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Diamond Design - Mathematical

In the survey of the history of diamond cutting, perhaps the most remarkable fact is that so old an art should have progressed entirely by trial and error, by gradual correction and slow progress, by the almost accidental elimination of faults and introduction of ameliorations.  We have traced the history of the art as far back as 1375, when the earliest recorded diamond manufacturers existed, and when the polishers had already attained a high degree of guild organisation.  We have every reason to believe that the process of diamond polishing was known centuries before.  And yet all these centuries, when numerous keen minds were directed upon the fashioning of the gem, have left no single record of any purposeful planning of the design of the diamond based upon fundamental optics.  Even the most bulky and thorough contemporary works upon the diamond or upon gems generally rest content with explaining the basic optical principles, and do no more than roughly indicate how these principles and the exceptional optical properties of the gem explain its extraordinary brilliancy ; nowhere has the author seen calculations determining its best shape and proportions.  It is the purpose of the present chapter to establish this shape and these proportions.  The diamond will be treated essentially as if it were a worthless crystal in which the desired results are to be obtained, i.e. without regard to the great value which the relation between a great demand and a very small supply gives to the least weight of the material.

It is useful to recall here the principles and the properties which will be used in the calculations.

Reflection

  1. The angles of incidence and of reflection are equal.
  2. The paths of the incident and of the reflected ray lie in the same plane.

Refraction

  1. When a ray of light passes from one medium into a second of different density, it is refracted as by the following equation :
      n sin r  =  n' sin i            (Eq. 3)

    where   r  =  angle of refraction.
            i  =  angle of incidence.
            n  =  index of refraction of the second medium.
            n' =  index of refraction of the first medium.

    If the first medium is air, n' = 1, and equation becomes
      n sin r  =  (sin i)             (Eq. 2)

  2. When a ray of light passes from one medium into another optically less dense, total reflection occurs for all values of the angle of incidence above a certain critical value.  This critical angle is give by equation
                   n
       sin i'  =  ---                 (Eq. 4)
                   n'

    Or, if the less dense medium be air,
                   1
       sin i'  =  ---                 (Eq. 5)
                   n'

  3. The paths of the incident and of the refracted ray lie in the same plane.

Dispersion

When a ray of light is refracted, dispersion occurs, i.e. the ray is split up into a band or spectrum of various colours, owing to the fact that each colour has a different index of refraction.  The dispersion is the difference between these indices for extreme rays on the spectrum.

Data

In a diamond :
Index of refraction :   n   =  2.417  (for a sodium light)
         dispersion : delta =  0.044
     critical angle :   i'  =  24 26'             (Eq. 7)

Determination of the Best Angles and Proportions

Postulate. -- The design of a diamond or of any gem-stone must be symmetrical about an axis, for symmetry and regularity in the disposition of the facets are essential for a pleasing result.

Let us now consider a block of diamond bounded by polished surfaces, and let us consider the effect on the path of light of a gradual change in shape ;  we will also observe the postulate and keep the block symmetrical about its axis.

fig. 20:  A plate has no brilliance.

Let us take as first section one having parallel faces (fig. 20), and let M M' be its axis of symmetry.  Let us for convenience place the axis of symmetry vertically in all future work, so that all surfaces crossing it are horizontal.

Consider a ray of light S P striking face A B.  It will be refracted along P Q and leave by Q R, parallel to S P (as we have seen in studying dispersion).  We also know that if N N' is the normal at Q, angles N Q P and Q P M' are equal.  Therefore, for total reflection
  Q P M'  ==  24 26'

but at that angle of refraction the angle of incidence S P M becomes a right angle and no light penetrates the stone.  It is thus obvious that parallel faces in a gem are very unsatisfactory, as all the light passing in by the front of a gem passes out again by the back without any reflection.

We can avoid parallelism by inclining either the top or bottom faces at an angle with the direction A B.  In the first case, we obtain the shape of a rose-cut diamond and in the second case that of a brilliant cut.  We will examine the rose cut in the first instance.

The Rose

fig. 21:  A rose has some brilliance.

Consider (fig. 21) a section having the bottom surfaces horizontal, and let us incline the top surface A B at an angle alpha with it.  To maintain symmetry, another surface B C is introduced.  We have now to find the value of alpha for which total reflection occurs at A C.  Now for this to be the case, the minimum angle of incidence upon A C must be 24 26'.  Let us draw such an incident ray P Q.  To ensure that no light is incident at a smaller angle, we must make the angle of refraction at entry 24 26' and arrange the surface of entry as shown, A B, for we know that then no light will enter at an angle more oblique to A B or more vertical to A C.  This gives us a value of twice the critical angle, i.e. 48 52'.  Such a section is very satisfactory indeed as regards reflection, as, owing to its derivation, all the light entering it leaves by the front part.  Is it also satisfactory as regards refraction ?

fig. 22:  A simple rose has no fire.

