Diamond Brilliance
Theories, Measurements and Judgement
Abstract:
The observation of brilliance in a diamond involves a complex interaction of the viewer, the illumination environment, and the manner in which light is processed by a diamond. In order to measure the brilliance of a diamond, a computer simulation of this interaction must model all three of these aspects. The success of any theoretical measure of brilliance is how well it agrees with human judgment.
A study and analysis is reported that contrasts the recent GIA 'analysis of brilliance' findings with the teaching of the GIA Diamond Course and the experience and theories of Tolkowsky and others in the diamond and gem trades. Three-dimensional computer modelling and diamond photography are used to gain a greater understanding of brilliance and arrive at suggestions for improving its measurement.
GIA, AGS and the American Ideal Cut
The Gemological Institute of America (GIA) and the American Gem Society (AGS) both have common roots in their founder Robert Shipley. For over half a century both have taught the concept of an 'Ideal' round brilliant cut diamond. The GIA attributed the mathematical computations of the best angles and proportions for what they termed the American 'Ideal' or the Tolkowsky cut diamond to Marcel Tolkowsky and his 1919 book Diamond Design. In his book, Tolkowsky stated that "the most vivid fire and the greatest brilliancy" is obtained with pavilion main angles of 40.75°, crown main angles of 34.5° and a table size of 53%.
GIA students have been taught: "Most cutters and other experts agree that even a two-degree deviation from Tolkowsky's theoretical pavilion angle will result in a less attractive diamond. An increase of two degrees in pavilion angle will result in a noticeable darkening of the stone and an obvious loss of brilliance ... a decrease of only two degrees in pavilion angle ... usually shows a reflection of the girdle in the table and is called a 'fish-eye' ... [and] also gives the stone a very 'glassy' appearance." [1]
In their latest revised course on diamond grading from 1993, the GIA adheres even more strictly to Tolkowsky's angles, e.g.: "Today most cutters and other diamond experts agree that varying more than one degree from a pavilion angle of about 41° reduces a diamond's optical efficiency, and thus its beauty ... A crown angle close to 34.5° is the best compromise between optical theory and economic reality."[2]
The AGS uses a 0-10 grading system for diamond cuts that is based upon the 'Ideal.' For discussion purposes, we will use their definition of 'Ideal', which is AGS 0. This AGS 0 grade requires fairly close adherence to Tolkowsky's pavilion and crown angles, but allows his 53% table to range from 52.5% to as large as 57.5%.
Diamond Brilliance and Weighted Light Return
Recently, Hemphill et al. (1998) reported ongoing research at GIA that has come to some conclusions that mark a significant break with the GIA Diamond Course teaching. The authors use a measure of brilliance known as Weighted Light Return (WLR), which they say "captures the essence of brilliance."[22] If brilliance were the only factor determining the beauty of a diamond, the GIA believes their measure has captured this essence of beauty. They qualify this by saying that remaining to be analyzed for a full understanding of diamond beauty are "fire and scintillation, and probably symmetry deviations and color." They also state that "... the relationship between brilliance and the three primary proportion parameters (crown angle, pavilion angle, and table size) is complex, and that there are a number of proportion combinations that yield high WLR values" [3] and "The results of this study suggest that there are many combinations of proportions with equal or higher WLR than 'Ideal' cuts."[4] Boyajian (1998) concluded that, "Although it is not the GIA's role to discredit the concept of an 'Ideal' cut, on the basis of our research to date we cannot recommend its use in modern times."
Focus on Diamond Brilliance and its Modeling by Computer
Is this GIA brilliance study cause for abandoning the Tolkowsky or American 'Ideal'? To answer this question, we have focused, as did the GIA study, on this single aspect of diamond beauty referred to as brilliance.
Brilliance is defined in the GIA Diamond Dictionary [5] as "the intensity of the internal and external reflections of white light to the eye from a diamond or other gem in the face-up position". The face-up position is the normal viewing position of a diamond with the viewer looking from a position that is approximately perpendicular to the gem's table.
The dual meaning of the normal viewing angle makes it a useful term for the face-up view of a gemstone. Mathematically, the normal direction to a surface is the perpendicular direction to the surface. The face-up view of a diamond is a view approximately perpendicular or normal to the surface of the diamond's table. In addition, this most important viewing angle is the usual or normal way a diamond is observed and evaluated for beauty.
Has the GIA captured the essence of brilliance in this single WLR measure? We will see that, in order to be useful, a measurement of brilliance such as WLR must agree with human judgment.
What are the necessary features that a computer model needs in order to measure brilliance in a way that is consistent with human judgement of brilliance? Since observation of brilliance in a diamond involves an interaction of the way in which light is processed by the diamond, the diamond's illumination, and the act of viewing by the observer, all three aspects of this interaction should be essential parts of a computer model of brilliance.
In particular, when brilliance only is being measured, as in the GIA study, a simulated illumination is needed that captures the properties of the typical lighting environments in which brilliance is normally judged. If the goal were a more complete computer analysis of diamond beauty, an illumination environment that enhances fire and sparkle as well as brilliance would be needed.
Jewellers know that it is important to display diamonds with illumination that best brings out a diamond's brilliance, fire and sparkle. Because point sources of light bring out a diamond's fire and sparkle as well as brilliance, most diamond-selling areas in jewelry stores have many bright spotlights to illuminate the diamond jewelry. Diamonds displayed in flat, diffuse, fluorescent illumination found in typical office environments may exhibit brilliance but display less fire and sparkle.
Examining the Three Main Features of the Diamond Brilliance Computer Model
Let us now examine the three main features of the computer model of diamond brilliance described by Hemphill et al. (1998). These features are the diamond, the illumination and the observer.
The GIA took great care to create a computer model of the way in which light is processed in a diamond that is more complete than any before it. Included are three-dimensional effects, a wavelength-dependent refractive index, and the accounting of secondary rays and light polarization. They state that their model differs from its predecessors in that it is three-dimensional and "uses the most detailed existing data on the properties of a colorless diamond"[6] It would be difficult to improve on this representation of how light is processed by a diamond.
It is necessary to point out that the modeling concerns a colorless, flawless round brilliant-cut diamond with mathematically perfect symmetry. Hemphill et al.[7] put this in context by saying, "Real diamonds will inevitably differ from the model conditions because of inclusions, symmetry deviations, and the like." Differences such as symmetry faults and inclusions could also be modeled, but are not addressed in this study. There are varying opinions as to the point at which these imperfections have an impact on diamond brilliance.
