Tolkowsky Revisited

Is there an ideal-cut diamond? GIA says no; EightStar Diamond says yes. The answer may depend on how you define "ideal."

By Gary Roskin, G.G., FGA, Senior Editor with additional contributions by Rob Bates, Senior Editor -- JCK-Jewelers Circular Keystone, 6/1/2002

Poor Marcel Tolkowsky. He's the Belgian diamond cutter (1899-1991) who, in 1919, wrote a treatise entitled "Diamond Design," which introduced what we now call the "ideal cut." The concept became popular in the United States and later in the Far East, and even Gemological Institute of America courses called diamonds with Tolkowsky's proportions (with a few modifications) the cream of the crop. With today's emphasis on cut, "ideals" are more popular than ever. Yet Tolkowsky's work is facing an unprecedented challenge. After embarking on an ambitious multi-year computer-simulation research project, GIA believes the term "ideal cut" should be abolished—mostly because it's irrevocably linked to Tolkowsky's measurements. "Furthermore, there is now even more evidence to support the conclusion that there is no one 'best' cut for a round brilliant diamond," says GIA president Bill Boyajian.

GIA now believes that other diamonds with very different measurements and proportions can equal the brilliance, dispersion, and scintillation of a Tolkowsky ideal cut. GIA researchers also believe that someday it will be possible to predict how a cutter might, for example, take a diamond with a 60% table and 60% total depth—major deviations from Tolkowsky proportions—and without too much repolishing or changing proportions, manipulate the facets to yield a diamond that's just as beautiful.

Tom Moses, vice president of the GIA Gem Trade Laboratory, anticipates the day when he can advise someone how to significantly improve a deficient stone by repolishing, with only a 1% or 2% weight loss. "We believe we will be able to provide information to the manufacturer who will then be able to decide whether or not to fix the stone," Moses says.

But Richard von Sternberg of EightStar Diamond Co., Santa Rosa, Calif., doesn't see how that's possible. "It would be especially important for cutters if this were so, to take dark stones and turn them into French poodles," he says. "But it doesn't seem to be the way we do things. I haven't found a way to do this yet."

There are a number of cutters who agree with GIA that there is no one ideal cut, but who think the concept shouldn't be thrown out. They say there may be several ideal cuts—and the way to prove it is to look past proportion percentages and angles and measure what diamonds are supposed to be about—beauty. They believe the term "ideal" should be redefined to mean "maximum brilliance and fire."

Optical vs. physical. "We've changed the metric from measurements and proportions to performance, and this is an important place in the history of diamond cutting," says von Sternberg, who believes the way to analyze performance is to view a diamond through a FireScope. This Japanese-developed device shows light leakage through a diamond's pavilion. (Imitations of the FireScope are the small Hearts & Arrows readers.) Seeing white in a FireScope image indicates leakage. Von Sternberg notes that the absence of light leakage occurs when the diamond has "optical symmetry."

This differs from "physical symmetry," which is what the trade typically measures. Physical symmetry means the facets are well shaped and uniform. Optical symmetry means the facets are placed in relation to each other such that the light striking them is optimally bounced to another facet or out through the stone's crown. Von Sternberg believes that a diamond's crystallography dictates the angles of maximum performance, for both brilliance and dispersion. Thus, the way to determine the ideal cut for a diamond is not to measure angles, but to observe a diamond's optical symmetry.

Failing grades. This contradicts not only GIA's research but also the thinking behind most gem lab cut grades. Most measurements are determined by using electronic optical devices such as a Sarin or Megascope, made by OGI Systems, OGItech, New York. These machines also are programmed to note whether or not a diamond's physical measurements reflect Tolkowsky's angles and proportions.

The AGS laboratory, which follows this methodology, is the only major lab issuing a cut grade for round brilliants. It considers a diamond ideal if its measurements are equal to or close to Tolkowsky proportions. For stones with other proportions, the cut grade is an educated guess at loss-of-beauty based on how far a diamond's measurements deviate from Tolkowsky's.

