Diamond Optics and Design
Introduction
The round brilliant cut diamond evolved over several hundred years of diamond cutting. This style of cut stands above all others in importance and popularity for one most important reason. In the round brilliant cut, the finest "make" (trade jargon for fashioning and craftsmanship) brings out the best in the attributes of diamond beauty - brilliance, fire and sparkle.
The greatest stride in the evolution of what today is known as the ideal cut took place in Boston around 1860 by the first American diamond cutter, Henry D. Morse. A half-century later in 1919, the Belgian diamond cutter, Marcel Tolkowsky, popularized this brilliant cut diamond with the publication of his landmark book, "Diamond Design". Since 1860 the terms used to signify the finest round brilliant cut are the scientific cut, perfect cut, ideal cut, high class brilliant, American Ideal, and the Tolkowsky Ideal.
These terms for finest have been used, often without definition, to promote not only the best of diamond cutting, but also variations that were far from best. The term perfect is no longer permitted to describe cut, because perfect cut was so exploited and misused. Because the term ideal has similarly been much abused, some have also abandoned its use in favor of less exploited terms for best.
However, all these terms are meant to indicate those round brilliants having angles and proportions that yield the greatest beauty in terms of brilliancy, fire and sparkle. Advocates of the ideal believe that these three attributes of diamond beauty are jointly maximized in this finest of round brilliant cuts. We will retain the term ideal cut (uncapitalized) with the understanding that we are referring, not to a single set of angles and proportions, but rather to the narrow range yielding the finest in round brilliant diamond cutting. Diamonds in a range of cut proportions seen as having ideal beauty possess the best combination of brilliance (in both its aspects of brightness and contrast), fire (rainbow hues) and scintillation (sparkle with movement).
The capitalized term Ideal Cut has come to mean the angles and proportions that the mathematician and diamond cutter, Marcel Tolkowsky determined and published in "Diamond Design". These are a pavilion main angle of 40.75°, a crown main angle of 34.5° and a 53% table. We will distinguish reference to the Tolkowsky Ideal, with this narrow and incomplete definition, from our above-defined usage by indicating ideal without capitalization. The above definition of the Tolkowsky Ideal is incomplete, because it only addresses 17 of the 57 important facets (compared to the minor culet and girdle facets) that define the round brilliant cut diamond.
Because of the historical overemphasis on Tolkowsky's theoretical angles of 40.75° and 34.5° in association with Ideal, it is of importance to note that the five diamonds that Tolkowsky listed in his book as examples of maximally brilliant diamonds had pavilion angles from 40°-41°, and crown angles from 33°-35°. Additionally, Tolkowsky notes in his book that American writers credit Henry D. Morse with first cutting for "maximum brilliancy". The angles that Morse first discovered that were said by American writers like Frank B. Wade ("Diamonds", 1916) and Herbert Whitlock ("Proportions of the Brilliant Cut", 1917) to yield an ideal make had a range that centered on a 41° pavilion and a 35° crown.
Frank Wade, one of the American diamond experts that greatly influenced the thinking about the ideal cut in his era, said of Morse's angles that "Within the limits of one or two degrees there is little variation in brilliancy" To a varying extent this accords with today's consensus that there is a range of appropriate angles producing ideal optical performance and beauty. Differences of opinion are principally in the amount of variation in angles and proportions from those of Morse and Tolkowsky that retain ideal brilliance, fire and sparkle. We will explore those variations and their impact on diamond beauty in this work.
Principles of optical science are used in this work to study the round brilliant cut. The insight gained from this study extends to all other styles of diamond fashioning. The understanding gained through study of the optics of the round brilliant cut diamond makes possible the understanding of all diamond cuts.
We begin by dissecting the brilliant cut diamond's anatomy, that is, its polished surfaces, called facets (from Old French for faces).
The 57 facet round brilliant cut
The round brilliant cut has two key parts, the top and the bottom known as the "crown" and "pavilion". The diamond's crown and pavilion are joined together at the "girdle", which is the thin part where the diamond is at its maximum width.
The Crown
Almost all the brilliance, fire and sparkle reflected to our eyes from within the finest round brilliant cut diamonds comes from light that entered the diamond through its top section. Fittingly called the crown, this is the diamond's light-gathering portion that sits atop the girdle. See Figure 1.
The largest facet, which is centered on the crown, is the octagon-shaped "table." Eight triangle-shaped facets called "stars" surround it. Next are the eight kite-shaped facets called "bezels" or "crown mains." The sixteen "crown halves" follow the mains. These are also called "upper girdle" facets, because one of their three sides forms the upper outline of the girdle.