A "diffuse hemisphere of even, white light" was selected "to best average the many different ambient light conditions in which diamonds are seen and worn, ... such as a common consumer experience of seeing a diamond worn outdoors or in a well lit room"[8]. This 'hemisphere' illumination provides even lighting from above the girdle but no lighting from below. Visualize the diamond in the centre of a white evenly-illuminated hemisphere mounted in a setting that blocks the light from entering below the girdle. The Figure 2a, computer-generated image of a diamond reveals the effect of this illumination environment[9]
Figure 1: Actual 'Ideal' cut photograph from Figure 1, lower left, of Hemphill et al., 1998.
Figure 2a: Virtual image of 'Ideal' cut from Hemphill et al., 1998, Figure 2. Figure 2b: Photo of 'Ideal' cut in hemisphere lighting from Hemphill et al., 1998, Figure 2.
Unlike the diamond photograph shown in Figure 1, the images obtained using hemisphere lighting look fairly evenly white except for some darker areas where the diamond is reflecting and refracting light from below the girdle where there is no light source.
To obtain a photograph that looked similar to the computer generated image of Figure 2a, Hemphill et al. (1998) photographed the diamond "in diffuse white light using a hemispherical reflector" and noted "diffuse illumination reduces the overall contrast" (see Figure 2b).
Comparing the images in Figures 1, 2a and 2b reveals that hemisphere lighting does not give a realistic presentation of the diamond brilliance observed in typical viewing circumstances. The reason for this is that most points on the diamond's crown are refracting and reflecting the same even light from above the plane of the girdle causing all these points to be bright. Consequently, differently proportioned diamonds under hemisphere lighting show smaller differences in brilliance than are seen by an observer in typical lighting environments. For example, the study results of Hemphill et al. (1998) indicate that with 'hemisphere illumination' the WLR of known high-brilliance diamonds and those of average brilliance only varies from .285 to .275, or about 4%.
When diamonds are judged for brilliance in typical viewing circumstances, the viewer's head and body interfere with the illumination that would otherwise be coming from behind the viewer. Diamond proportions that respond poorly under these circumstances are perceived to have low brilliance. Because the 'hemisphere' illumination does not incorporate this viewer interference, in some important instances these same diamond proportions may have high WLR.
In sections 5 and 6, evidence will be presented to show that greater consistency between the GIA WLR brilliance measure and human observation of brilliance can be obtained by taking explicit account of the interference in illumination resulting from the physical presence of the viewer.
In the Hemphill et al. (1998) study the brightness of 65,536 pixels (tiny areas) across the surface of the diamond image is evaluated. This requires up to 65,000,000,000 light rays traced from their hemisphere lighting source through the virtual diamond. This is similar in concept to the approach taken by most investigators from Tolkowsky in 1919 through to the present. Most of the reported research has concerned itself with whether light coming into a diamond's crown from all angles above the girdle (as in hemisphere illumination) is reflected and refracted back through the crown or is lost out of the pavilion. This approach does not concern itself with the impact of the physical presence of the viewer on the illumination environment, or whether the light returned through the crown is seen in the important face-up position by the 'normal' observer.
WLR Averaging Verses 'Snap Shot" Analysis of Brilliance
In the face-up observation of a diamond, the viewer sees the portion of light that exits the diamond at approximately 90° to the table. In answer to the question "Should a mathematical definition of brilliance represent one viewing geometry - that is a 'snapshot' - or an average over many viewing situations?", Hemphill et al. (1998) chose the average instead of the 'snapshot', as have previous researchers, but with an important difference. Their measure sums the light rays returned through the crown, multiplied by "the square of the cosine function". Recognizing the importance of the face-up viewing position they "wanted the contribution from rays that emerged straight up to be much greater."[10] Their weighting function emphasizes the face-up, 'normal' observer condition by giving light rays near 90° the greatest weight. They note: this "averaged observer condition ... takes into account the likeliest ways in which a diamond dealer or consumer looks at the stone."[11]
If the weight given to light rays near 90° were increased to the limit relative to other angles of observation, it would lead to a 'snapshot' of a diamond in the face-up position. Let us consider the suitability of a measure of brilliance using this and other 'snapshots'. Evaluating 'snapshots' of brilliance, especially the face-up, 'normal' viewing position has three important advantages over averaging:
(i) Averaging looses information
The single value of WLR obtained by an averaging of viewing angles has lost the detailed knowledge of the relative brilliance occurring at any particular angle of observation such as the important face-up position. Additionally, WLR has averaged out and lost the detailed information of the relative brilliance emitted at each point across the diamond's surface. 'Snapshots' retain this detail and can individually be used to assess brilliance at each point on the diamond and each viewing angle.
(ii) Economy of computation
By analysing the face-up 'snapshot' of a diamond for brilliance, a large economy of computation is realized. Analysing this one 'snapshot' greatly simplifies the search for the most brilliant diamond proportion parameters.
(iii) Ability to observe and measure contrast aspects of brilliance
There are aspects to the perception of brilliance that go beyond the amount of light returned from the crown of a diamond. Such aspects may be observed and measured from a 'snapshot'. A large amount of intensity variation or contrast between light and dark areas across the diamond gives it an aspect of brilliance that has been described as 'snappy', 'dramatic', 'hard' or 'sharp.'[12] This aspect is the opposite of the brilliance description of 'watery' and 'glassy' used in the GIA Course and the Diamond Dictionary[13] to describe a 'fish eye' diamond, which has a weak appearance due to the lack of contrast as well as lower light return.
This aspect is called 'contrast brilliance' to distinguish it from light return measures of brilliance.
This contrast aspect of brilliance has properties similar to the contrast variation from bright to dark that occurs with diamond movement called scintillation. Where scintillation is a dynamic contrast quality due to movement, the contrast quality of brilliance is the diamond's static contrast. The picture of contrast brilliance is one frame or snapshot of the moving picture of scintillation. The change in contrast brilliance from one moment to the next is scintillation, so contrast brilliance and scintillation are related in this way.
Additional aspects to the perception of contrast brilliance are the size and number of these contrasting light and dark areas and how evenly they are distributed over the diamond. A diamond exhibiting very efficient light return but having little contrast in light intensity from facet to facet over the diamond surface is perceived to be 'glassy' or lacking 'snap'. By analyzing a 'snapshot' of a diamond, more can be learned about these aspects of brilliance that are lost in a single measure that averages many viewing positions.
