This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.

The Evolution of Diamond Cutting

Wendi Clouse

Midterm

Math G

Due 03/18/02

Shrouded in mystery, intrigue and controversy, the diamond may be one of the most mysterious substances on earth. It is believed that the first diamond was discovered as early as 500 B.C., but geologists believe that the formation of diamonds occurred in the Earthís mantle at least one hundred million years ago. The diamond as a gem didnít make an appearance in jewelry until1074 A. D., and at that time it was used only in the natural crystalline form. It wasnít until 1916 that a systematic mathematical approach to diamond cutting was developed. The man who revolutionized the industry is Marcel Tolkowsky and he single-handedly changed the world of diamonds forever.

The natural crystalline form of a diamond is cubic; manifesting itself in a shape called an octahedron, which looks like two four sided pyramids stacked on top of one another, base to base. The cubic form has four distinct cleavage planes, each in a different direction. The cleavage plane is a weakness in the crystalline form where the crystal will break. Although a diamond is one of the hardest substances on earth (10 on the MOHS scale) a small amount of pressure at the proper point can separate a diamond crystal. Octahedron photo provided by http://www.drostes.com/pavilion_depth.html

The practice of cutting diamonds is a relatively new art form, only done within the last century. Up until the fifteenth century it was common practice to ìcleaveî a diamond by applying a chisel to one of the four perfect cubic planes in the crystalline structure, then striking it with a mallet. Unfortunately this sometimes resulted in the destruction of the gem due to miscalculation of the cleavage plane. If the angle of the chisel was wrong, the pressure from the mallet shattered the crystal. After cleaving was complete the diamond was then placed into an egg shaped tin cup and the edges were hit with another diamond until the desired shape was achieved. The shaping process was limited to the natural shape of the diamond crystal and was at best very rudimentary, producing clumsy looking diamonds with very little sparkle.

At the end of the fifteenth century, a diamond cutter in Antwerp named Lodewyk van Berken invented a machine called a scaif. The scaif was a manually operated polishing wheel, made of a large bronze disc imbedded with diamond dust. Using olive oil as a lubricant, the wheel was capable of grinding away flat symmetrical shapes called facets on the cleaved diamond crystal. The scaif made the cutting process so precise that the tradesman could now concentrate on the optics of the gem, producing stones that were livelier. Van Berkenís invention lured cutters from around Europe to Antwerp to study this new method, and the products that they produced quickly became popular with the aristocracy of Europe.

Although the popularity and demand for diamonds increased as cutting methods became more precise, there were no new innovations to the cutting world until the twentieth century when the diamond saw was invented. The diamond saw was a circular steel blade continually lubricated with diamond dust and oil, it was capable of cutting against the natural grain of the crystal without causing damage. Although it took a longer period of time to saw through a diamond, it was now possible to recut diamonds that had been damaged, and cut rough stones that were irregular in shape. Sawing was more expensive than cleaving due to the time constraints and the amount of diamond dust that it took to operate ìIt required about 1/10^th of a carat of diamond dust for every carat of diamond sawed through. And it was also a much slower process than cleaving a diamond with a single stroke. Indeed it took days to saw through a two-carat diamond. Despite such disadvantages, the diamond saw became the favored method of shaping diamondsÖ Since it was far easier to train workers to saw rather than cleave diamonds, it quickly transformed diamond cutting Antwerp from an esoteric craft to a semi-mechanized industryî[footnote 1]

Then came the largest change yet in the cutting world. Marcel Tolkowsky was born in 1899. His family had an established name in the diamond cutting and dealing industry. Educated first, at the German School in Antwerp, he studied at the Lycee FranÁois, and then later would receive a D. Sc in engineering from the University of London. In 1919 he published a book called Diamond Design. The book, only 104 pages in length had a profound affect on the diamond industry. At the age of 21, Marcel had managed to calculate a formula to maximize refracted light, with the least amount of sacrifice of reflected light, in other words he had calculated the parameters of cut proportion that would give a half and half ratio between brilliance and dispersion. Tolkowsky had in theory invented the modern round brilliant cut diamond, and his guidelines for proportion would become the defining factor in what the world would label ìa perfect cutî. Tolkowsky was not the first cutter to use this idea to improve the product, but he was the only one at the time that could provide the ìmathematical proofî for his work. For what others were attempting to do in practice, Tolkowsky had managed to prove on paper.

