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 ofDiamond Cutting

Wendi Clouse

Midterm

Math G

Due 03/18/02

Shroudedin mystery, intrigue and controversy, the diamond may be one of the mostmysterious substances on earth. Itis believed that the first diamond was discovered as early as 500 B.C., butgeologists believe that the formation of diamonds occurred in the Earth’smantle at least one hundred million years ago. The diamond as a gem didn’t make an appearance in jewelryuntil1074 A. D., and at that time it was used only in the natural crystallineform. It wasn’t until 1916 that asystematic mathematical approach to diamond cutting was developed. The man who revolutionized the industryis Marcel Tolkowsky and he single-handedly changed the world of diamondsforever.

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

Thepractice of cutting diamonds is a relatively new art form, only done within thelast century. Up until thefifteenth century it was common practice to “cleave” a diamond by applying achisel 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 dueto miscalculation of the cleavage plane. If the angle of the chisel was wrong, the pressure from the malletshattered the crystal. Aftercleaving was complete the diamond was then placed into an egg shaped tin cupand the edges were hit with another diamond until the desired shape wasachieved. The shaping process waslimited to the natural shape of the diamond crystal and was at best veryrudimentary, producing clumsy looking diamonds with very little sparkle.

Atthe end of the fifteenth century, a diamond cutter in Antwerp named Lodewyk vanBerken invented a machine called a scaif. The scaif was a manually operated polishing wheel, made of a largebronze disc imbedded with diamond dust. Using olive oil as a lubricant, the wheel was capable of grinding awayflat symmetrical shapes called facets on the cleaved diamond crystal. The scaif made the cutting process soprecise that the tradesman could now concentrate on the optics of the gem,producing stones that were livelier. Van Berken’s invention lured cutters from around Europe to Antwerp tostudy this new method, and the products that they produced quickly becamepopular with the aristocracy of Europe.

Althoughthe popularity and demand for diamonds increased as cutting methods became moreprecise, there were no new innovations to the cutting world until the twentiethcentury when the diamond saw was invented. The diamond saw was a circular steel blade continuallylubricated with diamond dust and oil, it was capable of cutting against thenatural grain of the crystal without causing damage. Although it took a longer period of time to saw through adiamond, it was now possible to recut diamonds that had been damaged, and cutrough stones that were irregular in shape. Sawing was more expensive than cleaving due to the timeconstraints and the amount of diamond dust that it took to operate “It requiredabout 1/10^th of a carat of diamond dust for every carat of diamondsawed through. And it was also amuch slower process than cleaving a diamond with a single stroke. Indeed it took days to saw through atwo-carat diamond. Despite such disadvantages, the diamond saw became thefavored method of shaping diamonds… Since it was far easier to train workers tosaw rather than cleave diamonds, it quickly transformed diamond cutting Antwerpfrom an esoteric craft to a semi-mechanized industry”[footnote 1]

Thencame the largest change yet in the cutting world. Marcel Tolkowsky was born in 1899. His family had an established name in the diamond cuttingand dealing industry. Educatedfirst, at the German School in Antwerp, he studied at the Lycee François, andthen later would receive a D. Sc in engineering from the University of London. In 1919 he published a book called DiamondDesign. The book, only 104pages in length had a profound affect on the diamond industry. At the age of 21, Marcel had managed tocalculate a formula to maximize refracted light, with the least amount ofsacrifice of reflected light, in other words he had calculated the parametersof cut proportion that would give a half and half ratio between brilliance anddispersion. Tolkowsky had intheory invented the modern round brilliant cut diamond, and his guidelines forproportion would become the defining factor in what the world would label “a perfectcut”. Tolkowsky was not the firstcutter to use this idea to improve the product, but he was the only one at thetime that could provide the “mathematical proof” for his work. For what others were attempting to doin practice, Tolkowsky had managed to prove on paper.

Inorder to understand Tolkowsky’s formula, we must first be familiar with theoptical properties that affect a diamonds appearance. The Gemological Institute of America defines opticalproperties as: characteristics of a gemstone that govern its interaction withlight. The most popular terms for these properties in relation to the diamondare dispersion, brilliance, and scintillation.

