How we measure

Principles

• Geometry first. Every stone is its facets and nothing else. The figures describe a cut’s design, not any real stone, and none of them is a grading report.
• Deterministic. Nothing is random between visits. Where we sample, the samples come from a fixed grid or a fixed seed, so the same cut always gives the same number.
• Disclosed simplifications. Each tool leaves some physics out, deliberately and for speed. Each section below says what, and What we do not model collects them in one place.
• Nothing is tuned to a result. Constants come from published material data, the geometry, or the calibration stated next to them.

Four light measures, side by side

The site traces light in four ways. They answer different questions, so their numbers are not interchangeable. The T73 round brilliant in diamond, for example, reads 93 for brilliance on the Cut Performance Wheel and 54% in the Cut Grading Tool: the first counts the share of near-vertical light returned, the second the energy returned from an evenly lit dome, head shadow included.

All four treat total internal reflection as lossless, and none carries polarisation from one bounce to the next.

The 3D models

Geometry and units

Each model is a closed, watertight solid of flat facets, in millimetres, with the table facing +Y and the girdle plane through the origin. Every cut except the heart is convex, so the tools describe it as an intersection of half-spaces, one per facet:

with n_i the facet’s unit outward normal and d_i its distance from the centre. A ray enters at the last plane it crosses going in and leaves at the first plane it meets going out. The heart is traced triangle by triangle instead.

The facet count on a cut page is the number of planar faces of the STEP solid. When the cutting design counts differently (a turned girdle, for instance, is one surface in the solid but many facets on the machine), the page gives both.

Volume and weight

Weight in carats follows from volume V in mm³ and specific gravity ρ in g/cm³:

because a carat is 0.2 g and a cubic centimetre is 1,000 mm³. Every weight on a cut page uses this. At a given shape, weight scales with the cube of size, so the simulator scales each model by

to reach the carat weight you choose, where V_ref is the volume at the model’s reference size. The build measures every model’s volume from its mesh each time, by the divergence theorem over its triangles,

and measures its size, length to width and depth from the same mesh, with the stone turned level in plan. Rounds keep their diameter across the girdle corners. These are the figures the cut pages print. The build also checks them against the measurements of the STEP solid, and stops if a mesh is not closed, with every edge shared by exactly two triangles.

Proportions

• Total depth is a percentage of the width, the shorter side in plan.
• Length to width is the ratio of the plan’s bounding box, to three decimals.
• Table is the table’s extent over the width. For rounds it is measured across the table’s corners: the largest of 64 directions.
• Crown and pavilion angles are the cutting design’s own main tiers. Where a design names none, they come from the model’s specification or, failing that, the area-weighted most common facet angle, in 0.5° bins, weighting each facet by its projected area A |n_y|. Angles are measured from the girdle plane: a facet’s angle is arccos |n_y|.
• Girdle facets are those within about 1° of vertical. Their thickness is sampled at every vertex column, and the page gives its minimum and maximum.

Cutting-file accuracy

Each cut page states how closely the cutting file matches the 3D model. Every facet in the .asc file, at angle a and index i on a gear of g teeth, becomes a normal:

We then turn the model about its axis in steps of a quarter index, in both handedness, and keep the alignment with the smallest area-weighted mean angle between each model facet and its nearest file facet. The page reports the largest angle and the mean, with the girdle given on its own. Facets smaller than 0.02% of the largest facet are left out as slivers. The check compares facet directions only, not where facets meet.

Gemstone constants

The gemstone pages and the simulator use these values. For stones with a range, the refractive index used is the lower end, which gives the stricter angle limits. Dispersion is the B to G interval (686.7 to 430.8 nm).

The Brightness & Tilt Analysis and the light census behind the Performance Wheel come from our research engine and use its constants, which differ slightly from the table above:

The fire spread is about 0.86 of the published dispersion. F₀ is the reflectance at normal incidence, ((n−1)/(n+1))².