Let us follow the path of a ray of light of any single colour of the spectrum, S P Q R T (fig. 22).  Let i and r be the angles of incidence upon and of refraction out of the diamond.

At Q, P Q N  =  R Q N, and therefore in triangles A P Q and R C Q
  angle A Q P  =  angle C Q R

Also by symmetry A = C,
therefore
  angle A P Q  =  angle C R Q

it follows that i = r.

As the angle i is the same for all colours of a white ray of light, the various colours will emerge parallel out of the diamond and give white light.  This is the fundamental reason of the unpopularity of the rose ;  there is no fire.

fig. 23:  A rose with steep sides leaks light out the sides.                   fig. 24:  A rose with flattened apex leaks light out the bottom.

This effect may be remedied to a small extent by breaking the inclined facet (figs 23 and 24), so that the angle be not the same at entry as at exit.  This breaking is harmful to the amount of light reflected whichever way we arrange it ; if we steepen the facet near the edge, there is a large proportion of light projected backwards and being lost, for we may take it that the spectator will not look at the rose from the side of the mounting (fig. 23).  If, on the other hand, we flatten the apex of the rose (fig. 24) (which is the usual method), a leakage will occur through its base.  There is, of course, no amelioration in the refraction if the light passes from one facet to another similarly placed (as shown in fig. 23, path S' P' Q' R' T').  Taking the effect as a whole, the least unsatisfactory shape is as shown in fig. 24, with the angles alpha about 49 and 30 for the base and the apex respectively.  The rose cut, however, is fundamentally wrong, as we have seen above, and should be abolished altogether.  It is the high cost of the material that is the cause of its still being used in cases where the rough shape is especially suitable, and then only in small sizes.  In actual practice the proportions of the cut rose depend largely upon those of the rough diamond, the stone being cut with as small a loss of material as possible.  Generally the values of alpha are much below those given above, i.e. 49 and 30, as where the material is thick enough to allow such steep angles it is much better to cut it into a brilliant.

The Brilliant

fig. 25:  The back of the brilliant -- first reflection.

Let us now pass to the consideration of the other alternative, i.e. where the top surface is a horizontal plane A B and where the bottom surface A C is inclined at an angle alpha to the horizontal (fig. 25).  As before, we have to introduce a third plane B C to have a symmetrical section.

First Reflection

Let a vertical ray P Q strike A B.  As the angle of incidence is zero, it passes into the stone without refraction and meets plane A C at R.  Let R N be the normal at that point, then, for total reflection to occur,
  angle N R Q  =  24 26'

But
  angle N R Q  =  angle Q A R  =  alpha
as A Q and Q R, A R and R N are perpendicular.

Therefore, for total reflection of a vertical ray,
  alpha  =  24 26'

Let us now incline the ray P Q so that it gradually changes from a vertical to a horizontal direction, and let P' Q' be such a ray.  Upon passing into the diamond it is refracted, and strikes A C at an angle Q' R' N' where R' N' is the normal to A C.  When P' Q' becomes horizontal, the angle of refraction T' Q' R' becomes equal to 24 26'.  This is the extreme value attainable by that angle ; also, for total reflection, angle Q' R' N' must not be less than 24 26'.  If we draw R' V, vertical angle V R' Q = R' Q' T' = 24 26', and
  angle V R' N'  =  V R' Q' + Q' R' N'
                 =  24 26' + 24 26'
                 =  48 52'

as before,
  alpha  =  angle V R' N'

and therefore
  alpha  =  48 52'                  (Eq. 9)

For absolute total reflection to occur at the first facet, the inclined facets must make an angle of not less than 48 52' with the horizontal.

Second Reflection

fig. 26:  The back of the brilliant -- second reflection of a vertical ray.

When the ray of light is reflected from the first inclined facet A C (fig. 26), it strikes the opposite one BC.  Here too the light must be totally reflected, for otherwise there would be a leakage of light through the back of the gem-stone.  Let us consider, in the first instance, a ray of light vertically incident upon the stone.  The path of the ray will be P Q R S T.  If R N and S N' are the normals at R and S respectively, then for total reflection,
  angle N' S R  =  24 26'

Let us find the value of alpha to fulfil that condition :
  angle Q R N  =  angle Q A R  = alpha
as having perpendicular sides.

  angle S R N  =  angle Q R N
as angles of incidence and reflection.
Therefore
  angle N R S  =  alpha

Now let
  angle N' S R  =  x

Then, in triangle R S C,
  angle S R C  =  90 - alpha
  angle R S C  =  90 - x
  angle R C S  =  2 * angle R C M
               =  2 * angle A R Q
               =  2 * (90 - alpha)

The sum of these three angles equals two right angles,
  90 - alpha + 90 - x + 180 - 2 alpha  =  180

or
  3 alpha + x  =  180
  3 alpha      =  180 - x

Now, x is not less than 24 26', therefore alpha is not greater than
             180 - 24 26'          
  alpha  =  ----------------  =  51 51'
                    3

fig. 27:  The back of the brilliant -- second reflection of an oblique ray.