Dispersionis simply the metamorphoses of white light, as it breaks into the spectral huesof color visible to the human eye. Dispersion is achieved through a process called refraction. Because a diamond is very denseatomically, it slows the velocity of traveling light. Andrew Cockburn in his recent National Geographic article, Diamonds The Real Story relates “they(diamonds) are so dense that they slow the speed of light by two-thirds”. This characteristic is the key inunderstanding the basis for Tolkowsky’s research. Refraction is what happens when light passes from one objectto another; the light will suddenly slow as it enters the second object, thechange in velocity causes the light to travel at a different angle thusbending. The angle of the benddepends on the angle of the light beam as it enters the denser object (angle ofincidence).

http://micro.magnet.fsu.edu/optics/timeline/people/snell.html“In 1621 a mathematician named Willebrord Snell discovered a ratio between theangle of incidence and the angle of refraction. Snell’s law [footnote 2] shows that every object has abending ratio; this is called the index of refraction”. The RI for diamond is 2.417, which isextremely high for a transparent substance. Light entering a diamond bends, the beam of light is thenseparated, because light waves travel at different speeds. Separation in the white beam causesminute flashes of color to travel back to the eye in the form of prismaticflashes of color. The eyeperceives 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 thefull range of spectral color from violet to red. The diamond is considered the most dispersive gems innature. Diagram to theright is an example spectral separation at the exit from the stone. Provided by the GIA website

Brillianceis the combination of white light reflecting from both the surface and theinterior of the diamond. Brilliance is defined as the brightness that lightens the stone. This optical characteristic is theresult of reflection, or the action of white light bouncing off of the surfaceof a diamond and traveling back to the eye intact (no color separationoccurring) [Footnote 3]. Brilliance also occurs in the interior of the diamond as light travelsin straight lines instead of bent lines, as it bounces off of the internalfacet pattern and returns to the eye as white light. Before Tolkowsky developed his formula it was thought that adiamond had to be cut to show either dispersion (refracted light) or brilliance(reflected light). Tolkowsky was thefirst to realize that you could have an equal balance of both if you controlledthe proportions of the diamond’s anatomy.

Scintillationis the “sparkle” of the diamond, or the tiny flashes of light viewed when thediamond, observer or light source moves. Scintillation is the result of small flashes of light that the diamondcollects and then distributes from the different light and dark values in theenvironment. This optical propertyis a source of great beauty in a diamond, but does not play a big role inTolkowsky’s formula. This diagram is from the website- http://www.accurateappraisal.com/estimati.htm is a dissected anatomy of a modernround brilliant. The diagram will help define the measurements discussed inTolkowsky’s proportional guidelines.

The table is the flat surfaceacross the top of the diamond. Thecrown is the set of facets that join the table to the girdle; the girdle is thethin set of facets or continual facet that encircles the diameter of thestone. The pavilion refers to thefacet pattern that originates from the lower edge of the girdle and terminatesat the point on the bottom. If thepoint of pavilion termination is faceted, the facet is called the culet, if thepavilion comes to a perfect point it is determined that the diamond has noculet.

Tolkowsky determined thatthe proportions for a “well made” diamond would fall into a specificrange. If a diamond were cutaccording to his parameters, light would both refract and reflect back to theeye. At the time of his research,he looked only at the behavior of refracted light as it exited thediamond. He did not take internaldispersion into consideration. Some experts in the gemological field consider his dismissal of internaldispersion to be an error, but the finished product speaks for itself. Even today there are few diamonds thatcan compare to those cut within the Tolkowsky proportions. Tolkowsky’s theoretical model outlinesthat 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 andthe pavilion angle of 40degrees and 40 minutes. Fourdiagrams of: Tolkowsky’s Ideal, proper light behavior, shallow cut and deep cutprovided by: http://www.jewelry1.com/diamond/Diamcut.htm

The following formula is takendirectly from http://www.folds.net/diamond_design/index.html#brilliance_and_fire It is a simplified version takenfrom Tolkowsky’s book Diamond Design. His formula, in this simplified form is the bestmathematical explanation of light behavior in the modern round brilliant. Although I have a thoroughunderstanding of the cause and affect of different variables that affect adiamond’s dispersion and brilliance, I do not have the mathematical experienceto define this formula on my own:

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

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

Step 2. We find out what fraction of the oblique rays are effective, and theiraverage angle:
                  CriticalAngle = arcsin(1 / 2.417) = 24° 26'23"
            EffectiveAngle    = arcsin(sin (alpha - 24° 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 - 24° 26' 23"
QRP = 90° - 2 * alpha + QPT
Q₂ED = 90° - 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 andthe vertical.
FirstAngle = 180° - 4 * alpha
SecondAngle = 2 * alpha- AverageRefractedAngle

Step 5. We calculate some ratios that make the calculations easier.
   f = 1 +(1 / tan QPT / tan alpha)
   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 +tan beta / 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/ cos alpha
SC = TC+ AP * (tan SPT) / (tan SPT + 1 / tan alpha) / cos alpha
Q₂C = (DE / 2) / cos alpha * (tan alpha -tan Q₂ED) / (tan alpha + 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 betastops changing.

