 23") * 2.417)
EffectiveFraction = [1/3  sin(EffectiveAngle)] * 3 / (2)
SPT = AverageRefractedAngle = arcsin(( (1/3) + sin(EffectiveAngle) ) / 2 / 2.417)
Step 3. We calculate angles of rays:
QPT = alpha  24° 26' 23"
QRP = 90°  2 * alpha + QPT
Q_{2}ED = 90°  2 * alpha + AverageRefractedAngle
Step 4. We calculate angles of typical rays before they leave the crown.
The FirstAngle is the angle between R'S' and the vertical.
The SecondAngle is the angle between Q_{1}R_{1} and the vertical.
FirstAngle = 180°  4 * alpha
SecondAngle = 2 * alpha  AverageRefractedAngle
Step 5. We calculate some ratios that make the calculations easier.
f = 1 + (1 / tan QPT / tan alpha)
g = (1 / tan QPT  tan QRP) / 2 / tan QRP
Step 6. The loop starts here.
We calculate the table ratio of a knifeedge diamond:
h = 1 + tan beta / tan alpha
t = g * h / (f + g * h)
Step 7. We calculate distances at the top of the diamond:
PM = (DE / 2) * t
AP = PM * f / g
Step 8. We calculate distances along the pavilion edge:
TC = PM / cos alpha
SC = TC + AP * (tan SPT) / (tan SPT + 1 / tan alpha) / cos alpha
Q_{2}C = (DE / 2) / cos alpha * (tan alpha  tan Q_{2}ED) / (tan alpha + tan Q_{2}ED)
Step 9. We calculate a new guess for beta (the crown angle).
FirstWeight = (TC * TC)
SecondWeight = EffectiveFraction *
(SC * SC  Q_{2}C * Q_{2}C)
beta = (FirstWeight * FirstAngle + SecondWeight * SecondAngle)
/ (FirstWeight + SecondWeight)
This gives us a new guess for beta.
The loop ends here.
We can repeat steps 69 until the guess for beta stops changing.
Step 10. Because Tolkowsky uses a knifeedge girdle,
we do NOT need to adjust the diameter and table ratio.
Step 11. Tolkowsky says that:
We change these angles, to have a 4sided kite facet.
Modern diamonds have longer lower girdle facets, so these angles are slightly different.
Step 12. The diamond total depth contains the crown and the pavilion.
Because it has a knifeedge girdle, there is no girdle thickness:
CrownHeight = diameter / 2 * (1  t) * tan beta
PavilionDepth = diameter / 2 * tan alpha
(CuletHeight) =((culet / 2) * cos(22° 30') * tan alpha)
TotalDepth = CrownHeight + PavilionDepth  CuletHeight
(Corrected) Best Proportions
Ed. Note 37 / Table IV:
Here are some combinations of alpha, beta, and t calculated by this method.
The total depth ratio is also shown. The total depth ratio
is the overall depth of the diamond divided by the diameter.
It includes an adjustment for the culet facet,
but not for the knifeedge girdle.
Pavilion Crown Table
Angle Angle Ratio Total Depth Ratio
(alpha) (beta) (t)
39° 38° 9' 0.534 0.588  culet height
40° 36° 30' 0.537 0.591  culet height
40° 45' 35° 1' 0.539 0.593  culet height
41° 34° 27' 0.539 0.593  culet height
42° 31° 41' 0.539 0.593  culet height
Caution: The model breaks down at both ends of this range.
Editor's notes 3034 show that some diamonds that match the formulas are NOT ideal.
Extrapolating to smaller pavilion angles makes
fisheyes.
Extrapolating to larger pavilion angles makes
nailheads.
Caution: These results use the same assumptions and logic as Tolkowsky 
including a knifeedge girdle.
FOLDS.NET has software that lets you adjust fig. 35.
You can see the effect of changing the pavilion angle
or the girdle thickness. You can view the calculations,
or just look at the results.
The MSU diamond study found similar results, with some noticeable differences.
The MSU diamond study was much more thorough than this model.
Errors and Inconsistencies
Ed. Note 38:
As shown in Table II,
Tolkowsky's rounding errors are small, but noticeable.
