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### Girdle Thickness Estimation Made Simple

This article was originally published by the author in Joe Tenhagen's Diamond Value Index, The industry's only diamond price guide which categorizes by cut grade. For information contact Joseph Tenhagen at 305-374-2411.

For years gemologists have had to use the Gemological Institute of America (GIA) charts to determine the qualitative measure of girdle thickness on diamonds. These charts presented a piecewise, or step function, correlating girdle thickness as a percentage of the base diameter (for round brilliants), in order to determine the subjective qualifications of Thin, Medium, Slightly Thick, Thick, Very Thick, and Extremely Thick. These data are presented in Figure 1 for illustrative purposes, the girdle thickness percentages being plotted against the given base diameter, with the lower step curve representing the Thin-Medium boundary and the upper curve the Extremely Thick boundary.

Figure 1 also illustrates a quadratic curve fit to base diameter for each of these curves, eliminating the discontinuities inherent in the GIA chart data. These smooth curves represent a continuous least squares approximation to the GIA girdle percentage thickness chart data, bisecting the discontinuities in that data, and eliminating the inherent ambiguity at the discontinuities.

Figure 1

Figure 2 represents the smoothed girdle percentage curves of Figure 1 multiplied by the base diameter to calculate a girdle thickness in millimeters as a function of base diameter. Surprise, these data indicate a CONSTANT level in millimeters for each of the boundaries, independent of base diameter!

The magic boundary numbers appear to be:

Thin: Less than 0.15 millimeters
Medium: between 0.15 and 0.20 millimeters
Slightly Thick: between 0.20 and 0.23 millimeters
Thick: between 0.23 and 0.33 millimeters
Very Thick: between 0.33 and 0.40 millimeters
Extremely Thick: greater than 0.40 millimeters

Figure 2

### What Tolkowsky Didn't Say

Have you ever heard the rumor story, started at one end of a line and listened to at the other end; every one adds their own spin and the original story, oft mistaken for fact, winds up quite differently; and fact and fiction become inter-twinned. Take the theoretical American Brilliant Cut Diamond, for example.

Every one in the jewelry industry has seen the Standard American Cut Brilliant picture (shown below), attributed to Tolkowsky, with a 53% Table, 34.5 degree crown angle and 43.1% pavilion. Added to that is a girdle thickness of 0.7% to 1.7% giving to those accepting what they see and read as "ideal", the perception that this is perfect cut or "ideal cut" or a "standard".
Ignoring, for the moment, any inaccuracy in the calculations Marcel Tolkowsky performed without the aid of a computer, Marcel Tolkowsky DID NOT SPECIFY GIRDLE THICKNESS. The origin of the 0.7% to 1.7% "ideal" girdle thickness, cannot be attributed to Tolkowsky, at least in the Mathematics portion, Part III (Chapter III) , of his original work, "Diamond Design"!

Tolkowsky's Ideal cut, 34.5 degree crown, 53% Table, 43.1 % pavilion depth, added up to 59.3%, period. The industry interpretation, however, took a different bent. Lost in the woodwork, was a 1979 statement in the GIA Diamonds Course that the ideal cut diamond "would be 61 to 62%", recognizing the fact that the 0.7% to 1.7% would give a "rather thin girdle". Indeed, mathematically, for the approximately (standard) 66 % pavilion girdle facet length, the 59.3% Tolkowsky brilliant, as defined, would have sixteen (16) knife edges forming the girdle ! Tolkowsky's picture of the "ideal" had knife edges.

If we look at the mathematics, the depth of the scallops have to be a MINIMUM 1.7%, between the WIDEST part of the girdle scallops, and in that condition, they meet at a POINT. Hardly an "ideal" cut!

If we change the pavilion girdle break facet length to approximately 80%, which appears the norm for today, this 1.7% limitation is reduced to approximately 1.5%.

Communication may be lacking in the industry, and everyone has their own interpretation, but the FACTS are, that the GIA, at least in their Diamond Grading Course(s), to the limits of my research, has never specified Total Depth Percentage!.

