GIA Study Finds Many Paths to "Ideal"

The Gemological Institute of America has announced the completion of its mathematical study on the effect of cut on the appearance of diamonds. The study appearing in the upcoming issue of Gems & Gemology. The controversial finding in the article is that many types of cut and varying combinations of proportions within the cut can give equal brilliance to the so-called "ideal" cut theorized by Tolkowski some eighty years ago.

The GIA study acknowledges many other factors in achieving an extraordinary appearance in round diamonds, among them scintillation and brilliance. The GIA will utilize the mathematical three-dimensional model to examine, with the most modern technology, the various appearance factors.

The DRB contacted the American Gem Society (AGS) in order to obtain a response to the GIA study and was told that AGS had not yet seen it. The AGS response to this study will be interesting to watch: Will AGS, the most muscular proponent of the "ideal" cut, change it’s standard in the face of a new scientific findings or will it continue to maintain that there is only one "ideal" ? Will it change the AGS parameter of the zero cut?

To understand what makes a diamond beautiful, the terms must first be defined. "Beauty", in diamond terms, encompasses brilliancy (intensity of internal and external reflections of white light to the eye from a diamond in the face-up position) and scintillation (alternating display of reflection from the polished facets of a gem seen by the observer as movement occurs or a lashing or twinkling of light). One of the results of the study has been, at least for GIA, to specify even more clearly the definitions of what is commonly termed the "brightness" of a diamond.

Within the study, "brightness" became a calculated numerical factor known as Weighted Light Return (WLR). According to GIA, values for WLR were determined for more than 20,000 combinations of proportions in diamond cuts. It is interesting to note that in one of the accompanying photos in the article, three stones were shown. Of the three, the "ideal" cut gem was less brilliant than those with larger tables.

Whether the GIA study will revolutionize the diamond business remains to yet to be seen. One wonders, though whether such an all encompassing change as this will be fought on the scientific front or as a showdown of the publicists. Certainly, jewelers and consumers alike should read the articles and consider the issues, but in the final analysis beauty is always defined by the eyes of the individual customer.

Let us now pass to the consideration of the other alternative, i.e. where the top surface is a horizontal plane AB and where the bottom surface AC is inclined at an angle (to the horizontal (fig. 25)). As before, we have to introduce a third plane BC to have a symmetrical section.
First Reflection
Let a vertical ray PQ strike AB. As the angle of incidence is zero, it passes into the stone without refraction and meets plane AC at R. Let RN be the normal at that point, then, for total reflection to occur, angle NRQ = 24°26'. But, angle NRQ = angle QAR - a as AQ and QR, AR and RN are perpendicular. Therefore, for total reflection of a vertical ray,
 

Fig. 25

a = 24°26'. Let us now incline the ray PQ so that it gradually changes from a vertical to a horizontal direction, and let P'Q' be such a ray. Upon passing into the diamond it is refracted, and strikes AC at an angle Q'R'N' where R'N' is the normal to AC. When P'Q' becomes horizontal, the angle of refraction T'Q'R' becomes equal to 24°26'. This is the extreme value attainable by that angle; also, for total reflection, angle Q'R'N' must not be less than 24°26'. If we draw R'V, vertical angle VR'Q' = R'Q'T' = 24°26', and angle VR'N' = VR'Q' + Q'R'N' =24°26' + 24°26' = 48°52' as before, a = angle VR'N', and therefore a = 48°52 (9).

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

Second Reflection

Fig. 26

When the tray of light is reflected from the first inclined facet AC (fig. 26), it strikes the opposite one BC. Here too the light must be totally reflected, for otherwise there would be a leakage of light through the back of gem stone. Let us consider, in the first instance, a ray of light vertically incident upon the stone. The path of the ray will be PQRST. If RN and SN' are the normals at R and S respectively, then for total reflection, angle N'SR = 24°26'. Let us find the value of a to fulfil that condition: angle QRN = angle QAR = a as having perpendicular sides. angle SRN = angle QRN as angles of incidence and reflection. Therefore angle NRS = a. Now let angle N'SR = c. Then, in triangle RSC,

angle SRC = 90° - a

angle RSC = 90° - c

angle RCS = 2 x angle RCM = 2 x ARQ = 2 (90° - a).

The sum of these three angles equals two right angles,

90° - a + 90° - c + 180° - 2a = 180°, or

3a+ c = 180° 3a = 180° - c.

Fig. 27


Now, c is not less than 24°26', therefore a is not greater than a = (180 - 24°26')/3 = 51°51'. Let us again incline PQ from the vertical until it becomes horizontal, but in this case in the other direction, to obtain the inferior limit. Then (fig.27) the path will be PQRS. Let QT, RN, SN' be the normals at Q, R, and S respectively. At the extreme case, TQR will be 24°26'. Draw RV vertical at R.

Then angle QRV = angle TQR = 24°26' angle VRN = a.
As before, in triangle RSC,

angle SRC = 90° - NRS = 90° - a - 24°26'

angle RCS = 2 a (90° - a)

angle RSC = 90° - c.

Then

90 - a - 24°26' + 180° - 2a + 90° - c = 180°

3a + x = 180° - 24°26' = 155°34'

3a = 155°34' - c.

In the case now considered,

c = 24°26'.
Then

3a = 155°34' - 24°26' = 131° 8'

a = 43°43' (10)

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