In order to understand Tolkowskyís formula, we must first be familiar with the optical properties that affect a diamonds appearance. The Gemological Institute of America defines optical properties as: characteristics of a gemstone that govern its interaction with light. The most popular terms for these properties in relation to the diamond are dispersion, brilliance, and scintillation.

Dispersion is simply the metamorphoses of white light, as it breaks into the spectral hues of color visible to the human eye. Dispersion is achieved through a process called refraction. Because a diamond is very dense atomically, it slows the velocity of traveling light. Andrew Cockburn in his recent National Geographic article, Diamonds The Real Story relates ìthey (diamonds) are so dense that they slow the speed of light by two-thirdsî. This characteristic is the key in understanding the basis for Tolkowskyís research. Refraction is what happens when light passes from one object to another; the light will suddenly slow as it enters the second object, the change in velocity causes the light to travel at a different angle thus bending. The angle of the bend depends on the angle of the light beam as it enters the denser object (angle of incidence).

http://micro.magnet.fsu.edu/optics/timeline/people/snell.html ìIn 1621 a mathematician named Willebrord Snell discovered a ratio between the angle of incidence and the angle of refraction. Snellís law [footnote 2] shows that every object has a bending ratio; this is called the index of refractionî. The RI for diamond is 2.417, which is extremely high for a transparent substance. Light entering a diamond bends, the beam of light is then separated, because light waves travel at different speeds. Separation in the white beam causes minute flashes of color to travel back to the eye in the form of prismatic flashes of color. The eye perceives color because white light is a combination of all light wavelengths, when the wavelengths separate as the light slows; the eye is able to see the full range of spectral color from violet to red. The diamond is considered the most dispersive gems in nature. Diagram to the right is an example spectral separation at the exit from the stone. Provided by the GIA website

Brilliance is the combination of white light reflecting from both the surface and the interior of the diamond. Brilliance is defined as the brightness that lightens the stone. This optical characteristic is the result of reflection, or the action of white light bouncing off of the surface of a diamond and traveling back to the eye intact (no color separation occurring) [Footnote 3]. Brilliance also occurs in the interior of the diamond as light travels in straight lines instead of bent lines, as it bounces off of the internal facet pattern and returns to the eye as white light. Before Tolkowsky developed his formula it was thought that a diamond had to be cut to show either dispersion (refracted light) or brilliance (reflected light). Tolkowsky was the first to realize that you could have an equal balance of both if you controlled the proportions of the diamondís anatomy.

Scintillation is the ìsparkleî of the diamond, or the tiny flashes of light viewed when the diamond, observer or light source moves. Scintillation is the result of small flashes of light that the diamond collects and then distributes from the different light and dark values in the environment. This optical property is a source of great beauty in a diamond, but does not play a big role in Tolkowskyís formula. This diagram is from the website- http://www.accurateappraisal.com/estimati.htm is a dissected anatomy of a modern round brilliant. The diagram will help define the measurements discussed in Tolkowskyís proportional guidelines.

The table is the flat surface across the top of the diamond. The crown is the set of facets that join the table to the girdle; the girdle is the thin set of facets or continual facet that encircles the diameter of the stone. The pavilion refers to the facet pattern that originates from the lower edge of the girdle and terminates at the point on the bottom. If the point of pavilion termination is faceted, the facet is called the culet, if the pavilion comes to a perfect point it is determined that the diamond has no culet.

Tolkowsky determined that the proportions for a ìwell madeî diamond would fall into a specific range. If a diamond were cut according to his parameters, light would both refract and reflect back to the eye. At the time of his research, he looked only at the behavior of refracted light as it exited the diamond. He did not take internal dispersion into consideration. Some experts in the gemological field consider his dismissal of internal dispersion to be an error, but the finished product speaks for itself. Even today there are few diamonds that can compare to those cut within the Tolkowsky proportions. Tolkowskyís theoretical model outlines that a round brilliant should have 58 facets that are symmetrical, 53% table; 60-61% total depth that includes the girdle thickness of 0.7 to 1.7%; 16.2% crown height; 43.1% pavilion depth, crown angle of 34 degrees 30 minutes and the pavilion angle of 40degrees and 40 minutes. Four diagrams of: Tolkowskyís Ideal, proper light behavior, shallow cut and deep cut provided by: http://www.jewelry1.com/diamond/Diamcut.htm

The following formula is taken directly from http://www.folds.net/diamond_design/index.html#brilliance_and_fire It is a simplified version taken from Tolkowskyís book Diamond Design. His formula, in this simplified form is the best mathematical explanation of light behavior in the modern round brilliant. Although I have a thorough understanding of the cause and affect of different variables that affect a diamondís dispersion and brilliance, I do not have the mathematical experience to define this formula on my own:

Start.
We choose alpha.
We start with a guess for beta (say, 35ƒ).