The dual meaning of the normal viewing angle makes it a useful term for the face-up view of a gemstone. Mathematically, the normal direction to a surface is the perpendicular direction to the surface. The face-up view of a diamond is a view approximately perpendicular or normal to the surface of the diamond's table. In addition, this most important viewing angle is the usual or normal way a diamond is observed and evaluated for beauty.
Because diamonds are evaluated for beauty in the face-up viewing position, the brilliance at this normal viewing angle is of paramount importance. The principal concern in deciding to use snapshots, especially this normal snapshot, for brilliance, instead of averaging viewing angles, is whether a diamond will retain its brilliance when it is tilted slightly from the perpendicular. To answer this concern, diamond proportions with the highest face-up brilliance should be evaluated at other viewing angles. Our experience has shown that diamonds with near Ideal proportion parameters maintain superior brilliance when viewed at angles off the perpendicular. Brilliance is essentially undiminished through 15° of tilt, only slightly at 25° and slightly more at 45°.
Owing to its relatively high refractive index (RI), maintaining brilliance from tilted viewing angles is one of the properties that distinguishes diamonds from white gemstones that have lower RI. The tendency of diamond imitations such as YAG, GGG, white topaz or spinel to lose brilliance when tilted can be used to separate them from diamonds.
Let us look at two views of an 'Ideal' cut diamond, one face-up and the other tilted 10° (Figures 3a and 3b). These were generated with the Russian DiamCalc software.[14]
Figure 3a: 'Ideal' cut diamond face-up view. Figure 3b: 'Ideal' cut diamond tilted 10°.
They illustrate that the brilliance of the 'Ideal' cut is retained at angles off the perpendicular. We will see later through the use of ray tracing diagrams, such as those of a 'nail head' diamond in Figures 5a and 5b of section 5, that certain paths of light from the illumination source back to the viewer remain basically unchanged in spite of normal amounts of diamond tilt.
Both examples illustrate that we are on safe ground by evaluating the face-up 'snapshot' of diamond brilliance.
It is clear from sections 2 and 3 that more can be learned about diamond brilliance if the analysis is done in a lighting environment that accounts for the viewer interference. The most important viewing angle is the face up, 'normal' viewing position. To study this single important viewing geometry, the 'normal snapshot', we only have to consider those light rays that emerge from the diamond's crown directly to our eyes along a path approximately perpendicular to the diamond's table. (The following discussion applies to all angles of observation.)
Light path Reversibility
This analysis makes use of a physical property of light reflection termed 'light path reversibility': Light reflecting internally in a diamond travels the same path through the diamond moving in either direction. (To be more complete and include the refraction and dispersion of light that occurs upon light entry and exit: Each spectral color or wavelength of light individually travels the same path into, through and out of the diamond moving in either direction)
Each pixel or tiny area of a diamond refracts and reflects light to our eyes from some angle and position in the space around the diamond. Note that this position varies slightly with the color and polarization of the light ray. There are additional secondary positions that also contribute to the light coming from each pixel.
Let us follow the path of light in reverse from the eye, along the perpendicular into the diamond through that tiny area in order to determine where it emerges. Because light follows the same path moving in either direction, that point of emergence and direction is the primary path along which light would have to enter the diamond in order to be seen in that pixel by the 'normal' viewer.
Figure 4: (a) This point on the diamond's table will be dark because little or no light enters below the girdle. (b) This point on the diamond will be bright if there is light in the direction of this ray.
If that path of light emerges from the diamond below the girdle where there is no light, as in Figure 4a, no light will be seen in that pixel's area. (Notice that a weaker, secondary path of light reflection also exits below the girdle.) If the light path emerges in a direction above the girdle, as in Figure 4b, that pixel will be bright if light exists in that direction, as it is bound to under hemisphere lighting. However, as will be seen in the 'nail head' diamonds of Figures 5a, 5b, 8c and 9b, if the viewer's head and body block some of the light, as they would in close-up examination of the diamond, the pixels reflecting light from the direction of the viewer will also be dark.
Primary and Secondary Ray Contributions to Brilliance
Because of multiple reflections and transmissions of each of these rays, there are other secondary paths from which light can reach the eye from any particular pixel (note Figure 4a for example). Strickland and Long's[15] measure of brilliance defines the sum of all these contributions as the brilliance for that pixel, and summing across all the pixel areas gives the total brilliance of a diamond for the 'normal' viewing position.
When evaluating diamond proportion parameters for brilliance, it is important to consider where light rays would have to originate in order to reach the viewer's eyes. Diamond proportion parameters that cause most of the tiny pixel areas to refract and reflect light from directions where illumination exists will result in a brilliant diamond. The next two sections will contain examples of diamonds that exhibit inferior brilliance owing to areas on the diamond that refract and reflect light to the observer from directions where little light exists.
A Computer Aided Approach and Methodology for Gemstone Cut Design
The methodology is to determine where each pixel or point on a diamond gets its light for each set of diamond proportions, and then to choose combinations of pavilion and crown angles that reflect the least light from the direction of the observer or from below the gemstone's girdle, where there is little or no source of illumination. This concept was first advanced by Harding (1975) and has had a significant impact on the determination of optimum angles for cutting gemstones. Glen and Martha Vargas published a portion of the work in chart form in Faceting For Amateurs (1977). However, Harding's concepts appear to have been largely overlooked, or not understood, by those in the diamond and jewelry industry concerned with optimum angles for cutting diamonds.
Harding's work did not have all the answers, but he showed that good brilliance depends on avoiding combinations of crown and pavilion angles that reflect light to the viewer from his/her direction, as well as from below the gemstone's girdle. Instead of searching for 'Ideal' angles, Harding eliminated combinations of crown and pavilion angles that were clearly not ideal from this perspective. This process of elimination is a useful tool in narrowing the search for the possible range of 'Ideal' diamond proportions.