GIA has defined a medium to slightly thick girdle to be a requirement for a Class 1 make. If one looks at the published GIA girdle thickness charts, curve fits through the inherent discontinuities, one finds that the implied recommended TOTAL DEPTH PERCENTAGE RANGE, for a 34.5 degree crown, 53% Table, and 43.1% pavilion depth to be at least 61.0% Total depth. IN FACT, for a theoretical one (1) carat Tolkowsky round brilliant, 6.53 mm in diameter, the acceptable range (interpolated) that GIA recommends, is 61.5 to 62.7 PERCENT TOTAL DEPTH. Industry publications suggest 60 to 61% TOTAL DEPTH for a CLASS 1 Cut, consistent with the third edition of the Diamond Dictionary which says that the Tolkowsky Ideal cut was 60-61% Total Depth.

Seems like a lot of Class 2 cuts become Class 1 cuts.

With all those deep pockets in the industry, most never did their homework (if any at all), or READ, and understood, the complicated (or at least background) issues. They probably tried, and knew they couldn't cut a "Tolkowsky Ideal", because, with a girdle that didn't have a KNIFE EDGE, it wouldn't add up to what was presented. So the "ideal", as defined by Tolkowsky, was apparently "modified" by the industry, and took form as spread tables, reducing the crown height, so that the 53-60 table/60-61 depth "ideal" relationship, so oft quoted, could be realized!

A simple mathematical proof is:

MINIMUM GIRDLE THICKNESS

Consider the girdle break facets, 16 in number, each subtending an arc (alpha) of 22.5 degrees (360/16). The chord of this arc (2a), and the length (b) from the center of the circular diamond outline are related by the trigonometric relationship

Cos(alpha/2)=b/r

Cos(22.5/2)=Cos(11.25)=0.98078=b/r

Therefore b ~ 0.98*r ~ 0.49*diameter ~ 0.49*D

And c = r - b ~ 0.50*D - 0.49*D ~ 0.01*D

The ungula cut by the girdle break facets form the familiar scallop on the girdle edge, the depth of which (h) is related to c, and the angle at which the girdle break facet (beta) cut the girdle by the trigonometric relationship

Tan(beta) = h / c

or h = c*Tan(beta)

The crown girdle break facets are at an angle GREATER than the crown mains (34.5 degrees) or approximately 39.45 degrees (for a 50% star facet length) and the pavilion break facets are at angle greater than the pavilion mains (40.76 degrees , zero culet, 43.1% pavilion depth) or approximately 41.4 degrees (for an 66% pavilion break facet length).

For the crown scallop depth (h1)

h1 = c*Tan (39.45) > c*Tan(34.5)

h1 = 0.01*D*0.82287 ~ 0.00822*D > 0.01*D*0.68728

For the pavilion scallop depth (h2)

h2 = c*Tan(41.4) > c*Tan(40.76)

h2 = 0.01*D*0.88162 ~ 0.0088*D > 0.01*D*0.86195

If we allow the crown scallop and the pavilion scallop to meet (at a knife edge) , the girdle thickness would be the sum of h1 and h2

h1+h2 ~ 0.0088*D + 0.0082*D ~ 0.017*D or 1.7%D

Any girdle depth less than 1.7%, for the Tolkowsky crown and pavilion, would result in not only knife edges at the girdle, but flattened girdle edges and not a true round, surely not an Ideal cut from the industries point of view.

The Ideal Tolkowsky cut therefore, has to have a MINIMUM girdle thickness (defined at the widest part of the scallops) of at LEAST 1.7%, and for a 16.2% crown and 43.1% pavilion, at LEAST 61% depth (16.2+1.7+43.1).

### 1993 Master Stone Study

The following report was published in the Summer 1993 edition of The Accredited Gemologist's Journal Cornerstone. Since then the author has done extensive investigations into spectrophotometry and it's application to the color grading of diamonds. Check this web site for more information.

The extinct (or nearly so) Shipley (AGS) Diamond Colorimeter, Part 2:

Tucson Test Results and Observations

by

Martin D. Haske, G.G., ISA, M.S.

This report and findings are the personal observations of the author, are not sanctioned or endorsed by the American Gem Society, the Gemological Institute of America or the Accredited Gemologists Association. This report is published solely for the exchange and stimulation of ideas.

The author wishes to express particular thanks to Mr. Richard Agnew, Ms. Daloma Armentrout, Mr. Serge Boro, Mr. Al Gilbertson, Ms. Anne Hawken, Mr. Phil Minsky, and the Accredited Gemologists Association for their contributions to this study.