Many find this inadequate, arguing that numbers and percentages can't tell you if a diamond looks good. (Even some Tolkowsky ideal cuts look better than others.) Michael Cowing, a gemologist and appraiser who works for von Sternberg, says that angles and percentages don't always correlate with each other—crown to pavilion, or even one side to the other side—to produce maximum brilliance and fire. Even if the vertical angle measurement is at Tolkowsky proportions, a facet may have to be twisted or rotated to line up optically with the other facets. The FireScope, according to von Sternberg, can determine if there's interaction among facets.

Interestingly, although EightStar doesn't cut specifically to Tolkowsky's calculations, the angles and proportions of its diamonds approach Tolkowsky's figures. "For the longest time, we were told by diamond dealers and retailers that we cut the best diamonds," says von Sternberg. "The truth is, we had no capacity to measure angles. All we were concerned about was reflective technology. And until we got a Sarin, eight years later, we didn't know just how close to Tolkowsky we were actually cutting."

What would Marcel say? There is evidence that Tolkowsky himself believed a diamond's look was more important than its measurements. "One thing Tolkowsky did to verify that his conclusions were correct was to ensure that his results agreed with what the best cutters had already known," says Cowing. "Tolkowsky was involved in the manufacture of some million pounds worth of diamonds, cut to obtain 'the liveliest fire and the greatest brilliancy.' The most brilliant larger stones were measured and Tolkowsky notes 'how remarkably close' these measurements came to his mathematically calculated values."

Moreover, Tolkowsky never said that one rigid set of numbers produced the best diamonds. His chart of the five best-cut diamonds, having maximum brilliance and fire, were averaged to his mathematically determined proportions for crown and pavilion angle. There was a two-degree variation in crown angle and a one-degree variation in pavilion angle among Tolkowsky's top picks—meaning there is a range, albeit a small one, of acceptable proportions.

Another cutter who adheres to performance-based, rather than proportion-based, standards for cutting diamonds is Baruch "Barry" Gutwein of SuperbCert, New York. Gutwein doesn't use EightStar's FireScope, but he does use GemEx's BrillianceScope to cut for brilliance and dispersion. "It helps me make a decision on where to place the facets in relation to each other," he says. "I also have the analog version of the B-Scope that I use, and I've built a prototype FireScope I call the ImageScope, all used to see where the light is concentrating in the stone, and how it comes out to your eye. There's an amazing relationship to the image of the ImageScope and [that of] the B-Scope."

Although the angles of Gutwein's stones are—like EightStar's—close to Tolkowsky, there seems to be a difference in performance, at least according to the BrillianceScope. "EightStar is cut to slightly different angles than a SuperbCert," notes Gutwein. "They look different on the B-Scope, but it's not a detriment. When you get down to the nitty-gritty, it's extremely subjective. One customer may love the look of an EightStar and another may love the look of a SuperbCert. One is not better than the other."

Von Sternberg has noted that the actual angles of an EightStar are not as exact as one might expect. That's why the FireScope is so important: It shows him how and where to cut. Gutwein concurs. "There's a window of manipulation," Gutwein says. "Based on my own cutting, I know that there are different ways to make a fine-looking stone without cutting exactly to Tolkowsky."

"Just as in any of the existing cut systems, there is a range of proportions that will be judged 'ideal,' " says Cowing. "The diamond's optical performance, as seen by your eye and judged in more detail by devices like the FireScope, should be the criteria for deciding what range of proportions is ideal. Richard [von Sternberg] has shown that he obtains ideal performance over a range of proportions, which should make GIA happy that there is not just one exact set of ideal proportions. On the other hand, Richard also has shown that there is an ideal performance that can be obtained by cutting with the FireScope. So there is still an 'ideal.' That should make AGS and G.G.s and former GIA students who were taught 'ideal' happy. What G.G.s learned about ideal cutting wasn't wrong in its conclusions, just inexact in the science of explaining it. The same can be said about Tolkowsky's work: It wasn't wrong in its conclusions of pavilion and crown angle, just inexact in the mathematics of explaining it."

At present, there is no outside authority to affirm that EightStar's or SuperbCert's formula leads to the ideal of ideal cuts. Until performance-based standards are studied by researchers, recognized by major gem labs, and practiced by more cutters, the debate over ideal cut will remain a debate, and beauty will remain subjective—a matter of personal taste and preferences.

For more information on ideal cuts, log onto www.eightstar.com, www.superbcert.com, www.gia.org, or www.gemex.com.