Let us again incline P Q from the vertical until it becomes horizontal, but in this case in the other direction, to obtain the inferior limit.

Then (fig. 27) the path will be P Q R S.  Let Q T , R N, S N' be the normals at Q, R, and S respectively.  At the extreme case, T Q R will be 24 26'.  Draw R V vertical at R.

Then
  angle Q R V  =  angle T Q R  =  24 26'
  angle V R N  =  alpha

As before, in triangle R S C,
  angle S R C  =  90 - angle N R S
               =  90 - alpha - 24 26'
  angle R C S  =  2 * (90 - alpha)
  angle R S C  =  90 - x

Then
  90 - alpha - 24 26' + 180 - 2 alpha + 90 - x  =  180
  3 * alpha + x  =  180 - 24 26'  =  155 34'

In the case now considered,
  x  =  24 26'

Then
  3 * alpha =  155 34' - x  =  131 8'
    alpha =   43 43'                                  (Eq. 10)

For absolute total reflection at the second facet, the inclined facets must make an angle of not more than 43 43' with the horizontal.

We will note here that this condition and the one arrived at on page 66 [Eq. 9 -- Ed.] are in opposition.  We will discuss this later, and will pass on now to considerations of refraction.

Refraction

fig. 28:  Refraction, if pavilion is shallower than 45 degrees.

First case :  alpha is less than 45

In the discussion of refraction in a diamond, we have to consider two cases, i.e. alpha is less than 45 or it is more than 45.  Let us take the former case first and let P Q R S T (fig. 28) be the path of the ray.  Then, if S N is the normal at S, we know that for total reflection at S angle R S N = 24 26'.  We want to avoid total reflection, for if the light is to be thrown back into the stone, some of it may be lost, and in any case the ray will be broken too frequently and the result will be disagreeable.

Therefore,
  angle R S N  <  24 26'                 (Eq. 11)

Suppose this condition is fulfilled and the light leaves the stone along S T.  It is refracted, and its colours are dispersed into a spectrum.  It is desirable to have this spectrum as long as possible, so as to disperse the various colours far away from each other.  As we know, this will give us the best possible " fire ".

This result will be obtained when the ray is refracted through the maximum angle.  By (11) the value for that angle is 24 26', and (11) becomes
  angle R S N  =  24 26' for maximum dispersion.

But then the light leaves A B tangentially, and the amount of light passing is zero.  To increase that amount, the angle of refraction has to be reduced :  the angle of dispersion decreases simultaneously, but the amount of light dispersed increases much more rapidly.  Now we know that the angle of dispersion is proportional to the sine of the angle of refraction.  It is, moreover, proved in optics that the amount of light passing through a surface as at A B is proportional to the cosine of the angle of refraction.  The brilliancy produced is proportional both to the amount of light and to the angle of dispersion, and therefore to their product, and (by the theory of maxima and minima) will be maximum when they are equal, i.e. when the sine and cosine of the angle of refraction are equal.  For maximum brilliancy, therefore, the angle of refraction should be 45.  This gives for angle R S N
                   sin 45       0.7071          
    sin R S N  =  ---------  =  --------  =  0.2930
                    2.417        2.417
therefore
  angle R S N  =  17 for optimum brilliancy     (Eq. 12)

Let (fig. 28) Q X and R Y be the normals at Q and R respectively, and let Z Z' be vertical through R.

We know that
  angle R Q X  =  angle P Q X  =  alpha
therefore
  angle P Q R  =  2 * alpha

Produce Q R to Q'.

Then, as P Q and Z Z' are parallel,
  angle Z R Q' =  angle P Q R  =  2 * alpha
Now, let
  angle R S N  =  x ( =  17 for optimum brilliancy)

Then, as Z Z' and S N are parallel,
  angle Z R S  =  x

As they are complements to angles of incidence,
  angle Q R C  =  angle S R B  =  i (say)
but
  angle Q' R B =  angle Q R C
therefore
  angle S R Q' =  2i

In angle Z R Q' we have
  angle Z R Q' =  angle Z R S + angle S R Q'
    2 * alpha  =  x + 2 * i           (Eq. 13)

In triangle Q C R
  angle R Q C  =  90 - alpha
  angle Q R C  =  i
  angle Q C R  =  2 * (90 - alpha)
therefore
  (90 - alpha) + i + 180 - 2 * alpha  =  180
or
            i  =  3 * alpha - 90

Introduce this value of i in (13),
    2 * alpha  =  x + 6 * alpha - 180
    4 * alpha  =  180 - x
and giving x its value 17,
    4 * alpha  =  180 - 17  =  163
        alpha  =   40 45'                    (Eq. 14)

If we adopt this value for alpha, the paths of oblique rays will be as shown in fig. 29, P Q R S T when incident from the left of the figure, and P' Q' R' S' T' when incident from the right.  Ray P Q R S T will leave the diamond after the second reflection, but with a smaller refraction than that of a vertically incident ray, and therefore with less " fire."  Oblique rays incident from the left are, however, small in number owing to the acute angle Q R A with which they strike A C ; the loss of fire may therefore be neglected.

fig. 29:  How to fix a spread stone.