The parameters that uniquely define the crown of the standard round brilliant are shown in Figure 2. They are the crown main angle, the table size, the angle of the crown halves, and the star angle. An alternative to listing the angle of the crown halves and the angle of the stars is to specify the star length percentage. The star length determines the angles of the stars and halves in the context of a specific crown main angle and table size.
The Pavilion
Below the girdle is the pavilion, which is the principal light-reflecting portion of the round brilliant. See Figure 3.
The pavilion is comprised of 16 pie-shaped "pavilion halves", also called "lower girdle" facets. Eight "pavilion main" facets follow the pavilion halves. A small, octagon shaped, 58th facet may be present at the tip of the pavilion. This tip is called the "culet" where the eight pavilion mains come to a point. Historically, the culet facet was big enough to be noticeable and was thought to increase brilliance. Today it is known to have a negative effect on diamond beauty and is minimized or absent.
All the brilliance, fire and sparkle our eyes see emerging from within the round brilliant through its crown is reflected from either the pavilion mains or the lower halves. We will find that changes in the sizes and angles of the pavilion mains and halves have the greatest effect on the diamond's beauty and optical performance.
The Girdle
The girdle is the thin section whose surface around the stone (perpendicular to the table) forms the diamond's perimeter. It separates the crown and pavilion. The upper and lower girdle facets, commonly called the "halves," form the girdle's scalloped boundaries. The girdle itself may be polished or unpolished. Today it is often faceted, as it is in Figures 1, 2 and 3.
Facet Alignment
In the round brilliant cut, the crown and pavilion halves are aligned across the girdle. The tail of the crown main "kite shaped" facet lines up directly across the girdle from the sharp point of the pavilion main facet.
A "Mosaic of Mirrors"
Light, in all its colors and intensities, radiates to the observer having been reflected by the diamond from a broad range of directions. These are not just the simple reflections from the diamond's surface. These are mostly the more significant reflections of light that have entered the diamond, been internally reflected multiple times, and finally emerged from the diamond's crown to the observer's eye, as in the example light ray of Figure 4. These are the important rays when designing a diamond cut for maximum beauty.
A small percentage of light is reflected from the diamond's surface. Surface reflection is called "luster." Luster can contribute to or detract from a diamond's beauty. Surface reflections are of less importance for several reasons. Among them, they are relatively weak, produce no dispersion, and cause glare, which diminishes the diamond's beauty.
A diamond's appearance is a pattern of internal mirror reflections, which resemble tiles of a mosaic. See the mosaic tile artwork in Figure 5. Figures 6 and 7 show with computer simulation and photography the face-up appearance of an ideal cut diamond's mosaic pattern of reflections. Unlike the static pattern in Figure 5, a diamond's mirror reflections are constantly changing with even slight movement of the diamond, observer or illumination. At any moment, each individual "mirror-tile" reflects a small portion of the surrounding panorama of light to the observer's eye.
As rays of light pass from the air into a diamond, they first bend (called refraction) upon entering this more optically dense medium. Then, as in Figure 4, the rays are internally mirrored (reflected) from multiple facets. Finally, they refract out of the diamond to the viewer's eye. In common lighting conditions, about ninety percent of the light leaving the diamond that reaches the observer does so after reflecting twice from the pavilion, as in the Figure 4 example. Figure 6 outlines as a black "wire framework" the mosaic-tile pattern resulting from this double reflection in a face up view of a round brilliant cut diamond. This mosaic pattern is seen as contrasting bright and dark reflections in the actual photograph of the diamond in Figure 7.
A diamond acts as an arrangement of many small "mirror-like tiles." We can think of a diamond as a miniature mosaic consisting of many of these "mirror-tiles". Each mirror-tile is actually made up of several mirrors acting as one (a compound mirror). These several mirrors, which act in unison, are the facets from which a ray of light internally reflects along its path inside the diamond. Each mirror-tile reflects to the viewer a small portion of the panorama of light surrounding the diamond.
"Virtual facets" and "compound mirrors" are two other terms used to describe these mirror-tiles of the diamond's mosaic pattern.
Any light or dark mirror-tile in the diamond is a reflection of a small part of the surrounding panorama of illumination. Whether the eye sees a bright or dark reflection depends on whether the part being reflected from the surroundings happens to be bright or dark. Most simply stated, reflections from bright directions are seen as bright spots and reflections from dark directions result in contrasting dark areas. Compare the diamond photograph of Figure 7 to the computer generated, double-reflection mosaic of Figure 6. Notice how very close the double-reflection pattern is to the actual diamond photograph. A closer comparison reveals additional detail in the reflection pattern in the photograph. For the present, suffice it to say that this added complexity is due to further reflections. All the light does not exit the diamond after two reflections. As Gabi Tolkowsky, grandnephew of Marcel Tolkowsky, likes to say "The light is having an eternal journey inside the diamond."