The computer faceting design software of Strickland, such as Gemcad, Gemray, Gemframe and Gemflick, can be used to simulate and measure diamond brilliance and explore these concepts. Strickland employs three-dimensional gemstone modeling with several illumination environments. His work and the work of others such as Long and Steele have taken faceting design to new levels of technological sophistication. In the present study, programs and work done by a group in Russia associated with Moscow State University have also been employed. This parallel effort in Russia has culminated in the computer aided, diamond cut design software called DiamCalc and the MSU Diamond Cut Study. Led by Sergey Sivovolenko, OctoNus Software, Yurii Shelementiev, Gemology Center of MSU, and Anton Vasiliev, this Russian effort grew from work by Vasiliev that expanded on Harding's original work.
The diamond cutting community could with advantage consider all these contributions and ideas in the quest for the proportions of the most beautiful round brilliant-cut diamonds.
Perhaps the best case to illustrate the need for incorporating the effect of the viewer's physical presence on brilliance is a diamond with pavilion main facets between 43° and 45°. This is known as the 'nail head' diamond owing to its dark appearance under the table relative to areas outside the table. Assignment 8 of the GIA Diamond Grading Course (1993), states: "If the pavilion is very deep, much of the light is leaking out. Then the table reflection and star facets look almost black, and the stone is called a 'nail head'."
Retro Reflection
Contrary to this usual explanation for the darkness in a deep cut diamond, a nail head with pavilion main facets of 45° mirrors light from above through the table in those main facets, and rather than leaking the light, reflects it straight back towards its source. In the optics field this is termed 'retro reflection'. A viewer of such a diamond could observe a mirror image of skin tones of him/herself in those pavilion main facets. Furthermore, the head obscures any illumination from behind, causing those main facets to darken under the table. The pavilion girdle facets, which are cut between 1° and 2° steeper than the mains, also darken under the table giving the whole table area a darkness relative to areas outside the table.
Figure 5: 'Nail head' diamond with pavilion angle of 45° (a) face-up and (b) tilted 15°.
Figures 5a and 5b, generated by the Russian computer software, illustrate light passage in a 'nail head' diamond with a pavilion angle of 45°. This 'Nail head' diamond retro reflects light from the direction of the viewer's head even when the diamond is tilted. Compare this to the diamond in Figure 4b, which has an 'Ideal' pavilion angle, causing light to reflect from an angle safely away from the viewer's head.
Much course and textbook literature attributes the undesirable 'nail head' appearance to light leakage out of the pavilion. Primary and secondary leakage (leakage at the first and second points of internal reflection) occurs to a greater extent in gemstones with lower refractive indices such as quartz, beryl or the plastic used in the GIA GEM Instruments' Proportion Comparator demonstration tool. Compare the Figure 6a photograph of the demonstration tool[16] and Figure 6b derived from the Russian computer software. They are similar and illustrate the secondary, pavilion light leakage that occurs with steep pavilion angles in the plastic demonstration tool.
Figure 6: (a) Demonstration tool showing pavilion light leakage in a 'nail head' diamond (see text); (b) Path of light in plastic with a lower RI; (c) Path of light in diamond, (RI 2.42).
In diamond with its relatively high refractive index, the pavilion angle would have to approach 52.5° before this type of leakage became apparent in the pavilion mains in the table in the face-up viewing position. The 'nail head' appearance is evident in diamonds with pavilion angles between 43° and 45°. Thus, as we see in Figures 5a, 5b and 6c, the dark 'nail head' appearance is due not to loss of light through the pavilion, as was commonly believed and taught. Rather, it is due to a steeper than 'Ideal' pavilion that is reflecting light to the 'normal' observer from the area of his head rather than from an unobscured source of illumination.
Observer's Influence on the Diamond's Illumination
A computer model of the 'nail head' diamond will not show the darkening caused by the observer's head if the illumination model does not take into account the way light is blocked by the physical presence of the observer. To support this, note that Hemphill et al.[17] state: "Pavilion Angle. This is often cited by diamond manufacturers as the parameter that matters most in terms of brilliance ... Images of virtual diamonds with low, optimal, and high pavilion angles (again, see Figure 5) are consistent with the appearances that we would expect for actual diamonds with these pavilion angles ('fish-eye', normal, and 'nail head')."
Figure 7: From GIA study Figure 5.[18]
Explanation of the Uncharacteristic Brightness of the GIA Nail Head
When we look at these virtual images in Figure 7, the 'nail head' example on the right does not appear "consistent with the appearance we would expect" of 'nail head' diamonds. Instead of being dark under the table, the virtual image shown is extremely bright in the center two-thirds of the table. Compare the virtual image of the 'nail head' diamond in Figure 8a, reproduced from Hemphill et al. (op. cit.) with the virtual image of the same 'nail head' diamond in Figure 8b that has been generated by a new version of Strickland's Gemray computer software and uses hemisphere lighting similar to that described by Hemphill et al.[19] Though Strickland's model uses only one RI and averages the ray polarization, the Figure 8a and 8b images depict a similar bright center that is unlike the appearance of a 'nail head' diamond.
Figure 8: 'Nail head' virtual image in hemisphere lighting: (a) Hemphill et al., 1998, (b) Strickland and (c) Strickland, including the interference effect of a viewer.
By introducing the effect of the viewer blocking some of the light from above, a new virtual image is generated (Figure 8c). This looks darker under the table and is much more like the familiar appearance of a 'nail head' diamond. Also the light return falls off more as expected when compared to the 'Ideal'. In hemisphere lighting, Hemphill et al. (1998) found only a 0.282 to 0.270 = 4.25% drop-off in WLR between an 'Ideal' cut and this 'nail head' diamond.
The truly dark 'nailhead' appearance is more apparent in diamonds with a pavilion depth of 48% to 50%, which corresponds to pavilion main angles of 44° to 45°. The 'nail head' discussed and pictured here, with 43° pavilion, is only beginning to darken under the table. It took a large amount of light blockage, simulating close-up inspection, to produce Figure 8c.
Diamonds in Hemisphere Light with and without Viewer Interference
To further demonstrate the importance of the type of illumination, we created a photographic set-up using three actual diamonds: a close to 'Ideal' cut and two 'nail heads'. Two lighting environments were used. The first approximates hemisphere lighting with diffuse illumination in a 180° hemispherical arc above the diamond's girdle plane. The second also approximates hemisphere lighting but with light blocked in an area above these diamonds to simulate the close-up viewing situation. In both cases, the three diamonds were photographed simultaneously. Interchanging them produced essentially no change in appearance, verifying that, for comparative purposes, each was illuminated in the same manner. (In both cases the diffuse illumination was not as even as in the computer model due to use of two diffused fiber optic light sources.)