Critical angle

Light inside the stone meeting a facet at more than the critical angle from its normal reflects totally:

Each gemstone page also gives the shallowest pavilion at which a round returns head-on light. In the meridian section, light arriving straight down meets the first pavilion main at the pavilion angle p, the opposite main at 180° − 3p, and the table at |180° − 4p|. It must reflect at the first and leave at the table:

That is 38.9° for diamond, 36.4° for ruby and sapphire, and the critical angle itself for emerald and amethyst. Faceters usually cut a degree or two steeper, for light that arrives off the axis.

Head-on light on cut pages

“Pavilion reflecting head-on light” is the share of the pavilion, by projected area, whose facets lie at or above the critical angle from the girdle plane. Light arriving straight down meets each such facet at at least θ_c and reflects. It is a first-reflection test only. The verdict reads “Holds head-on light” at 99% or more, “Minor windowing” from 90%, and “Steepen the pavilion” below that. A culet flat counts as not reflecting, and a cut with no table is not measured.

3D Gemstone Simulator

The simulator renders a stone the way a camera would see it, in a lit room. It is for seeing, not scoring: the only figures it reports are the Light Return shares, the cut grade proportions and the size.

The trace

Each pixel is ray-traced on the graphics card. The camera is a pinhole about five stone widths away. At the entry facet, a Fresnel fraction F reflects off the crown and the rest refracts in. Inside, the ray bounces up to 14 times. At each facet where it can escape, the transmitted fraction leaves and the reflected fraction carries on, so one branch is followed and the energy is split, not sampled. The path stops when less than 0.3% of its energy remains.

While the stone is still, the image refines over up to 64 samples per pixel, placed by the R2 low-discrepancy sequence. There is no random noise.

The room is a function of direction that turns with the camera: soft ambient light, a six-segment ring, lamps above and below, and small spot lights, in ten scenes from studio to candlelight. A head shadow blacks out the light within 6° of the line of sight and fades in by 10.5°. Colour temperature follows Tanner Helland’s fit, with 22% of the shift applied, because the eye adapts.

Fresnel and dispersion

Reflectance is the full unpolarised Fresnel equation, the mean of the two polarisations at each surface:

Fire comes from refracting the ray out three times, once per colour channel:

where D is the published B to G dispersion. Red and blue display primaries sit about 0.6 of that interval apart. A channel that reflects totally contributes nothing on that exit. Dispersion is applied at the exit facet only; inside the stone the three channels share one path.

Colour, clarity and fluorescence

Body colour is Beer–Lambert absorption per channel along each internal segment of length ℓ:

For the D to Z scale, σ grows as (grade/22)^1.25, from zero at D. Path lengths are in model units, so colour depth does not change with the carat weight you choose. Clarity grades place a fixed set of crystals, pinpoints and feathers, sized per grade, from a seed tied to the cut. Crystals and pinpoints end a path; feathers reflect part of it. Fluorescence adds a glow proportional to path length (up to 3 model units), under the UV scene only.

Polish is applied after the trace, as a loss of contrast and saturation. It does not scatter rays. Symmetry grades tilt each crown and pavilion facet by up to 0.25°, 0.6°, 1.2° or 2.0° in a seeded direction; the table and culet stay put.

Light Return

The Light Return map is traced face up and orthographic, with no Fresnel, no dispersion and no absorption, so it depends on the geometry alone. Each ray is classed by the direction its light came from, measured as elevation above the girdle plane:

Contrast (blue)

75° and above: within 15° of the line of sight, where your head is

Direct (red)

45° to 75°

Indirect (green)

below 45°

Leakage (black)

out through the pavilion, below the horizon, or still inside after 18 bounces

The four shares are the share of face-up pixels in each class, from a 160 × 160 render with one ray per pixel, rounded to whole per cent.

Cut grade

The simulator’s cut grade shows how a worse make of the same design looks. Each grade moves the design by a fixed schedule, and the weight is held constant by rescaling the stone:

Crown and pavilion facets turn about their girdle edges. The table and culet are rebuilt until their size, as the square root of the area ratio, meets the target. Volume is integrated over 512 horizontal slices by the trapezoid rule. The grade describes the make; it is not a measure of light return. House cuts, the heart and the Platonic solids are shown as designed.

Size

Measurements are the model’s reference dimensions multiplied by the scale factor k from Volume and weight, for the gemstone and carat weight you choose.