Defining 'Ideal'

ideal cut (also called "American cut") Those proportions and facet angles that were calculated mathematically by Marcel Tolkowsky to produce maximum brilliancy consistent with a high degree of fire in a round diamond brilliant are considered by many diamond men to constitute the ideal cut. These figures, computed as a percentage of the girdle diameter, are as follows: table, 53%; total depth, 59.3%, plus girdle thickness; crown depth, 16.2%; pavilion depth, 43.1%. The bezel angle is 34° 30' and the pavilion angle is 40° 45'.

—from the GIA Diamond Dictionary (first edition, 1960)

How the Labs Grade Cut

The leading gem labs grade the physical attributes of a diamond's polish and symmetry, using terms like "good," "very good," and "excellent" to describe how each facet is polished and how they fit together. In other words, the grader makes a judgment call on how good the cutter was.

The labs measure facet angles, height, width, and depth and calculate percentages, and they list some of these measurements and calculations on their diamond grading reports. The labs don't measure brilliance, fire, and beauty.

That may be changing. The American Gem Society and EightStar Diamond Co. of Santa Rosa, Calif., have entered into an agreement to help create a performance-based cut grade for fancy-shape diamonds. That will most likely alter the original round cut grade once performance measurement has been addressed.

The Architectural Argument

EightStar Diamond isn't the only company challenging GIA's pronouncements on ideal-cut diamonds, and its FireScope isn't the only machine being used to support the counter-arguments. GemEx's BrillianceScope examines and ranks diamonds against each other based on the amount of brightness and color flash apparent to the machine's internal camera. "GIA approaches cut from a theoretical standpoint," says Randy Wagner, president of GemEx and maker of the BrillianceScope. "We approached it from an empirical standpoint."

Wagner explains why he thinks there are different ideal cuts: "The architecture of a diamond performs differently at different proportions. Using 58 facets, for instance, is a particular architecture. When you lengthen the lower girdles, the geometry changes—that's an architectural change. When you change the depth, that's a change in architecture." Changing the number of facets also changes the architecture.

Even though GIA believes a 60/60 may perform well if cut to certain parameters, Wagner points out, "The proportions that result in the 'best' performance possible are different from [those that result in] 'acceptable' performance. Intuitively, you would expect that. Change the architecture, and you change the performance."

So is there a best cut for a given architecture? "Probably," says Wagner. "And there are many different architectures!"

AGS Panel Debates Cut

Cut was the subject of a lively "town meeting" at the American Gem Society Conclave in Vancouver, British Columbia, in April. Price sheet publisher Martin Rapaport dominated the panel, getting into verbal scraps with not only Hearts On Fire's Glenn Rothman but also the Gemological Institute of America's Ilene Reinitz.

At times, Rapaport seemed to be groping for a middle ground between the GIA and AGS approaches. He scoffed at the notion of a lab "setting policy" for the industry. "I don't want a lab telling me what is good and bad," he said. "People like Cadillacs, some people like Mercedes." But then he added, "We need [cut] standards for consumer protection, because people are getting ripped off. I hold the GIA responsible for all the consumers who bought off-makes off the Internet. I was disappointed that GIA hasn't instituted a cut grade, and I think they will eventually."

Reinitz noted that GIA is developing a classification system for cut. "The goal is to have something on the reports by early next year," she said. "It's not going to happen this year, because the trade wanted a more complete answer than what is available."

She also reiterated that GIA's cut study has found that no single set of proportions yields the most brilliance and fire. "We have no problem with a standard that you use as a yardstick," Reinitz said. "What we don't like is the concept that there is only one standard, and all else is inferior."

AGS lab director Peter Yantzer said he was proud that his lab had developed a standard for the industry. "We all have an emotional, instinctual desire to strive for an Ideal," he said. "Cutters, diamantaires, and consumers have validated the American Ideal through observation, consensus, and their pocketbooks."

Heads Up

In a study called "Diamond Brilliance: Theories, Measurement, and Judgment," published last year by the British Gemmological Association's Journal of Gemmology, Michael Cowing, MS, FGA, reviewed all the data on ray tracing and three-dimensional computerization to develop a formula for determining a diamond's light performance. Cowing concludes that, contrary to the Gemological Institute of America's ongoing cut study, Tolkowsky was pretty much correct.