Figure 9: 'Nail heads' vs. near 'Ideal' cut diamonds (a) in hemisphere lighting created by diffusing two fiber optic light sources, and (b) in hemisphere lighting partially blocked as in close-up inspection.
In the diffuse hemisphere lighting photograph (Figure 9a), all three diamonds have similar even brilliance. There is slightly more brilliance in the near 'Ideal' cut due to some dark areas in the outer table region of the 'nail head' diamonds. However, the two 'nail heads' are very bright in the middle portion of their tables, just as in Figures 8a and 8b. Contrast this with the dramatic darkening of the whole table and star facet areas of both 'nail head' diamonds in Figure 9b. This appearance is consistent with Strickland's virtual image of the 'nail head' diamond in Figure 8c, because it has accounted for the viewer blocking light directly over the diamond in the 'normal' viewing position.
This is photographic evidence that the typical 'nail head' appearance in a diamond with deep pavilion angles is not seen in hemisphere lighting. It is observable in lighting environments where little or no light is available in the area above the diamond such as occurs in the case of close-up inspection by the 'normal' viewer.
Color Coding Light Return with an Illumination-Source-Scope, (ISS)
The following demonstration was inspired by a jeweler's deduction that if his head were truly causing the darkness, rather than light leakage being the cause, looking close up at a 'nail head' diamond with a red bag over his head should turn the diamond's table to red instead of it simply looking dark.
Employing the Russian DiamCalc software, the Hemphill et al. (1998) example of a 'nail head' diamond has been illuminated with blue hemisphere lighting above the girdle. Instead of having no light below the girdle plane, we have added a lower hemisphere of green illumination. The effect of the jeweler's head, covered with the red bag, has been simulated by a circle of red illumination over the diamond. The pattern of colors seen in the computer simulation of the face-up appearance of the 'nail head' illuminated in this manner shows from where each point on the diamond is reflecting its light to its normal observer. This device that color codes the sources of the diamond's illumination is called an Illumination-Source-Scope (ISS).
A green table would verify the occurrence of light leakage from the pavilion, because the green illumination would follow the reverse path to the 'normal' observer through the area which was leaking and turn it green. A red table would verify that the viewer's head interference is the cause of the 'nail head' diamond appearance. Areas of blue would have neither of these problems. Color coding the illumination in this way reveals the cause of the 'nail head' table darkness that is apparent in the virtual image of Figure 8c.
Figure 10: GIA's 'nail head' example in the Illumination-Source-Scope compared to its virtual image in 8c.
As the jeweller with the red bag on his head learned, the table shows red rather than green (see Figure 10) providing further support for the cause of table darkening in a 'nail head' diamond. Outside the diamond's table there are green spots indicating light leakage in those regions of the diamond. The blue spots within the table show small regions that do not have either the problem of light leakage or viewer interference. In a 'nail head' cut with very good symmetry one would predict that these small spots within the table should be bright. If we refer again to the actual photograph of the 'nail head' diamond in the upper left of Figure 9b, those bright points, which this demonstration predicts, are apparent. The predictive ability of this Illumination-Source-Scope adds verification of its utility.
While examining many diamonds in various lighting environments, it has been noticed that diamonds with shallow crown angles below 33° are darker and less brilliant than an 'Ideal' cut when viewed close-up. This observation seems to conflict with GIA Gem Trade Lab Reports concerning crown angles.
In their diamond grading reports, the GIA Gem Trade Lab adds the comment, "crown angles less than 30 degrees" when appropriate. The Laboratory allows a 4.5° variation below Tolkowsky's 34.5° before this critical comment is included. The comment, "crown angles greater than 35 degrees" is added if they exceed 36°.[20] This only allows 1.5° of variation above the 'Ideal' and implies that as little as 0.5° upward variation above 34.5° is detrimental. To avoid these critical comments, diamond cutters and dealers must maintain the diamond's crown angle between 30° and 36°. This sends the message that relative to Tolkowsky's 34.5° 'Ideal' crown angle, shallower crown angles are more acceptable than steeper ones.
WLR Indicates the Shallower the Crown Angle the More Brilliant the Diamond
Hemphill et al. (1998) reinforce the idea that shallow crown angles are better in relation to a diamond's brilliance. Their study reported: "In general, WLR increases as crown angle decreases. ... These results suggest that, at the reference proportions, a diamond with a 23° crown angle is brighter than a stone with any other crown angle greater than 10°. ... Ironically, the highest WLR values are obtained for a round brilliant with no crown at all" (p.170).
Close up Observation and Diamond Photography Reveal the Opposite
These results run contrary to the present analysis of close-up viewing of diamonds with shallow crown angles between 28° and 32° compared to those between 33° and 36°. Diamonds with shallow crown angles appear darker and less brilliant when viewed close-up. Under the circumstances of close-up viewing, diamonds with as little as a 2.5° lower crown angle look less brilliant than those with crown angles of 34.5°. The same variation in crown angle in the opposite direction does not appear to produce this loss in brilliance compared to the 'Ideal'.
Watermeyer (1982) made the following observation when viewing a diamond's mains: "When the crown is cut on 30° the area outside the table reflection becomes a darker gray. At 29° the diamond appears blackish in color with only the table reflection remaining white."
In this paraphrase of his words, Watermeyer was referring to a diamond "when in full eight sides" (before the 40 star and girdle facets are cut). We will see that his observations still apply to a completed diamond in the pavilion main facet areas of the diamond.
To demonstrate this, two diamonds in the 'normal' viewing position under three different lighting conditions were photographed. The diamonds are a close match in most respects, except for crown angle. This presents a valuable opportunity to document how variations in the crown angle alone affect the brilliance in differing lighting environments. The left diamond in these photographs meets AGS 0 tolerances for 'Ideal', while the right diamond has an 'Ideal' pavilion angle and table size, but has a lower 32.6° crown angle.
The first lighting environment approximates hemisphere lighting, having diffuse illumination in a 180° hemispherical arc above the diamond's girdle. In this lighting both diamonds appear to the eye and the camera lens to be similar in brilliance (see Figure 11a).
Figure 11a: 'Ideal' vs. slightly shallow crown angle diamonds in hemisphere lighting created by diffusing two fiber optic light sources.