Ray P' Q' R' S' T' will strike A B at a greater angle of incidence than 24 26', and will be reflected back into the stone.  This is a fault that can be corrected by the introduction of inclined facets D E, F G ; ray P' Q' R' S' T' will then strike F G at an angle less than 24 26', and this angle can be arranged by suitably inclining F G to the horizontal so as to give the best possible refraction.  The amelioration obtained by thus taking full advantage of the refraction is so great that the small loss of light caused by that arrangement of facets is insignificant :  the leakage occurs through the facet C B, near C, where the introduction of the facet D E allows light to reach C B at an angle less than critical.  In a brilliant, where C B is the section of the triangular side of an eight sided pyramid, the area near the apex C is very small, and the leakage may therefore be considered negligible.

Second case :  alpha is greater than 45

In this case the path of a vertical ray will be shown by P Q R S T in fig. 30, and the optimum value for alpha, which may be calculated as before, will be
  alpha  =  49 13'             (Eq. 15)

As regards the vertical rays, this value gives a fire just as satisfactory as (14) (alpha = 40 45'); let us consider what happens to oblique rays.

Rays incident from the left as p q r s t u may strike B C at an angle of incidence less than the critical, and will then leak out backwards.  Or they may be reflected along s t, and may then be reflected into the stone.  Both alternatives are undesirable, but they do not greatly affect the brilliancy of the gem, because, as we have seen, the amount of light incident from the left is small.

That incident from the right is, on the contrary, large.

fig. 30:  Steep stones lose fire.

Let us follow ray P' Q' R' S' T'.  It will be reflected twice, and will leave the diamond after the second reflection, like the vertically incident ray, but with a smaller refraction, and consequently less fire ;  most of the light will be striking the face A B nearly vertically when leaving the stone, and the fire will be very small.  This time it is impossible to correct the defect by introducing accessory facets, as the paths S' T' of the various oblique rays are not localised near the edge B, but are spread over the whole of the face ;  we are therefore forced to abandon this design.





Summary of the Results obtained for alpha

We have found that--

  • For first reflection,  alpha must be greater than 48 52'.
  • For second reflection, alpha must be less than 43 43'.
  • For refraction, alpha may be less or more than 45.  When more, the best value is 49 15', but it is unsatisfactory.  When less, the best value is 40 45', and it is very satisfactory, as the light can be arranged to leave with the best possible dispersion.

Upon consideration of the above results, we conclude that the correct value for alpha is 40 45', and 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.  In all future work upon the modern brilliant we will therefore take
  alpha  =  40 45'

The editor's notes show that Tolkowsky's analysis is still valid even for diamonds that have a different pavilion angle. Tolkowsky provides a way to calculate how the best crown angle and table size depend on each diamond's actual pavilion angle. -- Ed.
fig. 31:  Cross-section

When arriving at the value of alpha  =  40 45', we have explained how the use of that angle introduced defects which could be corrected by the use of extra facets.  The section will therefore be shaped somewhat as in fig. 31.  It will be convenient to give to the different facets the names by which they are known in the diamond-cutting industry.  These are as follows :--



A C and B C are called pavilions or quoins
(according to their position relative to the axis of crystallisation of the diamond).

A D and E B are similarly called bezels or quoins.

D E is the table.

F G is the culet,
which is made very small and whose only purpose is to avoid a sharp point.

Through A and B passes the girdle of the stone.

We have to find the proportions and inclination of the bezels and the table.  These are best found graphically.  We know that the introduction of the bezels is due to the oblique rays ;  it is therefore necessary to study the distribution of these rays about the table, and to find what proportion of them is incident in any particular direction.

fig. 32:  Intercepted light

Consider a surface A B (fig. 32) upon which a beam of light falls at an angle alpha.  Let us rotate the beam so that the angle becomes beta (for convenience, the figure shows the surface A B rotated instead of A' B, but the effect is the same).  The light falling upon A B can be stopped in the first case by intercepting it with screen B C, and in the second with a screen B C' where B C C' is at right angles to the direction of the beam.  And if the intensity of the light is uniform, the length of B C and B C' will be a measure of the amount of light falling upon A B and A B' respectively.