Brightness & Tilt Analysis

The Brightness & Tilt Analysis measures how much of a dome’s light a cut returns to the viewer, and how that changes as the stone tilts. It uses the same tracing for its pictures and its figures, in two separate passes: the graphics card draws the four windows, and the figures are computed on the processor.

The four light models

Each path is followed back from the eye, through the stone, to the direction its light came from. Only that direction’s elevation e above the horizon matters. The three dome models give it a weight:

and W = 0 below the horizon or within h degrees of the zenith, where h is the head shadow you set. ISO, COS and SC2 follow the definitions in Robert W. Strickland’s GemRay user guide; the implementation is our own. The fourth window, Light Return, uses the zones in Light Return. None of the four depends on a lighting scene: they are properties of the geometry.

The figures

Rays start on a 132 × 132 grid, straight down, one per cell centre. Of those that hit the stone, each contributes the light reflected off the crown plus the light that passes through:

R is Schlick’s approximation, R(θ) = F₀ + (1 − F₀)(1 − cos θ)⁵, at the entry facet and at the exit facet. Bounces in between are total internal reflections and lose nothing. A figure of 100% would mean every ray returned all its light at full weight. The table-only figure is the same sum over the rays that enter through the table.

Leakage counts paths, not energy: the share of rays that leave below the horizon, find no exit, or are still inside after 12 bounces.

It does not depend on the head shadow. The “vs T57” column is a relative difference, 100 (a/b − 1), and for leakage a difference in percentage points.

Because partial internal reflections are not counted, these figures sit above those of a full physical simulation, and the dip near face-up is shallower than GemRay shows.

Tilt

The stone rotates about one horizontal axis while the dome and the viewer stay fixed, as on a hand. The chart traces 13 tilts from −30° to 30° in 5° steps, on an 84 × 84 grid. The windows draw a reduced set of planes for T105 and DG-Vitruvio, because the graphics card holds 96 at most; the figures always use the full set. Only cuts whose planes agree with their STEP solid to within 1 percentage point on every light model are included.

Cut Grading Tool

The Cut Grading Tool grades a round brilliant from its proportions. It is for education: not a grading standard, and not a substitute for a laboratory report.

The grade

The grade is read from GIA’s published Cut Grade Estimation Tables for Standard Round Brilliant Cut Diamonds (revision 12-09): 18 tables, one per table size from 50% to 67%, each a grid of 37 crown angles (22.0° to 40.0°, in 0.5° steps) by 22 pavilion angles (38.8° to 43.0°, in 0.2° steps). That makes 14,652 graded cells.

We read the tables from the published document’s vector data, not by eye. Each cell’s grade was taken twice, from its fill colour and from its printed label, and the two agree on every cell. A separate pass over a 300 dpi image of each page found no differing cell.

Your inputs are snapped to the reporting steps (table 1%, crown 0.5°, pavilion 0.2°, star and lower half 5%, girdle 0.5%), rounding half up. There is no interpolation. The grade is the worst of:

• the grid cell for the table, crown and pavilion angle;
• the limits on each proportion, crown height and total depth, girdle, star and lower half;
• culet, polish, symmetry, painting and digging.

The breakdown lists every component at that worst grade. Outside the tables the grade is the worse of the nearest cell and the individual limits. The tables assume star 55%, lower half 80% and girdle 3.0%; other values can shift the grade in ways the limits do not capture, and the tool says so. Digital Gemology is not affiliated with GIA, and the tool’s result is not a GIA grade.

Derived proportions

With the diameter set to 100 and the table measured corner to corner:

for table t, crown angle c, pavilion angle p and girdle g, all in per cent of the diameter. The −0.25 offset is calibrated. Of seven candidate formulas, it is the only one that reproduces both depths worked in the published document (62.8% at a 3.0% girdle and 63.3% at 3.5%) without contradicting the grids. Crown height is limited at its value rounded to 0.5, the only rounding with no conflicts. The T73 as cut (53 / 34.5° / 40.75° / 2.5%) gives 16.15 + 2.5 + 43.16 − 0.25 = 61.6%.