BACKGROUND

The original article published in this series (Reference 1) introduced us to the Shipley Diamond Colorimeter, Serial #5. Shortly after publication, and as a direct result of some limited success in understanding the design theory of that particular machine, the author received four additional Shipley Diamond Colorimeters. Of the four colorimeters received, one --- Serial #30 --- was used for the Tucson study. The other machines are in varying stages of reconstruction.

One of the particular things the author did during examination of these machines was to remove and photocopy each of the "circular slide rules" affixed to the front of the machine. To his surprise, the author discovered that the slide rules were all different. Figure 1 illustrates the conversion of the "slide rule" coordinates on four of the Shipley colorimeters to an equivalent Yellow/Blue reading ratio, that ratio effectively being utilized by the "slide rule" to convert the colorimeter Yellow and Blue transmission data to AGS colorimeter values. This conversion was accomplished by laboriously measuring chord lengths between the SET point on the "slide rule" and the "colorimeter reading", converting the chord lengths to angular distances, and then relating the angular distances logarithmically (Base 10) to Yellow/Blue ratios.

Figure 1:"Slide rule" coordinates converted to Y/B ratios

The complexity of this procedure was reflected in the necessity to return each colorimeter to the GIA when a bulb burned out. Apparently, after replacing the photobulb, a GIA/AGS reference set of master stones used for calibration were tested, a new calibration curve determined, and a new slide rule (upper part only) constructed for the instrument. My complements for their ingenuity! Rather than relying on, most probably, unattainable consistency in the selenium photocell's sensitivity, coupled with the problem of repeatability in the illuminating bulb's operating temperature, and hence spectral output, the designers took the nuts and bolts approach and attacked the problem head on.

REFERENCE CALIBRATION

The author's diamond master stone set, GIA set #4693, was used prior to, during, and after Tucson testing to establish a reference for the study. The set consists of 5 stones: 1) F-G master (colorimeter equivalent 1.25), 2) H-I master (colorimeter equivalent 2.25), 3) J-K master (colorimeter equivalent 3.25), 4) L-M (colorimeter equivalent 4.25) and 5) N (colorimeter equivalent 5.00).

GIA certified masters not having assigned AGS colorimeter readings were assigned, for the purpose of this study, specific colorimeter readings in accordance with page 27 of Reference 2. GIA split grade masters, ie: an F-G master, are interpreted by the author as lying halfway between the start of the F color range and the beginning of the G color range, hence it would be a colorimeter equivalent 1.25.

These series of calibration readings were taken on at least six different days, both at the author's facility, and at Tucson, and have been presented in their entirety in Figure 2, shown against the original calibration of Shipley colorimeter serial #30. With the exception of one low reading on the author's F-G master stone, the vertical dispersion of the data, if interpreted against the horizontal axis, imply a min-max repeatability of perhaps 0.2 colorimeter units. The slope of the original calibration curve for colorimetric readings below 2.0 would expand this repeatability to perhaps 0.5 colorimeter units. This is the expected accuracy for subsequent tests. This repeatability is not bad, considering the following prime impediments to consistency:

A) Cleanliness of the master stones

C) Variability in the power source

D) Drift in the output

E) Cut and clarity of the masters

Figure 2: Master Set #4693 vs Colorimeter #30 Calibration
TUCSON TEST RESULTS

Tucson testing was initiated after some hasty field repairs to a broken wire; attesting to Murphy's first law, "if anything can go wrong, it will, and at the worst possible moment." Other than that, before presenting the Tucson data we must add to the above variability factors one more affecting the interpretation of the test results:

F) Variability in the master stones themselves

A) Cleanliness of the master stones: Somewhat subjective, but each stone at Tucson was wiped clean with a diamond cloth, in the same manner as the author used to clean his own master stones for the pre and post Tucson calibrations.

B) Variability in grader's technique: The author conducted all testing, eliminating this as a consideration for these test results.

C) Variability in the power source: Of prime concern, the one limiting condition which has to be addressed. Even though the Shipley colorimeter was driven by a Wang regulated power supply, having a 0.4KVA capacity while the illuminating bulb draws 0.075KVA --- well under the capacity of the power supply --- the inherent instability of the driving voltage to the Shipley can be, and most probably is, a contributing factor to the Tucson test results. The effect of ambient electrical power fluctuations could be seen both at the author's facility and at Tucson, even with a regulated, supposedly stable power supply. The author sees the effect of refrigeration compressor voltage draw down during on-off cycles in his facility, and saw the same --- if not greater --- effects in Tucson due to HVAC cycling. The author addressed this instability by rechecking the blue and yellow transmission readings during the measuring process and not accepting them unless it was felt that they were stable. The data presented in figure 2 reflect voltage instabilities both at the author's facility and at Tucson.

