Charles Moon told me that on his visit to Australia he found that some of the top cutters there did not understand what all those symbols on the design sheet, like L/W, P/W etc. stood for, and when he tried to explain that they were ratios, they did not know what ratios were. I expect that we in the U.S. are not so very different, and that there are quite a few cutters here in the same boat, although they may not acknowledge it to anyone. In seems that quite a few people, including college graduates, do not understand ratios, even though they run into them every time they pay the sales tax on items they purchase. If your state sales tax is 6%, it means you pay six cents tax per every hundred cents of the item’s base cost. Percent means per hundred. Other words using “cent” to mean a hundred or a hundredth are “century”, “centennial”, centimeter, and centigrade. In money the dollar is the basic unit, to which all other amounts are compared.
If you have learned to hate math in school, or find that abstract symbols like L, W, X, Y are slippery, then bear with me. For cutting stones there are only a few such symbols, and they occur over and over. If you can learn them, they can furnish you with a lot of information you would otherwise be missing.
In general math and geometry the basic unit is 1 (one). If board A is twice as long as board B, we say that A:B is 2:1, which is a short way of saying that A is to B as 2 is to 1. More commonly this ratio is expressed as A/B=2. We could write A/B=2/1, but since anything divided by 1 is itself, and the general math unit is always 1 of something, it is not necessary to indicate anything divided by 1, so it is left out. L/W=1.5 means that the stone’s length is one and one-half times longer than its width. On faceted stones and diagrams for them, the width is the basic unit to which all other measurements are compared. On older designs, designers used to compare the height of the crown, or pavilion, or girdle thickness to the total height of the stone, but since the stone’s total height cannot be known until the stone is finished and it is too late to make any changes, that practice has been abandoned.
W, the width, for any design, is wherever it is shown on the design sheet, but it is usually the shorter distance across the stone as shown in the plan view, and usually passes through the stone’s center. L, or length, is, whenever possible, at right angles to W, and also passes through the center of the stone. We could easily categorize people by their H/G, which is their Height divided by their Girth–their circumference around the waistline. So if you were 36 inches around the middle and six feet tall your HI G would be 2.0. On gemstones the most critical dimensions are the vertical ones–C/W, P/W, and H/W–because we are usually trying to get the largest possible plan-view size stone from a piece of rough, and in doing so we are limited by the height requirements. So having the height ratios available, it is wise to use them. Cutting by pure guesswork will cost you money in loss of stone size and value.
Trigonometry is all about ratios. “Tri” means three, gon means angle, and metry means measure, but although trigonometry is the measurement of the three angles of a triangle, it is more frequently the ratios of the sides of a triangle that we deal with. The sine, cosine, and tangent are nothing more than the ratios of the lengths of one side of the triangle to another side.
If you are looking at the crown main facet on a SRB stone from a sideways position, along the plane of the facet so that it appears as a straight line, then the height of the facet (H) divided by its base (B) is the tangent of that facets angle. If its angle is 45 degrees then the height and base are the same, so the tangent would be 1.0. If the angle is greater than 45 the tangent will be greater than 1, and if the angle is less that 45 the tangent will be less than 1. When you change all the angles of a crown by the tangent-ratio formula, you are changing only the height of each facet in relation to its base; the base of each facet remains the same, so the plan view appearance is not changed.
About the stone’s coordinate system: All design drawings done in the U.S. are based on the Rectangular, or Cartesian, system of coordinates, where the X axis runs horizontally across the paper, and the Y axis runs vertically down the paper. (Piet Van Zanten tells me that in the Netherlands the axes are just the opposite. I guess any people who build dikes and defy the Atlantic ocean must be a bit contrary). The two axes are at 90 degrees to each other, and where they cross is the center of the system, also called the “Point of Origin”, where X=0, Y=0, and Z=O, which corresponds to the center of the stone. In a plan view the Z axis is coming straight up at you from the center of the picture.
The SLOPE of a line on the plan view is always relative to the X axis. The SLOPE of every girdle facet and of every facet around a flat table is directly the result of the index number used to cut it. If you’re using a 96 gear, for example, your DPT (degrees per tooth) is 360/96 or 3.75 degrees, and if you cut a star facet at index 4, the slope of the line made with the table will be 4 X 3.75, or 15 degrees. If you cut a star facet at index 96 (same as zero) your line at the table will be zero degrees (parallel to the X axis), and if you cut a star facet at index 24 your line at the table will be 90 degrees, or parallel to the Y axis.
The terms “Facet” and “Plane” are often used interchangeably, but strictly speaking a Facet is a very limited portion of the plane it lies on. Planes, like Lines, have limitless extension. It takes three planes to make a point; two planes make a line. The direction, think SLOPE, that any line runs on your stone is the result of the Angles of the two adjacent planes and their index numbers. Suppose you are cutting a rectangular stone and making a scissor crown. You have cut all the girdle break facets around the stone, and are ready to cut the wing facets. If your gear is 64 you will use index 1 or 2, but if it is 96 you will use index 2 or 3. Using a 96 gear, if you use index 3 you must expect to make the wing facet’s angle lower to get the slope of the line where you want it, than you would if you used index 2. The rule is: the more index separation you have between two facets, the greater the angle difference must be to get the same slope for the line between them.
At some time or another you have all seen ECED on a design sheet. This means that all girdle facets are Equal Center to Edge Distance, which is the shortest distance from the center of the stone to the girdle facet, and which means that all girdle facets can be cut at a single mast height setting.
Lastly, on my designs, you may see a little circle around a line connected to an alphabetical letter. This symbol is called a “lassoo” and the letter it is connected to is part of a ratio. A/W, for example, is the ratio between the length of the lassooed line ON the finished stone, to the stone’s Width. This is given in certain designs where there is no point on the stone to cut a facet to, and you would otherwise have to guess at your depth of cut.