Step 1. We look at the girdle:
DE = diameter of a knife-edge diamond.(1 mm is easiest.)

Step 2. We find out what fraction of the oblique rays are effective, and their average angle:
                  CriticalAngle = arcsin(1 / 2.417) = 24ƒ 26' 23"
            EffectiveAngle    = arcsin(sin (alpha - 24ƒ 26' 23") * 2.417)
            EffectiveFraction = [1/3 - sin(EffectiveAngle)] * 3 / (-2)
SPT =      AverageRefractedAngle = arcsin(( (1/3) + sin(EffectiveAngle) ) / 2 / 2.417)

Step 3. We calculate angles of rays:
QPT = alpha - 24ƒ 26' 23"
QRP = 90ƒ - 2 *alpha + QPT
Q₂ED = 90ƒ - 2 *alpha+AverageRefractedAngle

Step 4. We calculate angles of typical rays before they leave the crown.
The FirstAngle is the angle between R'S' and the vertical.
The SecondAngle is the angle between Q₁R₁ and the vertical.
FirstAngle = 180ƒ - 4 *alpha
SecondAngle = 2 *alpha-AverageRefractedAngle

Step 5. We calculate some ratios that make the calculations easier.
   f = 1 + (1 / tan QPT / tanalpha)
   g = (1 / tan QPT - tan QRP) / 2 / tan QRP

Step 6. The loop starts here.
We calculate the table ratio of a knife-edge diamond:
   h = 1 + tanbeta / tan alpha
   t = g * h / (f + g * h)

Step 7. We calculate distances at the top of the diamond:

PM = (DE / 2) * t
AP = PM * f / g

Step 8. We calculate distances along the pavilion edge:
TC = PM / cosalpha
SC = TC + AP * (tan SPT) / (tan SPT + 1 / tanalpha) / cos alpha
Q₂C = (DE / 2) / cosalpha * (tan alpha- tan Q₂ED) / (tanalpha + tan Q₂ED)

Step 9. We calculate a new guess for beta (the crown angle).
FirstWeight = (TC * TC)
SecondWeight = EffectiveFraction* (SC * SC - Q₂C * Q₂C)
beta = (FirstWeight * FirstAngle + SecondWeight * SecondAngle) / (FirstWeight + SecondWeight)
This gives us a new guess for beta.
The loop ends here. We can repeat steps 6-9 until the guess for beta stops changing.

Step 10. Because Tolkowsky uses a knife-edge girdle, we do NOT need to adjust the diameter and table ratio.

Step 11. Tolkowsky says that:

Modern diamonds have longer lower girdle facets, so these angles are slightly different.

Step 12. The diamond total depth contains the crown and the pavilion. Because it has a knife-edge girdle, there is no girdle thickness:
CrownHeight = diameter / 2 * (1 - t) * tan beta
PavilionDepth = diameter / 2 * tan alpha
-(CuletHeight) =-((culet/ 2) * cos(22ƒ 30') * tanalpha)
TotalDepth = CrownHeight + PavilionDepth - CuletHeight

Shallow Cut:

If the diameter becomes to large for the depth of the stone, both reflected light and refracted light will exit the diamond via the pavilion, returning a very small amount of light to the eye. When this happens the affect is called a fish eye, and the value of the diamond is decreased because its ìcorrected carat weightî[Footnote 5] is much smaller that the actual carat weight purchased. Photos of fisheye (right) and deep cut below are provided by www.pricescope.com

Deep Cut:

When the diameter of the stone is too small for its total depth, light is thrown out of the pavilion at odd angles instead of being reflected back to the eye. A deep cut is a very poor value because the diameter of the stone is smaller than the cut should carry; therefore the total weight of the diamond is heavier than its appearance. Because diamonds are priced according to weight, jewelers can make a greater amount of profit selling stones that have a deep cut. They buy the poorly cut stones at discounted rates, and then sell them, because of the higher carat weight at the same price as a diamond that is cut to proper proportions. Take for example two diamonds that weigh 1 carat each, and are the same color and clarity- one is cut to proper proportions and one is cut deep. The one that is cut well will measure approximately 6.5 mm in diameter, however the diamond that is cut poorly will only measure 5.75mm in diameter. Three quarters of a millimeter may not seem like much difference, but the diamond that measures smaller will look the same as a .75-carat stone. If you had purchased a .75 ct instead of 1 carat you could have saved as much a $4,000.00 in todayís market. Not to mention that the .75ct that is cut well will be more beautiful than the one carat with poor proportions. Compare the behavior of light in the following diagrams:

WELL CUT DIAMOND

The diagrams above is how light should behave in a diamond cut properly. Photo above provided by GIA

SHALLOW CUT DIAMOND

The diagram above is an example of a shallow cut.

DEEP CUT DIAMOND This diagram is an example of a deep cut.

When examining the different examples of cut proportion, you can easily visualize why diamonds cut close to Tolkowskyís proportions manage to return a large amount of light to the eye, but it should also raise a question. If Tolkowskyís proportions are best for light return, why arenít all diamonds cut in this fashion? In order to answer this question we need to consider several factors: the diamond industry is profit driven, diamonds are a commodity priced according to weight and rarity, the industry itself can not agree on terminology for categorizing cut, most jewelry purchases are based on emotion instead of logic, and most sales people do not take the time to educate the consumer properly on this grading aspect.

The diamond industry (cutters, designers, manufacturers) buys diamonds by the carat [Footnote 4] and sells by the carat. If a cutter buys 100 carats of rough crystal, but only yields 40 carats of finished goods his profit is much lower than if he yields 60 carats of finished goods. Due to the natural shape of the diamond crystal (octahedron) it is more profitable for a cutter to produce a diamond with a deep cut because they realize higher yield from their initial investment. Retail establishments want a high profit margin; therefore they purchase diamonds at the lowest cost they can find. Often their inventory is purchased sight unseen, with cost being the driving factor. Low employment cost usually accompanies a higher profit margin so employee-training courses are substandard; most employees are trained only marginally about proportion and given none of the mathematics involved. The diamond industry itself cannot agree on a specific grading policy for cut. The industry relies on profit, and if profit decreases because the end customer demands a better product, the company coffers diminish. The consumer has very few educational tools available, and in all honesty most consumers make their diamond purchases on a whim. Jewelry purchases are luxury items, purely emotional, very few people who are purchasing that tenth anniversary gift want to explore ìmathematicsî when they are shopping. Weight diagram provided byhttp://www.drostes.com/pavilion_depth.html

In conclusion there are many factors that can change both the value and esthetics of a diamond. Tolkowskyís formula is one way to define beauty in mathematical language. However, as with art many people interpret beauty in different ways. A mathematical proof may provide us with the understanding of the scientific behavior, but it does not provide the only definition of beauty. Ultimately beauty is in the eye of the beholder.

Footnote 1-The Rise and Fall of Diamonds the Shattering of a Brilliant Illusion

Page 105, paragraph 1.

Footnote 2- the formula for the law of refraction is as follows:

The ratio of the sine of the angle of incidence i to the sine of the angle of refraction r is equal to the ratio of the speed of light in the original medium,V; to the speed of light in the medium, Vr. Or sin i/sin r = Vi/Vr. Snellís law is often related in refractive indexes instead of the speed of light in the two mediums.

Footnote 3- Reflection according to the GIA Diamond Dictionary is the bouncing back of light when it strikes a polished surface. Approximately 17 percent of the light striking the external surface of a polished diamond vertically is reflected back into the air; the greater part enters the stone. Light striking an internal surface of a polished diamond at an angle greater than the critical angle (24 degrees 26 minutes) is reflected back into the diamond (total internal reflection).

Footnote 4- Carat is a weight measurement equaling 200 milligrams. The carat is broken into 100 units of measure called points, thus a 50 pointer equals ‡ carat.

Footnote 5- Corrected carat weight is the weight a diamond would have been if cut to correct proportions.

Bibliography

The Rise and Fall of Diamonds the Shattering of a Brilliant Illusion

National Geographic Periodical, March 2002, Diamonds The Real Story, written by Andrew Cockburn. Page 2-35