D) Output drift: Operation of the Shipley colorimeter requires a stabilization period (Reference 3 and Reference 4) after switching between blue and yellow filters, to allow the photocell to integrate to a stable reading. The tendencies of the system were to increase the yellow reading over time, shifting the results towards a higher Yellow/Blue ratio. All testing performed allowed this stabilization to occur.

E) Cut and Clarity: Cut and clarity of the master stones were not specifically addressed in this study, although each stone was loupe checked by the author for obvious problems with table centered inclusions: none were evident. It has been commented that cut grade can effect Shipley colorimeter test results; however, the author can find no specific reference in the limited literature available.

F) Variable Masters: The master stone variability question must also be addressed. All results presented in this report were based on master stones represented as graded by the GIA/GTL, either to GIA or to AGS nomenclature. The author is in possession of photocopies of the appropriate certifications supplied by the owners of the master stones. The author cannot firsthand certify that the master stones supplied by the participants to this study are the originals graded by GIA, as measurements and weights were not crosschecked on every stone (most participants had their masters but not their documentation available at Tucson). The sets tested are those supplied, and attested to, by the participants as masters corresponding to the documentation forwarded after the testing. It might be noted that, in the past, stones have been known to have been inadvertently switched, hence I repeat my suggestion from reference 1, "get your masters laser engraved".

Figure 3 presents all data accumulated in this study with the exception of the author's master stone set data presented separately in Figure 2. The vast majority of the data represents a single test on a stone. In some cases, stones were retested, and the original and retest data is included in this sample set.

Figure 3: Tucson Test Results, Total Data Distribution
Figure 4 presents multiples, ie: Tucson test results where two to three tests were conducted on the same stone, again excluding the author's master stone set. The maximum variation in these data occurred in the data pair at colorimeter grade 5.25, the spread being 0.057 colorimeter units, attributed to a dirty stone in the first reading.
Figure 4: Multiples Tests on Single Stones, Data Spreads
WHAT CAN BE CONCLUDED

From the data gathered, as well as from visual comparisons made in Tucson (in less than ideal lighting conditions), it would appear that --- qualitatively --- all master stones are not created equal. At least, not equal with respect to colorimetric consistency as measured by the Shipley Diamond colorimeter. The data presented here (see Figure 3) obviously shows too wide a vertical (Y/B ratio) dispersion, in comparison with expectations (see Figure 2), to be consistent with the hypothesis that all master stones are created equal. It cannot be known which generation of GIA masters was used to check each master stone in the study, although with GIA's help correlations could be established, as well as any possible correlation between graders.

The author was taught in GIA classes that master stones are absolute --- a master is a master --- not that master stones may just represent a more "limited range of color" to be used as a reference. Yet, the master sets tested obviously represent a range of color rather than color points; this was observed in the data set published last fall as well (Reference 1). Conversations with individuals having first hand knowledge of GIA/GTL grading procedures have provided general comment support these conclusions. It behooves the GIA to address this issue. The owners of the master stones used in this study, representing 15 sets, have been notified of their individual stones' placement with respect to other masters tested; those anomalous readings should be rechecked by GIA.

THE FUTURE:

The author is now in the process of reconditioning and testing three more AGS colorimeters. These colorimeters will be calibrated against master stone set #4693. If any reader is in the possession of an unusable AGS colorimeter, please contact the author, or Mr. Phil Minsky, and we will try to put it in operational order and accumulate some more data.

References

1) Haske, Martin D. , "The Nearly Extinct Shipley (American Gem Society) Diamond Colorimeter: Some Current Results and Observations," Accredited Gemologists Association Journal, Cornerstone (Winter 1993).

2) American Gem Society / Gemological Institute of America, "Diamond Grading Standards Manual", 1991

3) Shipley, Robert M. , "Electronic Colorimeter for Diamonds," Gemological Institute of America, Gems and Gemology (Spring 1958).

