MODELING the APPEARANCE of the ROUND BRILLIANT CUT DIAMOND: an ANALYSIS of BRILLIANCE by T
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MODELING THE APPEARANCE OF THE ROUND BRILLIANT CUT DIAMOND: AN ANALYSIS OF BRILLIANCE By T. Scott Hemphill, Ilene M. Reinitz, Mary L. Johnson, and James E. Shigley Of the “four C’s,”cut has historically been the he quality and value of faceted gem diamonds are most complex to understand and assess. This often described in terms of the “four C’s”: carat article presents a three-dimensional mathemat- weight, color, clarity, and cut. Weight is the most ical model to study the interaction of light with Tobjective, because it is measured directly on a balance. a fully faceted, colorless, symmetrical round- Color and clarity are factors for which grading standards brilliant-cut diamond. With this model, one have been established by GIA, among others. Cut, however, can analyze how various appearance factors (brilliance, fire, and scintillation) depend on is much less tractable. Clamor for the standardization of proportions. The model generates images and a cut, and calls for a simple cut grading system, have been numerical measurement of the optical efficien- heard sporadically over the last 25 years, gaining strength cy of the round brilliant—called weighted light recently (Shor, 1993, 1997; Nestlebaum, 1996, 1997). Unlike return (WLR)—which approximates overall color and clarity, for which diamond trading, consistent brilliance. This article examines how WLR val- teaching, and laboratory practice have created a general con- ues change with variations in cut proportions, sensus, there are a number of different systems for grading in particular crown angle, pavilion angle, and cut in round brilliants. As discussed in greater detail later in table size. The results of this study suggest that this article, these systems are based on relatively simple there are many combinations of proportions assumptions about the relationship between the proportions with equal or higher WLR than “Ideal” cuts. In and appearance of the round brilliant diamond. Inherent in addition, they do not support analyzing cut by one examining each proportion parameter indepen- these systems is the premise that there is set (or a nar- dently. However, because brilliance is just one row range) of preferred proportions for round brilliants, and aspect of the appearance of a faceted diamond, that any deviation from this set of proportions diminishes ongoing research will investigate the added the attractiveness of a diamond. In this article, we present effects of fire and scintillation. and discuss our findings with regard to the rather complex relationship between cut proportions and brilliance. Diamond manufacturing has undergone considerable ABOUT THE AUTHORS change during this century. For the most part, diamonds are cut within very close proportion tolerances, both to save Mr. Hemphill is research associate, and Dr. Shigley is director, of GIA Research, Carlsbad, California. weight while maximizing appearance and to account for Dr. Reinitz is manager of Research and Develop- local market preferences (Caspi, 1997). As shown in figure 1 ment at the GIA Gem Trade Laboratory (GIA GTL), and table 1, however, differences in proportions can produce New York. Dr. Johnson is manager of Research and Development at GIA GTL, Carlsbad. noticeable differences in appearance in round-brilliant-cut Please see acknowledgments at the end of the diamonds. Within this single cutting style, there is substan- article. tial debate—and some strongly held views—about which Gems & Gemology, Vol. 34, No. 3, pp. 158 –183 proportions yield the best face-up appearance (Federman, © 1998 Gemological Institute of America 1997). Yet face-up appearance depends as well on many intrinsic physical and optical properties of diamond as a 158 Modeling Brilliance GEMS & GEMOLOGY Fall 1998 Figure 1. These round-bril- liant-cut diamonds illus- trate how cut affects face- up appearance. Of the three larger stones (1.07–1.50 ct), the one on the lower right (F color) is obviously less bright than the other two (above, H color, and lower left, E color). All three 0.35–0.38 ct diamonds in the inset are brighter, on average; but the F-color diamond on the lower right, which could be marketed as an “Ideal” cut, is less bright than the F- and G-color stones with larger tables and small crown angles. See table 1 for the propor- tions and other data on these diamonds. Photo © GIA and Tino Hammid. material, and on the way these properties govern used to solve sets of relatively simple equations that the paths of light through the faceted gemstone. described what was considered to be the brilliance (Also important are properties particular to each of a polished round brilliant diamond. (Tolkowsky stone, such as polish quality, symmetry, and the did include a simple analysis of fire, but it was not presence