Why does Cowing believe Tolkowsky and not GIA? Cowing noticed that GIA's lighting experiments and computer simulations did not take into account the light blockage from a viewer's head. So he devised a study to demonstrate that such light blockage dramatically reduces the variable options for creating an ideal-cut diamond. Cowing suggests that Tolkowsky arrived at his ideal numbers in part because cutters' heads blocked the overhead light as they examined diamonds.

Cowing credits Bruce Harding as the first to determine "where each point on the diamond gets its light and then researching and mathematically determining pavilion and crown angles which avoid light's being drawn from the observer's head or from below the gemstone's girdle." Harding's work was published in an article entitled "Faceting Limits," in the Fall 1975 issue of GIA's Gems & Gemology. "This work has had a significant impact on the design of optimum angles for cutting colored gemstones," says Cowing.

Unfortunately, Cowing notes, these concepts have been overlooked or misunderstood by diamond cutters and the jewelry industry.

Robert Strickland's computer-aided design (CAD) programs show three-dimensional gemstone modeling with illumination. Strickland, along with fellow CAD designers Robert Long and Norman Steele, also subscribes to the idea that a viewer's head blocks light from above.

Cowing's study uses the two diagrams at right as evidence that Tolkowsky's angles are correct. "Compare figures 1a and 1b, derived from Bruce Harding's ray-tracing computer program from 1986," says Cowing. "Figure 1a shows 'ideal' pavilion and crown angles causing a light ray to reflect to the viewer's eye from an angle of 45°, just as Tolkowsky determined. Changing the pavilion angle from 40.75° to 45° causes the light ray to reflect from 90° where the viewer's head is located, as is shown in Figure 1b."

When real diamonds are judged for brilliance by means of careful examination in real lighting environments, the viewer's head and body interfere with illumination from behind and in line with the gem. Diamond proportions that respond poorly to real lighting conditions with real viewers are perceived to have low brilliance.

Light leakage has always been blamed for making these deep stones dark (see Fig. 2a and Fig. 2b). In reality, however, the pavilion angle of a diamond would have to exceed 52.5° before this type of leakage took place. Thus, as seen in Fig. 2c and Fig. 1b, the cause of the dark "nailhead" appearance is not because light is lost through the pavilion, but because the viewer's head blocks the light from that direction.

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°.

Cowing suggests an experiment: "Stick a red paper bag over your head, poke a couple of holes for your eyes, and look at a diamond. You will see your head [the red bag] wherever light would have been coming from if your head wasn't there."

Tolkowsky was right. "Marcel Tolkowsky still rules," says Cowing, who lists four reasons why:

1. With the diamond's pavilion angle held close to Tolkowsky's ideal, crown angles below Tolkowsky's ideal yield decreasing brilliance under close-up observation in spite of the fact that they show increasing weighted light return (WLR) in GIA's hemisphere lighting conditions.

In side-by-side comparison in normal close-up viewing and lighting conditions (e.g., with someone's head in the way of the overhead light), this diamond is nowhere near as brilliant as Tolkowsky's.

2. Although there have been some changes in the design of the "ideal cut" (e.g., increases in table size and girdle thickness, lengthening of the lower girdle facets), it has yet to be shown that Tolkowsky's recommended crown and pavilion angles can be surpassed for brilliance and beauty.

3. To be valid, a computer-generated measure of brilliance, such as GIA's weighted light return, should agree with human judgment. Yet GIA's observerless, multi-angled measurements of brilliance give high measurements that do not match observations of the effects of diamond proportions on brilliance made by diamond cutters from Tolkowsky to Basil Watermeyer (one of the world's current leading experts in diamond cutting)—or even those of consumers.

4. Once the effect of the observer is factored into the illumination portion of GIA's computer model, the range of acceptable deviations from ideal proportions for producing optimal brilliance narrows considerably. By employing typical overhead rather than omnidirectional illumination environments, and by allowing for the physical presence of an observer, a measure of brilliance can better mirror real-world differences between diamonds with "ideal" vs. those with poor proportions.

Cowing realizes the subject is becoming very technical, but he persists in telling his story: "I am still in the process of finding ways of communicating to interested gemologists the fundamental reasons why there really is an ideal cut."