Now
  B C  =  A B sin alpha
  B C' =  A'B sin beta  =  A B sin beta

Therefore, other things being equal, the amount of light falling upon a surface is proportional to the sine of the angle between the surface and the direction of the light.  We can put it as follows :--

If uniformly distributed light is falling from various directions upon a surface A B, the amount of light striking it from any particular direction will be proportional to the sine of the angle between the surface and that direction.

If we draw a curve between the amount of light striking a surface from any particular direction, and the angle between the surface and that direction, the curve will be a sine curve (fig. 33) if the light is equally distributed and of equal intensity in all directions.

fig. 33:  Sine curve showing 1/3 of rays are incident at less than 19 1/2 degrees.

For calculations we can assume this to be the case, and we will take the distribution of the quantity of light at different angles to follow a sine law.

It is convenient to divide all the light entering the diamond into three groups, one of vertical rays and two of oblique rays, such that the amount of light entering from each group is the same.  Now in the sine curve (fig. 33) the horizontal distances are proportional to the angles between the table of a diamond and the direction of the entering rays ;  the vertical distances are proportional to the amount of light entering at these angles.  The total amount of light entering will be proportional to the area shaded.  That area must therefore be divided into three equal parts ;  this may be done by integrals, or by drawing the curve on squared paper, counting the squares, and drawing two vertical lines on the paper so that one-third of the number of the squares is on either side of each line.

By integrals,
                      /
           area  =   |  sin x dx  =  - cos x
                    /

                    [        ] 180
  The total area =  [ - cos x]       =  1 + 1  =  2
                    [        ]   0
therefore
     (1/3) area  =  2/3

The value of alpha corresponding to the vertical dividing lines on the curve is thus given by
  cos x  =  1 - (2/3)  =    1/3
  cos x  =  1 - (4/3)  =  - 1/3
therefore
      x  =   70 approximately
and
      x  =  109

Taking the value x = 90 as zero for reckoning the angles of incidence,
      i  =  90 -  70  =   19
and
      i  =  90 - 109  =  -19

The corresponding angles of refraction are
             sin i       sin 19       0.3333
  sin r  =  -------  =  ----------  =  --------  =  0.1377
               n          2.417         2.417

      r  =  7 52'

The range of the different classes is thus as follows :--

  Angle of incidence:
     vertical rays  -19 to +19
      oblique rays  -90  to +19
          and       +19 to +90

  Angle of refraction:
     vertical rays   -7 52' to  +7 52'
      oblique rays  -24 26' to  +7 52'
          and        +7 52' to +24 26'

The average angle of each of these classes may be obtained by dividing each of the corresponding parts on the sine curve into two equal parts.  The results are as follows :--
  Angle of incidence:
     vertical rays    0
      oblique rays  -42
          and       +42

  Angle of refraction:
     vertical rays    0
      oblique rays  -16
          and       +16

For the design of the tables and bezels, we have to know the directions and positions of the rays leaving the stone.  The values just obtained would enable us to do so if all the rays entering the front of the gem also left there.  We have, however, adopted a value for alpha (alpha = 40 45') which we know permits leakage, and we have to take that leakage into consideration.

The angle where leakage begins is inclined at 24 26' to the pavilion (fig. 25).  We have thus
    Q' R' N'  =   24 26'
therefore
    Q' R' A   =   90 - 24 26'   =   65 34'

Now in triangle A Q' R',
    Q' R' A  + A Q' R' + R' A Q'  =  180
therefore
    A  Q' R'  =  180 - 65 34' - 40 45'
              =   73 41'

The limiting angle of refraction R' Q' T is thus
              =   90 - 73 41'   =  16 19'
corresponding to an angle of incidence of
       sin i  =  n sin r   =  2.417 sin 16 19'
              =  2.417 * 0.281  =  0.678
           i  =  42

fig. 34:  Sine curve INCORRECTLY showing incident rays that leak.

Editor's note 9 shows Tolkowsky made a rounding error. The correct angle is 42. Also, fig. 34 is wrong. The 42 is an angle of incidence, so it is measured from the vertical. In fig. 34, 0 and 180 are horizontal, and 90 (line M M') is the vertical. Using the coordinates of fig. 34, the shading should be between 0 and 47. -- Ed.

Upon referring to the sine curve, we find that the area shaded (fig. 34), which represents the amount of light lost by leakage, although not so large as if the same number of degrees leakage had occurred at the middle part of the curve, is still very appreciable, forming as it does about one-sixth of the total area.  Just under one-half (exactly 0.493) of the light incident obliquely from the right (fig. 25) is effective, the other half being lost by leakage.  Still, the sacrifice is worth while, as it produces the best possible fire.

Editor's note 12 shows that the fraction depends on alpha. For alpha = 40 45', the fraction = 0.518. Tolkowsky rounds it off to 0.493. -- Ed.

The oblique rays incident from the right range therefore 19 to 42, with an average (obtained as before) of 30 15'.  The corresponding refracted rays are 7 52',    16 19', and 12  0'.