The second lighting environment (Figure 11b) has more the flavor of jewelry store lighting. In addition to hemisphere lighting, it contains two 'hot spots'. Like jewelry store spotlights, these improve dispersion or fire in the diamond. In this lighting, both diamonds appear to the eye to be equally brilliant, although the photography makes the 'Ideal' cut (left) look slightly better. Both diamonds exhibit similar dispersion.
Figure 11b: 'Ideal' vs. slightly shallow crown angle diamonds in hemisphere lighting with two 'hot spots'.
The third lighting environment approximates hemisphere lighting but with little light in an area directly above to simulate a close-up viewing situation (Figure 11c). There is now a dramatic decrease in brilliance in the diamond with a shallower crown angle compared to the 'Ideal'. The right diamond's eight main facets have gone dark everywhere except for the central bright circular area, which is the table reflection. This is the area where light, which has entered the table, reflects to the 'normal' viewer. The pattern of darkness in the main facet areas is just as observed by Watermeyer (1982) when he described the appearance of diamonds with crown mains cut on 30° and below.
Figure 11c: 'Ideal' vs. slightly shallow crown angle diamonds in hemisphere lighting partially blocked as in close-up inspection.
This is photographic documentation that when viewed close-up in real lighting environments, shallow crown angles produce less brilliance than Tolkowsky's 34.5°, just as diamond cutters such as Watermeyer and others have observed. Photography or computer modeling can only capture this fact using an illumination environment where little or no light is available directly above the diamond to simulate the circumstances of close-up inspection.
In this case of shallow crown angles, expert observation and photographic comparisons reveal the opposite of the WLR brilliance measure. The Hemphill et al. (1998) finding that brightness increases as the crown angle decreases is a consequence of their hemisphere lighting model not accounting for the effect of the viewer's physical presence on a diamond's brilliance. When the viewer's presence is included in the lighting model there is a brilliance decrease as the crown angle decreases. This is a wake up call for the necessity to use a lighting model that is representative of normal viewing circumstances including the impact of the viewer's head and torso.
The red head, Illumination-Source-Scope, (ISS), can be employed to further document the cause of the darkening of the eight mains in the diamond with shallower than 'Ideal' crown angles. As observed in Figure 12, the red areas show that the close-up viewer's head interference is the cause of the darkening in the main facet areas outside the table reflection.
Figure 12: Illumination-Source-Scope image of simulation of right diamond in Figure 11c with 32.6° crown angle.
The blue table reflection correctly predicts that the diamond will remain bright in that inner circle as it does in the diamond of Figure 11c (right) with a halo of darkness surrounding it.
As the viewer looks at an 'Ideal' cut diamond, from closer distances, there is a point where it too will darken in the main areas. As the crown angle is decreased, the point at which the viewer's interference causes the mains to darken occurs at farther viewing distances.The shallower the crown angle the further the observer must be from the diamond to avoid the darkening in the main areas.
The relative absence of illumination in the area over the diamond where the close-up viewer's head is located has been demonstrated to be the cause of darkening in the diamond's mains. It is important to note, if there is relative absence of illumination in this area for any reason the darkening will occur no matter what the observer's viewing distance.
(A shallow crown angle can be compensated to minimize viewer interference to some extent by cutting a slightly steeper than Ideal pavilion angle. This cutting technique employs the widely discussed inverse relationship between crown and pavilion angle. Compensating deviations from Ideal crown and pavilion angles using this inverse relationship can retain high diamond brilliance over a small range of proportion variation from Ideal. More than a degree of variation in pavilion angle from 41 degrees has negative consequences that cannot be alleviated by inverse adjustments to the crown angle.)
Comparison of the Illumination-Source-Scope and the Firescope®
Darkening of the mains due to the blocking of light from above the diamond is also illustrated with devices such as the Firescope®. This diamond viewing instrument was developed in Japan in 1977 to demonstrate diamond brilliance. This device reveals the well-known, eight-rayed arrows pattern which characterizes the Eightstar® and subsequent 'hearts and arrows' type diamond cuts. Figure 13 (courtesy of Eightstar® Diamond Co.) shows a photograph of the Firescope® view of the arrow pattern in an 'Ideal' cut diamond.
Figure 13: Firescope® image of "Ideal' cut diamond with 34.5° crown angle (courtesy of Eightstar® Diamond Co.)
In place of the red head of the model is the darkness due to the Firescope'sŪ viewing lens. Instead of green illumination from below, the Firescope® has white light, and red illumination exists in place of the blue computer illumination. Besides colour, the main causes of the differences in the pattern of these images are the diamonds' proportion parameters and the proximity of the viewer interference.
This comparison illustrates a similarity in properties between the Illumination-Source-Scope and the Firescope®. The Firescope® may be used to analyze light reflection in diamonds in a fashion similar to the Illumination-Source-Scope. Extensions of this hardware device for more detailed analysis of diamond light return was the idea of gemologist A. Gilbertson in 1994, who is using multi-colored lighting devices to study the efficiency of light return in diamonds and other gemstones.[21]
Shallow Crown Angle Parker's Cut Compared to the American Ideal
Computer modeling programs, such as those of Hemphill et al. (1998), Strickland and Sivovolenko et al., are useful tools for exploring the effects on diamond brilliance of changes in diamond proportions such as shallow crown angles. For a further illustration supporting this, the Parker's cut is examined. This was the 25.5°, very shallow crown angle diamond that Hemphill et al. (1998, p.178) calculated to have "the highest WLR (0.297)". The diamond images of Figures 16 and 17 were generated with the Russian DiamCalc software using a representation of jewelry store lighting, which included the effect of a close-up viewer.
Figure 16: Parker's cut with a 55.9% table, 25.5° crown angle, and 40.9° pavilion angle.
Figure 17: 'Ideal' cut resulting from increasing the Parker's cut crown by 9°.
The shallow crown angle Parker's cut appears in Figure 16. By increasing the crown angle by 9°, the Parker's cut becomes the AGS 0 American Ideal cut pictured in Figure 17. Visual comparison of these leave little doubt as to which is more brilliant in this lighting environment. The viewer obstruction has greatly reduced the brilliance of the Parker's cut, while the same viewer obstruction leaves the Ideal cut unaffected. This demonstrates that the Ideal cut is a much more brilliant and dispersive diamond in usual viewing conditions.
These images further support the idea that aspects of brilliance including the amount of light return and 'contrast brilliance' can be effectively studied with computer modeling using face-up and angled 'snapshots' generated using illumination representative of usual viewing conditions.