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

Step 11. Tolkowsky says that:

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

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

Shallow Cut:

If thediameter becomes to large for the depth of the stone, both reflected light andrefracted light will exit the diamond via the pavilion, returning a very smallamount of light to the eye. Whenthis happens the affect is called a fish eye, and the value of the diamond isdecreased because its “corrected carat weight”[Footnote 5] is much smaller thatthe actual carat weight purchased. Photos offisheye (right) and deep cut below are provided by www.pricescope.com

Deep Cut:

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

WELL CUT DIAMOND

The diagrams above is howlight should behave in a diamond cut properly. Photo aboveprovided by GIA

SHALLOW CUT DIAMOND

Thediagram above is an example of a shallow cut.

DEEP CUT DIAMOND Thisdiagram is an example of a deep cut.

When examining thedifferent examples of cut proportion, you can easily visualize why diamonds cutclose to Tolkowsky’s proportions manage to return a large amount of light tothe eye, but it should also raise a question. If Tolkowsky’s proportions are best for light return, whyaren’t all diamonds cut in this fashion? In order to answer this question we need to consider severalfactors: the diamond industry isprofit driven, diamonds are a commodity priced according to weight and rarity,the industry itself can not agree on terminology for categorizing cut, mostjewelry purchases are based on emotion instead of logic, and most sales peopledo 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] andsells by the carat. If a cutterbuys 100 carats of rough crystal, but only yields 40 carats of finished goodshis profit is much lower than if he yields 60 carats of finished goods. Due to the natural shape of the diamondcrystal (octahedron) it is more profitable for a cutter to produce a diamondwith a deep cut because they realize higher yield from their initialinvestment. Retail establishments want a high profit margin;therefore they purchase diamonds at the lowest cost they can find. Often their inventory is purchasedsight unseen, with cost being the driving factor. Low employment cost usually accompanies a higher profit marginso employee-training courses are substandard; most employees are trained onlymarginally about proportion and given none of the mathematics involved. The diamond industry itself cannotagree on a specific grading policy for cut. The industry relies on profit, and if profit decreasesbecause the end customer demands a better product, the company coffersdiminish. The consumer has veryfew educational tools available, and in all honesty most consumers make theirdiamond purchases on a whim. Jewelry purchases are luxury items, purely emotional, very few peoplewho are purchasing that tenth anniversary gift want to explore “mathematics”when they are shopping. Weight diagram provided by http://www.drostes.com/pavilion_depth.html

In conclusion there aremany factors that can change both the value and esthetics of a diamond. Tolkowsky’s formula is one way todefine beauty in mathematical language. However, as with art many people interpret beauty in differentways. A mathematical proof mayprovide us with the understanding of the scientific behavior, but it does notprovide the only definition of beauty. Ultimately beauty is in the eye of the beholder.

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

Page 105, paragraph 1.

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

The ratio of the sine of theangle of incidence i to the sine of the angle of refraction r is equal to the ratio of the speed of lightin the original medium,V; to the speedof light in the medium, Vr. Or sin i/sin r = Vi/Vr. Snell’s law is often related in refractive indexes insteadof the speed of light in the two mediums.

Footnote3- Reflection according to the GIA Diamond Dictionary is the bouncing back oflight when it strikes a polished surface. Approximately 17 percent of the light striking the external surface of apolished diamond vertically is reflected back into the air; the greater partenters the stone. Light striking an internal surface of a polished diamond atan angle greater than the critical angle (24 degrees 26 minutes) is reflectedback into the diamond (total internal reflection).

Footnote4- Carat is a weight measurement equaling 200 milligrams. The carat is broken into 100 units ofmeasure called points, thus a 50 pointer equals ½ carat.

Footnote5- Corrected carat weight is the weight a diamond would have been if cut tocorrect proportions.

Bibliography

The Rise and Fall ofDiamonds the Shattering of a Brilliant Illusion

National GeographicPeriodical, March 2002, Diamonds The Real Story, written by AndrewCockburn. Page 2-35