Editor's note 37 shows that for a pavilion angle of 40° 45',
the best crown angle is 35°, with a table ratio of 0.54 .
Tolkowsky's various rounding errors cause him to estimate
that for a pavilion angle of 40° 45',
the best crown angle is 34° 30', with a table ratio of 0.53 .
Notably, these errors do not change the crown height.
Also, these errors partly offset the error from not including the girdle thickness.
Tolkowsky does not include the thickness of the girdle.
The girdle adds considerable thickness
 because of both the nominal "girdle" thickness,
and because of the waviness of the girdle.
The waviness is about 1.7% of the diamond's diameter.
This model assumes a pointed culet.
Ed. Note 39:
Tolkowsky's model does not follow all of the rays that strike the diamond.
It emphasizes rays that enter through the table, which are about 28% of the total.
It pays less attention to rays that enter or exit through
the star facets, upper girdle facets, and lower girdle facets.
It does not explain if (or how)
changing the pavilion angle should change
the slopes of these facets.
It assumes that all rays that exit the bottom of the diamond are lost.
It assumes that either no rays enter the bottom of the diamond,
or that they do not make it out the top of the diamond.
Tolkowsky does not consider how the thickness of the girdle
changes the paths of the light. In reality, the optimum crown angle and table ratio
are functions of both the pavilion angle and the girdle thickness.
It is not a 3D model. It only follows rays that bounce around in a single plane.
Although very few rays only bounce around in a single plane,
these rays provide a good average for the behavior of the rays that cross them
in all directions.
Ed. Note 40:
Tolkowsky studies brilliance and fire,
but not scintillation.
Tolkowsky gives brilliance and fire equal weight,
by multiplying them together.
 Brilliance measures how bright the diamond is.
Of the rays that enter the crown, how many return to the observer?
 Fire measures how much the diamond separates white light into colours.
When the diamond shifts slightly,
how much of a light show does the observer see?
 Scintillation measures contrast.
When the diamond shifts slightly,
how much variation between light and dark does the observer see?
Computer Models
Ed. Note 41:
A computer performed the calculations in this appendix.
Jasper Paulsen (the editor of this book) created the software, and posted it at FOLDS.NET.
You can use "Diamond CrossSection" to adjust fig. 35.
You can see the effect of changing the pavilion angle
or the girdle thickness. You can view the calculations,
or just look at the results.
This software can be used to calculate other combinations of proportions
that also satisfy Tolkowsky's model. Unfortunately, some of these proportions
do not result in ideal diamonds. Two problems are fisheye diamonds
and nailhead diamonds.
Ed. Note 42:
Sergey Sivovolenko (of OctoNus Software),
Yurii Shelementiev (of the Gemology center of Moscow State University),
and Anton Vasiliev (of the LAL company)
have created a 3D model of a diamond.
Just like Tolkowsky, they gave brilliance and fire equal weight
by multiplying them together.
They found a similar relationship between optimum crown angle
and actual pavilion angle.
(Their simulations suggest slightly smaller pavilion angles.
The difference is about the same as Tolkowsky's rounding error.)
Ed. Note 43:
The Gemological Institute of America has extensively studied diamond cut.
The GIA has created another 3D model of a diamond.
They do not report as clear a relationship between optimum crown angle
and actual pavilion angle.
Ed. Note 44:
Garry Holloway has studied the effects of many facet proportions on
diamond beauty. He conducted these studies using MSU and Octonus
'DiamCalc' software and actual diamonds.
Leonid Tcharnyi wrote a computer program that uses
Holloway's data. The program produces a numerical score (HCA) and written
descriptions of a round brilliant cut diamond.
Ed. Note 45:
In the real world, extremely well cut diamonds are rare.
Of course, it is almost impossible to make every diamond
match just a single set of parameters.
But even if one includes diamonds close to the optimum ranges
in this study, the MSU study, and the GIA study,
very few diamonds are extremely well cut.
In order to purchase a diamond that is extremely well cut,
a buyer will need to make tradeoffs of
cut parameters, symmetry, polish, colour, fluorescence,
size of crystal defects (clarity),
carat weight, cost, and availability.