Can the optimum range of pavilion, crown and table proportions be worked out with a simpler model than the complete 3D model? Can a 2D model such as Tolkowsky's establish the range of ideal combinations of crown, pavilion and table proportions, or was his analysis inadequate because it was only two-dimensional?
To begin answering these questions, compare the diamond model images published by Tolkowsky in 1919 with those appearing in recent times (see Figures 14a, b and 15a, b).
Figure 14a: Tolkowsky's drawings from Diamond Design (1919).
Figure 14b: Tolkowsky's drawings of the round brilliant-cut top and bottom views from Diamond Design (1919).
Figure 15a: Profile diagram of a modern round brilliant.
Figure 15b: Top and bottom diagrams of modern brilliant.
There are three principal differences reflecting the changes in 'Ideal' diamond cutting from Tolkowsky's time until today. First, Tolkowsky did not consider girdle thickness but assumed a knife-edge girdle. Secondly, his table is 53%, which is smaller than is normally cut today. Thirdly, the pavilion (or lower) girdle facets extend only 50% of the way to the culet of Tolkowsky's drawing from 1919, while they extend at least 75% down the pavilions of diamonds cut today.
It is important to understand that the three parameters (pavilion main facet angle, crown main facet angle and table percentage) were all that Tolkowsky was attempting to optimize for brilliance. Other proportion parameters such as total depth percentage are a result of the choice of these three and the choice of a reasonable girdle thickness. Tolkowsky knew, as do most gemstone cutters, that the interaction of the pavilion angle, crown angle and to a lesser extent the table percentage has the greatest influence on brilliance. The other proportions, such as the lower-girdle facet angles (which are cut between 1° and 2° steeper than the pavilion main angle), are chosen relative to these three parameters.
Thus, obtaining optimum brilliance and beauty in a round brilliant-cut diamond simplified to finding the best combination(s) of these three most important cutting parameters. This is what Tolkowsky endeavored to accomplish. Because his model was two-dimensional, he only considered a cross-section or plane through the diamond perpendicular to the pavilion mains and crown mains, so only rays of light traveling in that plane were studied. This covers a more significant portion of the diamond than one might imagine. Because the round brilliant-cut has four-fold, mirror image symmetry, that cross-section repeats in eight positions around the diamond. In Figure 11c, for example, the eight dark main facet areas in the right diamond are where Tolkowsky's analysis would apply, since his two-dimensional model plane was a slice through the mains and table. Notice from Figure 14b that these eight main facet areas covered a greater portion of Tolkowsky's diamond due to the shorter smaller lower girdle facets compared to those of today shown in Figure 15b.
Figure 12: Illumination-Source-Scope image of simulation of right diamond in Figure 11c with 32.6° crown angle.
Tolkowsky is credited with revolutionizing diamond cutting with publication of his book Diamond Design in 1919, and his crown and pavilion angles are still considered 'Ideal' today. Both are indications of the validity of his conclusions relating to those two angles. Tolkowsky (1919) says: "The gradual shrinking-in of the corners of an old-cut brilliant necessitated a less thickly-cut stone with a consequent increasing fire and life, until a point of maximum brilliancy was reached. This is the present-day brilliant", and he goes on to say, in a footnote, "Some American writers claim that this change from the thick cut to that of maximum brilliancy was made by an American cutter, Henry D. Morse." Then he says: "In the next chapters the best proportions for a brilliant will be calculated without reference to the shape of a rough diamond and it will be seen how startlingly near the calculated values the modern well-cut brilliant is polished."
While many credit Tolkowsky with the development of the 'Ideal' cut diamond, we see from his own words that diamond cutters such as Henry D. Morse had been cutting maximum brilliance diamonds (as defined by Tolkowsky) for years before he wrote his book. We also see that Tolkowsky placed great importance on ensuring that his results agreed with what the best cutters of diamonds had been practicing for years before his book. When his mathematical analysis verified these proportions he declared the following in his Mathematics Chapter:
"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 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 [improvement] and rule-of-thumb correction, come to the calculated values."
Tolkowsky's words indicate that he was acutely aware and in awe of the diamond cutters' skill in developing the proportions that had been in use for many years to produce optimum brilliance and dispersion. Because his mathematics confirmed these proportions he concluded: "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 theoretical values as is possible in practice, and gives a magnificent brilliancy to the diamond."
The last words in his book are: "It seems likely that the brilliant will be supreme for, at any rate, a long time yet." Although there have been changes such as increases in table size and girdle thickness and lengthening of the lower girdle facets, his basic findings concerning the best pavilion and crown angles have held true for eighty years.
The GIA is making a concerted effort with its computer modeling to explore the extent to which the proportion parameters may be varied and still retain or exceed the beauty of the current 'Ideal'. Hemphill et al. (1998) state that their study results "do not support the idea that all deviations from a narrow range of crown angles and table sizes should be given a lower grade".
The study, computer modelling and diamond photography presented above demonstrate that, with an illumination that accounts for interference from the 'normal' viewer, the possible range of deviations from 'Ideal' proportions can be narrowed. In summary:
(i) A measure of brilliance must agree with human judgment. Observations of the effects of diamond proportions on brilliance by diamond cutters from Tolkowsky to Watermeyer, and observations by people in the diamond trade and consumers provide the litmus test for conclusions drawn from computer modeling of brilliance in diamonds.
(ii) An illumination that takes into account the observer's physical presence is necessary to reveal the loss in brilliance in the 'nail head' and diamonds with shallow crown angles.
(iii) Diamond cutters and the GIA Diamond Course are correct in their adherence to close to a 41° pavilion angle as the single most important proportion. The present work shows that the 43° to 45° pavilion angles lower brilliance under close-up inspection by a greater amount than hemisphere lighting revealed. (There are further reasons why the pavilion angle should have a smaller tolerance than 2° around 41° that may be demonstrated in a continuation.)
(iv) With the pavilion angle held close to 41°, crown angles below Tolkowsky's 34.5° yield decreasing brilliance under close-up observation in spite of the fact that they show increasing WLR in hemisphere illumination. With a pavilion angle near 41°, shallow crown angles are not a direction to go in search of greater brilliance.