4) American Gem Society, "Operating and Maintenance Instructions, American Gem Society Electronic Colorimeter", (1956 ?).

### Master Diamond Colorimetry Study

We have seen in the last few years, the introduction of both bistimulous and tristimulous colorimeters for use by the jewelry industry, initially overly priced above the cost of a good 5 stone set of 30 point diamond masters. The author has been quoted in the trade press as having the opinon that a set of diamond masters were a better investment for the jeweler than the electronic devices offered at the time. In barely a year from the publication of an article in JCK on electronic colorimeters, the retail price of these "new" devices have been cut in half, in a dumping mode. Those who spent \$7500 dollars on such a device have seen their "investment" go down in value, as newer and better devices come on the market. Kind of disappointing for those who took a 3 year payment plan! Sort of like the used car market.

Since the 1993 Shipley Colorimeter study the author has been involved in the development of an highly accurate visable range spectrophotometer, usable by the jewelery industry for both grading diamonds and colored stones. The data presented below illustrate the information content attainable with such a device and some of the questions associated with the use of electronic colorimetry in diamond grading.

Early colorimetry suffered from both electronic stability problems as well as limited dynamic range. Bistimulous and tristimulous devices grouped data by integrating signals over two or three broad wavelength ranges, and then relied on a calibration of the electronic signals by either ratioing, in the bistimulous case, or converting to universal CIE coordinates and equating the "CIE color" to the diamond color ranges or grades, with mixed results. It appeared that one still needed a set of reference diamond master stones to either calibrate the machine or confirm a "outlier" reading.

A prototype Diamond Spectrophotometer, under development by Sarasota Instruments, was used to further qualify the author's set of diamond masters used in the Tuscon Shipley study. Additional stones were added to the original split grade set #4693, for a total of nine diamond masters split into two sets, #4693 and #4981. This was necessary because the GIA/GTL would not allow spacing of a half grade within one "certified" master stone set. The diamond master stones are tabulated below and crossed referenced to equivalent "AGS" colorimeter readings. All diamonds are Class 1 cut grade from Lazare Kaplan, each weighing approximately 0.3 cts. The object of this study was to ascertain the linearity and spacing of these sets, as well as to try to explain the consistant "misgrading" of the authors I/J master (stone#5) as an "H to H/I" by three Gran Colorimeters, tests conducted in Tuscon and NYC. Additionally the author's GIA "certed" I/J master was visually graded against Lazare Kaplan's master set and Lazare Kaplan cut masters in use by IGI in New York. Both visual tests substantiated the I/J split grade assigned by GIA.

1) E #4981-1 (AGS Colorimeter 0.50)
2) F/G split #4693-1 (AGS Colorimeter 1.25)
3) H #4981-2 (AGS Colorimeter 2.00)
4) H/I split #4693-2 (AGS Colorimeter 2.25)
5) I/J split #4693-3 (AGS Colorimeter 2.75)
6) J/K split #4693-4 (AGS Colorimeter 3.25)
7) K/L split #4693-5 (AGS Colorimeter 3.75)
8) L-M split #4693-6 (AGS Colorimeter 4.25)
9) N #4981-3 (AGS Colorimeter 5.00)

Figure 1 illustrates the normalized "transmittance" (transmittance rather than "transmission", as we do not a constant pathlength within the diamond for each wavelength) ratios versus wavelength for the master stones in these sets. One can easily see in this graph that the I/J (visual) split master did not follow the characteristics of the rest of the "cape series" masters in the two sets.

Figure 1: Normalized Transmittance Ratios For Sets 4693 and 4981

Using the AGS Colorimeter scale as a "linear" reference, Figure 2 illustrates the CIE coordinates obtained by numerical integration of the "transmittance" data shown in Figure 1 with a CIE illuminant C source. The numerical conversion to CIE coordinates of the "transmittance" data gives, for the I/J split master (at {AGS) X axis 2.75), CIE coordinates surprising not unlike the H master (at {AGS} X axis 2.0). As mentioned before, Gran colorimeter readings for this I/J master stone were in the H to H/I range but visual grading gave a consistant I/J split.

However, the full range Sarasota Instruments prototype diamond spectrophotometer used in this study, showed the abnormal "transmittance" characteristics for this diamond, as it did not rely on global "black box" colormetric conversions.

Figure 2: CIE Coordinates For Sets 4693 and 4981
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