of inclusions.) central to his model and it will not be discussed at Diamond appearance is described chiefly in any length in this article.) For the most part, the terms of brilliance (white light returned through the existing cut grading systems are based on crown), fire (the visible extent of light dispersion Tolkowsky’s research. into spectral colors), and scintillation (flashes of We believe that diamond cut, as a matter of such light reflected from the crown). Yet each of these importance to the trade, deserves a more thorough terms represents a complex appearance concept that and thoughtful investigation. The issues raised can has not been defined rigorously, and that cannot be only be resolved by considering the complex combi- expressed mathematically without making some nation of physical factors that influence the appear- assumptions and qualifications (see below). ance of a faceted diamond (i.e., the interaction of Despite the widespread perception in the trade light with diamond as a material, the shape of a that diamond appearance has been extensively given polished diamond, the quality of its surface addressed, there is limited information in the litera- polish, the type of light source, and the illumination ture, and some aspects have never been examined. and viewing conditions), and incorporating these Several analyses of the round brilliant cut have been into an analysis of that appearance. published, starting with Wade (1916). Best known The initial goal of this research project was to are Tolkowsky’s (1919) calculations of the propor- develop a theoretical model for the interaction of tions that he believed would optimize the appear- light with a faceted diamond that could serve as the ance of the round-brilliant-cut diamond. However, basis for exploring many aspects of the effect of cut Tolkowsky’s calculations, as well as most others on appearance. Computer graphics simulation tech- since then, involved two-dimensional images as niques were used to develop the model presented graphical and mathematical models. These were here, in conjunction with several years of research Modeling Brilliance GEMS & GEMOLOGY Fall 1998 159 below. Although other factors (e.g., bodycolor or TABLE 1. Proportions and calculated WLR values for the diamonds photographed in figure 1. inclusions) may also influence how bright a particu- lar round brilliant appears, light return is an essen- Position Color Weight Table Crown Pavilion Calcu- tial feature of diamond brilliance. (ct) size angle angle lated (%) (°) (°) WLR a In future reports on this project, we plan to address how fire and scintillation are affected by Main photo proportions. We also intend to examine how sym- Top H 1.21 62 29.4 41.7 0.279 metry, lighting conditions, and other factors affect Lower F 1.50 63 39.8 41.7 0.257 all three of these appearance concepts. The overall right goal of this research is to provide a comprehensive Lower E 1.07 57 34.6 40.9 0.282 left understanding of how cut affects the appearance of a faceted diamond. Inset Top G 0.38 60 26.5 42.6 0.288 BACKGROUND Lower F 0.35 56 34.7 41.2 0.281 right Early History. Diamond faceting began in about the Lower F 0.35 59 27.0 41.4 0.290 1400s and progressed in stages toward the round left brilliant we know today (see Tillander, 1966, 1995). a WLR, our metric for overall brightness, is calculated from the given In his early mathematical model of the behavior of crown angle, pavilion angle, and table size, using our standard refer- light in fashioned diamonds, Tolkowsky (1919) used ence proportions (given in table 4) for the other five parameters. principles from geometric optics to explore how light rays behave in a prism that has a high refrac- tive index. He then applied these results to a two- on how to express mathematically the interaction dimensional model of a round brilliant with a knife- of light with diamond and also the various appear- edge girdle, using a single refractive index (that is, ance concepts (i.e., brilliance, fire, and scintillation). only one color of light), and plotted the paths of Our model serves as a general framework for exam- some illustrative light rays. ining cut issues; it includes mathematical represen- Tolkowsky assumed that a light ray is either tations of both the shape of a faceted diamond and totally internally reflected or totally refracted out of the physical properties governing the movement of the diamond, and he calculated the pavilion angle light within the diamond. We plan to analyze the needed to internally reflect a ray of light entering appearance aspects one at a time and then, ulti- the stone vertically through the table. He followed mately, assemble the results in order to examine that ray to the other side of the pavilion and found how proportions affect the balance of brilliance, fire, that a shallower angle is needed there to achieve a and scintillation. second internal reflection.