(v) When a diamond is graded for cut, crown and pavilion angles are the important proportions that should be measured along with table size, rather than total depth or even crown height and pavilion depth percentages. As gemstone cutters know, these angles most directly affect the gemstone brilliance. Many diamonds that possess an 'Ideal' total depth have thin crown heights and shallow crown angles with a deep pavilion depth to compensate. This yields an 'Ideal' depth even though the brilliance and beauty of the diamond is negatively affected owing to the faults discussed in section 5 and section 6. Also, a greater than 'Ideal' pavilion angle, producing a deep pavilion, can be made to measure within AGS tolerance of the 'Ideal' pavilion depth by cutting a medium culet facet.
(vi) The illumination of the diamond has as much influence on the measure of brilliance as the diamond's proportions do. In Figures 1 and 2b, the differing illumination environments have caused the diamond in 2b to have greater light return, but the diamond in Figure 1 would normally be judged to be more brilliant because of the 'contrast brilliance' or 'snappy' contrast between its facet reflections. Although both of these diamonds are 'Ideal' cut, the illumination has made them appear dissimilar. Illuminating a diamond from enough different angles can cause even the most poorly proportioned diamonds to have high light return. By employing typical rather than averaged illumination environments, and by including consideration of the physical presence of the viewer, a measure of brilliance can better separate diamonds of 'Ideal' proportions from those of poor proportion.
(vii) Computer modeling programs, such as those of Hemphill et al. (1998), Strickland and Sivovolenko et al., are effective tools for exploring diamond brilliance. For a final illustration supporting this the Parker's cut is examined. This was the diamond that Hemphill et al. (1998, p.178) calculated to have "the highest WLR (0.297)". The diamond images of Figures 16 and 17 were generated with the Russian DiamCalc software using a representation of jewelry store lighting, which included the effect of a close-up viewer.
Figure 16: Parker's cut with a 55.9% table, 25.5° crown angle, and 40.9° pavilion angle. Figure 17: 'Ideal' cut resulting from increasing the Parker's cut crown angle by 9°.
Parker's cut appears in Figure 16. By increasing the crown angle by 9°, the Parker's cut becomes the AGS 0 'Ideal' pictured in Figure 17. Visual comparison of these leave little doubt as to which is more brilliant in this lighting environment. The viewer obstruction has greatly reduced the brilliance of the Parker's cut while the Ideal cut is unaffected and is a much more brilliant and dispersive diamond.
These images further support the idea that aspects of brilliance including the amount of light return and 'contrast brilliance' can be effectively studied with computer modeling using a face-up 'snapshot' generated using illumination representative of normal viewing conditions.
(viii) Finally, in answer to the opening question: Is the GIA brilliance study cause for abandoning the Tolkowsky or American 'Ideal'? This work and its example demonstrations and conclusions suggest otherwise. The American Gem Society agrees.
The AGS has stated that advanced optical technologies mandate the refinement of parameters defining the American Ideal Cut. However, nothing they have seen justifies elimination of this lofty standard.
In their widely published, cut research update AGS goes on to say: "Cutters, optical physicists, gemologists and mathematicians have labored to better understand proportion, polish and symmetry parameters in their search for the most beautiful diamond. The goal that drove the evolution of the American Ideal was to find proportions that would maximize the diamond's beauty.
The light that the face-up or normal viewer sees in a diamond has reflected and or refracted from both crown and pavilion facets, so the interrelationship of all the facets together determine the paths of light through the diamond to the observer. This is a valid reason for criticism of cut grading systems that grade a diamond cut by specifying tolerances of each proportion parameter independently of the others. It is necessary to observe the result of the interaction of all diamond facets, in order to thoroughly evaluate optical performance, and assist the industry and the consumer with their understanding of the relationship of light and beauty.
Continuing research at the American Gem Society is keeping the AGS Ideal 0 cut grading system current with advancing knowledge of diamond optics and performance as it relates to diamond beauty. Just as in the current AGS American Ideal 0 cut-grading system, there is a range of proportions that will produce an Ideal result. The diamond's optical performance, as seen by the trained eye, aided by performance assessment devices, will become the final quality control measure that will be used to refine the range of 'Ideal' proportions.
A noble purpose in any and every endeavor is mankind's quest for the ideal. Our goal is to refine the AGS Ideal 0 using the most advanced diamond cutting knowledge and science of the day. This is an essential mission of the American Gem Society and the American Gem Society Laboratories."
Thank you to Bruce Harding, William Day, Robert Strickland, Martin Haske, Ilene Reinitz, Mary Johnson and Donald Dietz for many hours of intense discussion, peer review, ideas and suggestions gratefully received which contributed greatly to this work. Thank you to Sergey Sivovolenko and Anton Vasiliev for personal communication of their work and all at Octonus for custom modifications to their software product enabling the fine computer imaging.
References
Boyajian, W., 1998. Editorial. Gems & Gemology, 34(3), 157
Bruton, E., 1978. Diamonds. Chilton Book Co., PA, 532pp
Eulitz, W., 1975/76. The variable effects of faceted gemstones. Gems & Gemology, 15(4), 98
Gemological Institute of America, 1979 and 1994. Diamond Grading Courses
Harding B., 1975. Faceting limits. Gems & Gemology, 15(3), 78
Hemphill, R., Reinitz, I., Johnson, M., and Shigley, J., 1998. Modeling the appearance of the round brilliant cut diamond: an analysis of brilliance. Gems & Gemology, 34(3), 158-83
Gaal, R., 1977. The Diamond Dictionary. Gemological Institute of America, California, 342 pp
Sivovolenko, S., OctoNus Software, and Shelementiev, Y., Gemology Center of MSU, and Vasiliev, A., 'LAL' Optics, 'MSU Diamond Cut Study', personal communication, March, 2000. DiamCalc Software and Internet Site http://www.gemology.ru/octonus
Strickland, R., 1992. GemCad User's Manual. (c) 1992 Robert W. Strickland, 69 pp
Strickland, R., 1993. Gemray, GemFrame and GemFlick Users' Guide. (c) 1992, 1993 Robert W. Strickland, 12pp
Tolkowsky, M., 1919. Diamond Design. Spon & Chamberlain, New York, 104 pp
Vargas, G. and M., 1977. Faceting For Amateurs. 2nd edn. Published by the authors, California, 345 pp
Watermeyer, B., 1982. Diamond Cutting. 2nd edn. Centaur, Johannesburg, 406 pp
Footnotes:
