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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 19.)

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  B-line   687   nm
  Yellow   2.417     24 26'   Sodium D-line   589.3 nm
  Violet   2.452     24  4'   Solar  G-line   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 42.)

Ed. Note 10:
One-third 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 Q1R1 and the vertical.
It is also the angle between Q2E and the vertical,
as well as the angle between Q3R3 and the vertical.
  SecondAngle   =  2 * alpha - 12 05'.
(Tolkowsky rounds this to 69.)

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 - Q2C * Q2C) is proportional to the area.
  SecondWeight      =  (SC * SC - Q2C * Q2C) * 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 P2Q2 and the vertical.
This ray is incident at:
  AverageEffectiveAngle  =  arcsin(( (1/3) + sin EffectiveAngle ) / 2)
This ray is refracted to follow P2Q2:
  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').


Picture used to find formulas for AM, PM, SC, and TC.

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

Using ray Q2E to find Q2C

Ed. Note 27:
Fig. 39 is a simple version of fig. 35. We will use ray Q2E to find Q2C.

Let point J be where the girdle DE crosses the axis MC.
Let point K be where ray Q2E crosses the axis MC.
Let point L be the point on the axis MC at the same height as Q2.
(Q2L is perpendicular to MC).
Then,
  JC  =  JE * tan alpha
  JK  =  JE * tan Q2ED
  KL  =  Q2L * tan Q2ED
  LC  =  Q2L * tan alpha
  JC  =  JK + KL + LC
LC + KL =  JC - JK
 Q2L * (tan alpha + tan Q2ED)  =  JE * (tan alpha - tan Q2ED)
 Q2L  =  JE * (tan alpha - tan Q2ED) / (tan alpha + tan Q2ED)
 Q2C  =  Q2L / cos alpha
      =  JE / cos alpha * (tan alpha - tan Q2ED) / (tan alpha + tan Q2ED)
  JE  =  (DE / 2)
 Q2C  =  (DE / 2) / cos alpha * (tan alpha - tan Q2ED) / (tan alpha + tan Q2ED)

Ed. Note 28:
We can use line Q2KE to solve for angle Q2ED.
Angle Q2ED is between Q2E and the horizontal.
The SecondAngle is between Q2E and the vertical.
The AverageRefractedAngle is SPT in figs. 35 and 38.
         Q2ED  =  90 - SecondAngle
  SecondAngle  =        2 * alpha - AverageRefractedAngle
         Q2ED  =  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.

Fish-eye 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 fish-eye diamonds and nail-head diamonds. Editor's notes 30-32 are based on the explanation of fish-eyes in the PriceScope fish-eye tutorial.

Fish-eye 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 fish-eye.

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 fish-eye, 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 fish-eye.

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     fish-eye
  38        39 35'   0.531       0.584 - culet height     fish-eye

Nail-head diamonds

Ed. Note 33:
A nail-head 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 nail-head 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 209-227. 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 nail-head 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     nail-head
  44        20 30'   0.52        0.572 - culet height     nail-head

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 30-34 explain that some combinations of proportions that satisfy this model make fish-eye or nail-head 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 knife-edge 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 knife-edge 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
Q2ED  =  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 Q1R1 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 knife-edge 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
 Q2C  =  (DE / 2) / cos alpha * (tan alpha - tan Q2ED) / (tan alpha + tan Q2ED)

Step 9. We calculate a new guess for beta (the crown angle).
   FirstWeight  =  (TC * TC)
  SecondWeight  =  EffectiveFraction * (SC * SC - Q2C * Q2C)

  beta = (FirstWeight * FirstAngle + SecondWeight * SecondAngle) / (FirstWeight + SecondWeight)

This gives us a new guess for beta.
The loop ends here. We can repeat steps 6-9 until the guess for beta stops changing.


Step 10. Because Tolkowsky uses a knife-edge girdle, we do NOT need to adjust the diameter and table ratio.

Step 11. Tolkowsky says that:

We change these angles, to have a 4-sided 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 knife-edge girdle, there is no girdle thickness:
  CrownHeight   =  diameter / 2 * (1 - t) * tan beta
  PavilionDepthdiameter / 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 knife-edge 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 30-34 show that some diamonds that match the formulas are NOT ideal. Extrapolating to smaller pavilion angles makes fish-eyes. Extrapolating to larger pavilion angles makes nail-heads.

Caution: These results use the same assumptions and logic as Tolkowsky -- including a knife-edge 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 3-D 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 Cross-Section" 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 fish-eye diamonds and nail-head 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 3-D 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 3-D 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 trade-offs of cut parameters, symmetry, polish, colour, fluorescence, size of crystal defects (clarity), carat weight, cost, and availability.