Analyzing this mirror-tile pattern in well-proportioned round brilliant cuts is the key to understanding the beauty of this and all styles of diamond fashioning. We are going to analyze this mirror-mosaic pattern of reflections using two groupings. The first is a radial grouping into three concentric rings that resemble an eye - "the eye of the diamond." The second is an axial grouping like spokes in a wheel.
The Eye of the Diamond
From the diamond's center to its girdle, the reflection pattern can be seen to have three rings of light reflection resembling an eye - the "eye of the diamond". The first of these three rings is to be found if you look in the center of the table. Here you will see a small, octagonal, pie-like grouping of 8 reflections (colored blue in Figure 6.) It turns out this octagonal grouping is a reflection of the octagonal table in the pavilion main facets. This inner ring is the pupil of the diamond's eye. The remainder of the reflection pattern that is seen in the table is the middle ring and is colored green in Figure 6. Think of this as the iris. The third, or outer ring is all the reflections seen outside the table. This includes both the reflections seen in the star facets colored yellow and the surrounding reflections, shown in white. This is the "white of the eye." Pairs of star facets and their reflections, colored yellow, form "bow ties." They mark the eight-sided boundaries of both the middle ring, table and the inner ring, table reflection.
There is significance in this analogy of the three rings of reflection to an eye. The innermost ring, which is the table's reflection, dilates like the human eye's pupil when the diamond is cut with a pavilion that is steeper than ideal. When a diamond's pavilion is ideal cut, its pupil, or innermost ring of reflections will be about a third the size of the table. We will demonstrate that if this section is larger than one half the table, the balance of brilliance, fire and sparkle is upset and part of the "essence of ideal" is lost
Reflections from the Pavilion Mains and Halves
In the axial direction, the, reflection pattern mosaic separates into eight, arrow-shaped groupings, colored blue in Figure 8. These are reflections from the eight pavilion main facets. Between these "arrow-like" main reflections are 16 groups of reflections from the 8 pairs of pavilion halves. This axial grouping into reflections from the mains and those from the halves is apparent in the ideal round brilliant diamond in Figure 9, because of the sharp contrast exhibited by this particular diamond and its illumination.
The properties of reflections from each of these radial and axial groupings directly relate to the diamond's optical performance and thus to its beauty. These reflection properties include their relative sizes, distribution, and number. These properties determine the diamond's beauty and appeal.
Before proceeding, let us comment upon some details, which will be important when we take this model and understanding of the diamond to a higher plane. The interaction of light with the diamond's facets is more similar to that of a two-way mirror than it is to a simple mirror. Where a simple mirror has full reflection, a two-way mirror partially reflects and partially transmits incident light.
Total reflection only occurs inside the diamond, and then only when the light ray is incident (meets the facet) at greater than the critical angle of 24.4°, as it does in Figure 10b, where the white cone outline indicates 24.4° from the facet perpendicular. A ray of light striking a facet externally only partially reflects from the diamond's surface. Most of the light refracts (bends) into the diamond, as the entering beam does in Figure 10a. Light rays incident to an internal surface at less than the critical angle will, similarly, mostly refract out of the stone with a portion reflecting and staying inside, as in Figure 10d. This remaining light continues reflecting and refracting out of the diamond until it has completely dissipated.
Because of the diamond's dispersion property, each wavelength of light (each color) is refracted into and out of the diamond at slightly differing angles. As a result white light is dispersed, often appearing as rainbow colors called the diamond's "fire".
Surface reflections from the diamond's crown facets also contribute to the diamond's reflection pattern. These simple mirror reflections called luster can often result in undesirable glare, especially from the table. This glare reduces the sharp, high-contrast appearance of the internal reflections from the diamond. The author's opinion is that such reflections need not be considered in diamond design, as they have little to do with designing for the attributes of diamond beauty - brilliance, fire and scintillation.
Marcel Tolkowsky expressed a similar opinion about surface luster in the following excerpt from his book Diamond Design:
"In a diamond, the amount of light reflected from the surface is much smaller than that penetrating into the stone; moreover, a diamond is practically perfectly transparent, so that all the light that passes into the stone has to pass out again. This is why luster may be ignored in the working out of the correct shape for a diamond, and why any variation in the amount of light reflected from the exposed surface due to a change in that surface may be considered as negligible in the calculations."
Tolkowsky goes on to make another point central to diamond cutting for best light performance:
"The brilliancy or, as it is sometimes termed, the " fire " or the " life " of a gem thus depends entirely upon the play of light in the gem, upon the path of rays of light in the gem. If a gem is so cut or designed that every ray of light passing into it follows the best path possible for producing pleasing effects upon the eye, then the gem is perfectly cut. The whole art of the lapidary consists in proportioning his stone and disposing his facets so as to ensure this result."