We have now all the information necessary for the design of the table and the bezels. 

fig. 35:  Geometric model for calculating crown angle and table ratio, given the pavilion angle.

Design of Table and Bezels (fig. 35)

Let us start with the fundamental section A B C symmetrical about M M', making the angles C A B and A B C 40 45'.

The bezels have been introduced into the design to disperse the rays which were originally incident from the right upon facet A B.  To find the limits of the table, we have therefore to consider the path of the limiting oblique ray. We know that this ray has an angle of incidence of 42 and an angle of refraction of 16 19'. Let us draw such a ray P Q:  it will be totally reflected along Q R, if we make P Q N  =  N Q R, where Q N is the normal.  Now Q R should meet a bezel.

If the ray P Q R was drawn such that M P  =  M R, then P and R will be the points at which the bezels should meet the table.  For if P Q be drawn nearer to the centre of the stone, Q R will then meet the bezel, and if P Q be drawn further away, it will meet the opposite bezel upon its entry into the stone and will be deflected.

The first point to strike us is that no oblique rays incident from the left upon the table strike the pavilion A C, owing to the fact that the table stops at P.  We will, therefore, treat them as non-existent, and confine our attention to the vertical rays and those incident from the right.

Let us draw the limiting average rays of these two groups, i.e. the rays of the average refractions 0 and 12 passing through P, P S, and P T.  The length of the pavilion upon which the rays of these two groups fall are thus respectively C S and C T.

The rays of the first group P' Q' R' S' are all reflected twice before passing out of the stone, and make, after the second reflection, an angle of 17 with the vertical (as by eq. (12)).  Of the rays of the second group, most are reflected once only (P1 Q1 R1) and make then an angle of 69 with the vertical.  (This angle may be found by measurement or by calculation).  Part of the second group is reflected twice (P3 Q3 R3 S3), and strikes the bezel at 29 to the vertical.  This last part will be considered later, and may be neglected for the moment.

We have to determine the relation between the amount of light of the first group and the of the first part of the second group.  Now we know that the amount of oblique light reflected on pavilion A C is 0.493 of the amount of vertical light reflected (cp. fig. 34 and context).  If we take as limit for the once-reflected oblique ray the point E (as a trial) on pavilion B C, i.e. if it is at E that the girdle is situated, then the corresponding point of reflection for that oblique ray will be Q2 (fig. 35).  The surface of pavilion upon which the oblique rays then act will be limited by S and Q2, and as in a brilliant the face A C is triangular, the surface will be proportional to
  (SC * SC) - (Q2C * Q2C)

Similarly, the surface upon which the vertical group falls will be proportional to
         TC * TC

Thus we have as relative amounts of light--
    for vertical rays    (TC * TC)
    for oblique rays     0.493 * ((SC * SC) - (Q2C * Q2C))

The first group strikes the bezel at 17 to the vertical, and the second at 69 to the vertical.  The average inclination to the vertical will thus be

   17 * (TC * TC) + 69 * 0.493 * ((SC * SC) - (Q2C * Q2C))
  ------------------------------------------------------------
         (TC * TC)     +    0.493 * ((SC * SC) - (Q2C * Q2C))

Let us draw a line in that direction (through R, say), and let us draw a perpendicular to it through R, R E ;  then that perpendicular will be the best direction for the bezel, as a facet in that direction takes the best possible advantage of both groups of rays.

If the point E originally selected was not correct, then the perpendicular through R will not pass through E, and the position of E has to be corrected and the corresponding value of C Q2 correspondingly altered until the correct position E is obtained.

For that position E (shown on fig. 35), measures scaled off the drawing give
  CS  =  2.67       CS  * CS  =  7.12
  CT  =  2.13       CT  * CT  =  4.54       (CS * CS) - (CQ2 * CQ2)  =  4.57
  CQ2 =  1.60       CQ * CQ2 =  2.56

Therefore the average resultant inclination will be

   17 * (CT * CT) + 69 * 0.493 * ((CS * CS) - (CQ2 * CQ2))
  ----------------------------------------------------------
         (CT * CT)     +    0.493 * ((CS * CS) - (CQ2 * CQ2))              

   17 * 4.54 + 69 * 0.493 * 4.57
  ----------------------------------  
         4.54     +    0.493 * 4.57              

   77.2 + 156.2       233.4
  ----------------  =  --------  =  34.45  =  34
    4.54 +   2.24         6.78

to the vertical.

By the construction, the angle beta, i.e. the angle between the bezel and the horizontal, has the same value
  beta  =  34

The small proportion of oblique rays which is reflected twice meet the bezel near its edge, striking it nearly normally :  they make an angle of 29 with the vertical.  Facets more steeply inclined to the horizontal than the bezel should therefore be provided there.  The best angle for refraction would be 29 + 17  =  46, but if such an angle were adopted most of the light would leave in a backward direction, which is not desirable.  It is therefore advisable to adopt a somewhat smaller value ; an angle of about 42 is best. 

Editor's note: The inclination of the upper girdle facets must be less than 42 to the horizontal, to have a 4-sided kite facet and follow Tolkowsky's other recommendations. An upper girdle facet angle of 39 is possible.

Faceting

The faceting which is added to the brilliant is shown in fig. 37 Near the table, " star " facets are introduced, and near the girdle, " cross " or " half " facets are used both at the front and at the back of the stone.

We have seen that it is desirable to introduce near the girdle facets somewhat steeper than the bezel, at an angle of about 42, by which facets the twice-reflected oblique rays might be suitably refracted.  The front " half " facets fulfill this purpose.

We have remarked that the angle (42) had to be made smaller than the best angle for refraction (46) to avoid being sent in a backward direction, where it is unlikely to meet either a spectator or a source of light.

Editor's note: Actually, the angle is even smaller (39) so that the kite facet can have 4 sides. -- Ed.

To obviate this disadvantage, a facet two degrees steeper than the pavilion should be introduced near the girdle on the back side of the stone ; for then the second reflection of the oblique rays will send them at an angle of 25 to the vertical (instead of 29), and the best value for refraction for the front half facets will be between
  25 + 17  =  42

These values are satisfactory also as regards the distribution of light ; for now the greater part of the light is sent not in a backward, but in a forward, direction.

The facet two or three degrees steeper than the pavilion is obtained in the brilliant by the introduction of the back " half " facet, which is, as a matter of fact, generally found to be about 2 steeper than the pavilion in well-cut stones.  Where the cut is somewhat less fine and the girdle is left somewhat thick (to save weight), that facet is sometimes made 3 steeper, or even more, than the pavilion.

The " star " facet was probably introduced to complete the design of the brilliant, which without its use would be lacking in harmony, but which its introduction makes exceedingly pleasing from the point of view of the balance of lines.

Let us examine the optical consequences of the use of " star " facets.

On the one hand, their inclination -- about 15 to the horizontal -- permits a certain amount of light to leave the stone without having been sufficiently refracted.  On the other, they diminish the area of the bezels and consequently decrease the leakage of light which occurs through the bezel and the opposite pavilion (owing to the surfaces being nearly parallel).  They also cause a somewhat better distribution of light, for they deflect part of the rays which would otherwise have increased the strength of the spectra refracted by the bezels, and create therewith spectra along other directions ; it is true that, as seen above, these spectra will be shorter.  But they will be more numerous ;  and though the " fire " -- as consequent from the great dispersion of the rays of light -- will be slightly diminished, the " life " -- if we may term " life " the frequency with which a single source of light will be reflected and refracted to a single spectator upon a rotation of the stone -- will be increased to a greater degree.  And if we take into account the decrease in the leakage of light, we may conclude that the introduction of the stars, on the whole, is decidedly advantageous to the brilliant.

Editor's note: The inclination of the star facets must be greater than 15 to the horizontal, to have a 4-sided kite facet and follow Tolkowsky's other recommendations. A star facet angle of 19 or 20 is possible.

Best Proportions of a Brilliant

We have thus as best section of a brilliant one as given in fig. 35, A B C D E, where

             alpha  =  40 45'
              beta  =  34 30'

Editor's notes:
        star facet  =  15       inclination from girdle
                                 -- but kite facet has 6 sides
                    =  19 - 20 -- if  kite facet has 4 sides
upper girdle facet  =  42       inclination from girdle
                                 -- but kite facet has 6 sides
                    =  39       -- if  kite facet has 4 sides
lower girdle facet  =  42 45'   declination from girdle      
                       (2       steeper than alpha)
-- Ed.

D E is obtained from P R in fig. 35.

Editor's notes:
Tolkowsky has just renamed the points.
The new points A D M E B C are the same as fig. 35's D P M R E C.
The new A B is the diameter in the girdle plane.
The new A B is obtained from D E in fig. 35.
The new D E is the corner-to-corner diameter of the table.
The new D E is obtained from P R in fig. 35.
M C is the gem axis, as shown in fig. 35.
M' is the intersection of the gem axis and the girdle plane. -- Ed.

If we make the diameter A B of the stone 100 units, then the main dimensions are in the following proportions (fig. 35) :--

Diameter A B                  100   %
Table D E                      53.0 %
Total thickness MC             59.3 %
Thickness above girdle M  M'   16.2 %
    "     below    "   M' C    43.1 %

Editor's note:
Tolkowsky omits the girdle and the culet.
My article on girdles shows that the girdle adds 1.7% plus the nominal girdle thickness to the overall depth. On the other hand, part of this extra thickness is offset by a smaller optimum crown angle and a larger optimum table ratio. -- Ed.

fig. 36:  Proportions of diamond with knife-edge girdle. fig. 37:  Top and bottom of diamond.

Fig. 36 shows the outline of a brilliant with these proportions.

These proportions can be approximated as follows :--

In a well-cut brilliant the diameter of the table is one-half of the total diameter, and the thickness is six-tenths of the total diameter, rather more than one-quarter of the thickness being above the girdle, and rather less than three-quarters below.

It is to be noted here that a different proportion is generally stated for the thickness above the girdle (" one-third of the total thickness "), both in works upon diamonds, and by diamond polishers themselves.  It is true that diamonds were cut thicker above the girdle and with a smaller table before the introduction of sawing, for then the table was obtained by grinding away a corner or an edge of the stone, and the loss of weight was thus considerable, and would have been very much greater still if the calculated proportions had been adopted.  With the use of the saw, the loss in weight was enormously reduced and the manufacture of sawn stones become therefore much finer and more in accordance with the results given above.  It is a remarkable illustration of conservatism that although diamonds have been cut for decades with (approximately) of the thickness above the girdle, yet even now the rule is generally stated as 1/3 of the thickness.

Stones are still cut according to that rule, but then they are not sawn stones as a rule, and the thickness left greater to diminish the loss of weight.  The brilliancy is not greatly diminished by making the stone slightly thicker over the girdle.

Comparison of the Theoretically Best Values with Those Used in Practice

In the course of his connection with the diamond-cutting industry the author has controlled and assisted in the control of the manufacture of some million pounds' worth of diamonds, which were all cut regardless of the loss of weight, the only aim being to obtain the liveliest fire and the greatest brilliancy.  The most brilliant larger stones were measured and their measures noted.  It is interesting to note how remarkably close these measures, which are based upon empirical amelioration and rule-of-thumb correction, come to the calculated values.

As an instance the following measures, chosen at random, are given (the dimensions are in millimetres) :--

Table I

       Dia 1   Dia 2   Dia 3   Dia 4   Dia 5
alpha  40    40    40     41     41
beta   35     35     34    33     34
A B     7.00    7.08    6.50   21.07    9.12
M C     4.12    4.35    3.61   12.34    5.47
M M'    1.08    1.32    0.85    3.31    1.61

These measures, worked out in percentage terms of A B, give :--

Table II
                                                            Rounded    1% Girdle
      Dia 1   Dia 2   Dia 3   Dia 4   Dia 5  = Average  vs. Theory   ~ Theory

alpha  40    40    40     41     41   =  40 42' vs.  40 45' ~  40 45'
beta   35     35     34    33     34   =  34 18' vs.  34 30' ~  33 53'
A B   100     100     100     100     100    = 100      vs. 100      ~ 100
M M'   15.7    18.6    13.3    15.7    17.8  =  16.2    vs.  16.2    ~  15.0
M'C    43.0    42.8    42.1    42.8    42.2  =  42.6    vs.  43.1    ~  43.1
M C    58.7    61.4    55.4    58.5    60    =  58.9    vs.  59.3    ~  60.9
                                                                       -culet ht

Editor's note 37 shows that the best crown angle and table ratio depend on the actual pavilion angle. The column headings, "=", "vs.", and "~" notes, and "1% Girdle Theory" column are new in this edition. The order of the last two rows has been changed for easier reading.

The "1% Girdle Theory" column is based on my article,
"Diamond Girdles -- How Adding a Girdle Changes Marcel Tolkowsky's Diamond Design". I do not know if Tolkowsky's sample diamonds had girdles, let alone how thick they may have been. I do not know what the best girdle thickness is. It might be 1% for some diamonds; it might a larger percentage for smaller diamonds. (I measure the "girdle" by averaging the thin parts of the girdle.)

Editor's note 29 shows that the "culet height" is about one-half the "culet" (diameter) reported by Sarin machines. -- Ed.

In the seventh column, the averages of the measures are worked out, and the eighth gives the calculated theoretical values.  It will be noted that the values of alpha, beta, and M M' correspond very closely indeed, but that M C and M' C are very slightly less than they should be theoretically.

The very slight difference between the theoretical and the measured values is due to the introduction of a tiny facet, the collet, at the apex of the pavilions.  This facet is introduced to avoid a sharp point which might cause a split or a breakage of the diamond.

What makes the agreement of these results even more remarkable is that in the manufacture of the diamond the polishers do not measure the angles, etc., by any instrument, but judge of their values entirely by eye.  And such is the skill they develop, that if the angles of two pavilions of a brilliant be measured, the difference between them will be inappreciable.

We may thus say that in the present-day well-cut brilliant, perfection is practically reached :  the high-class brilliant is cut as near the theoretic values as is possible in practice, and gives a magnificent brilliancy to the diamond.

That some new shape will be evolved which will cause even greater fire and life than the brilliant is, of course, always possible, but it appears very doubtful, and it seems that the brilliant will be supreme for, at any rate, a long time yet.