Light maps

There are 814 light maps, one for each crown and pavilion cell at 53% table, plus the T73 as cut. All are traced on the T73 (star 50%, lower half 77%, girdle 2.5%, no culet), with only the crown and pavilion angles changed. They use the zones in Light Return, face up and orthographic, at index 2.417, with no Fresnel, no dispersion and no head shadow. Each is traced at 1024 × 1024 and stored at 512 × 512 without loss. The shares are fractions of the face-up area, with rounding remainders assigned so the four sum to exactly 100.

Brilliance, fire and scintillation

Brilliance is traced. It is the ISO figure from the Brightness & Tilt Analysis, with a 23° head shadow, averaged over 13 tilts:

over the tilts τ = −30°, −25°, …, 30°, each on an 84 × 84 grid. The even dome and the 23° head shadow follow Hemphill et al. (1998) and Moses et al. (2004), in Gems & Gemology; 23° was not fitted to the grade tables. Values run from 41.7% to 65.1%, with a sampling noise of about ±0.2 points. Brilliance is traced on the T73 base for every cell of all 18 tables, so it does not follow the star, lower half, girdle or culet controls.

Fire and scintillation in this tool are modelled, not traced. They are smooth functions of the crown angle c and pavilion angle p, peaking at 100:

They show where each effect is strongest across the grid; the numbers are not measurements.

Cut Performance Wheel and cut-page scores

The Cut Performance Wheel, the chart on every cut page and the top ten on every gemstone page share one set of scores, computed once by our face-up light census and stored unrounded. Pages round half up for display.

The light census

Each cut is scaled to unit size in plan. For each gemstone, 3,200 rays from a fixed seed fall on the stone from above: their directions are cosine-weighted within 0.12 rad (6.9°) of vertical, and their starting points cover a disc larger than the stone. Rays that miss are not counted, and every ray that enters is fully transmitted: there is no Fresnel loss. Inside, a ray reflects totally or leaves through the first facet that lets it out, for up to 12 bounces. Each entering ray ends in one class:

Useful

out through the crown, after at least one pavilion reflection, within 64° of vertical

Blink

out through the crown without touching the pavilion

Leak

out through the pavilion, or through the crown heading down

Grazing

out through the girdle

Lost

out of the viewing cone, or still inside after 12 bounces

A useful ray also counts as fire when refracting it out at n − Δ and at n + Δ sends the two colours more than 0.6° apart. The census runs twice: face up, and tilted 20° about x (2,200 rays).

The eight axes

With N the rays that enter, f the cut’s active facets and depth in per cent:

• Brilliance and fire are traced. Fire is a share of the useful light, not of all light, so a dim stone can still score high on fire.
• Scintillation is not traced. It is a saturating function of the number of facets that reach the surface; buried planes do not count. It is the same in every gemstone.
• Tilt is retention: how much of its face-up light a cut keeps at 20°. It is capped at 100, so a dim cut cannot score well for having little to lose.
• Symmetry groups the pavilion facets into angle families, split wherever consecutive angles differ by more than 0.45°. σ is the size-weighted mean, over families, of the standard deviation of their angles plus 57.3 times the relative standard deviation of their distances from the centre, both in degrees. A multi-angle pavilion is a design, not an asymmetry.
• Spread is face-up size per carat against a 61.5% deep stone. Weight goes with the cube of the diameter, hence the cube root.
• Shape-norm compares brilliance with a ceiling for the cut’s family: 93 for round brilliants, 86 for fancy brilliants, 76 for step cuts and 64 for the rest.

Spread and Shape-norm are the only axes that can pass 100. The wheel draws them on the 100 ring and the list gives the true figure. Every other axis is an absolute figure, not scaled to any reference. The T73 round brilliant in the same gemstone is the comparison: the dashed outline on the wheel and the thin mark on each cut-page bar. In diamond it reads 93 brilliance, 78 fire and 80 scintillation.

House cuts

DG-M5 and DG-Rafaello use research plane sets that match our models to within 0.09°. DG-Miliano, DG-Elongated Hexagon and DG-Octagonal Kite are measured on our own models, placed as the census places every stone. Before those scores are accepted, six cuts with known figures are re-measured the same way, and each must land within 0.5 of its Global score. Three cuts are not measured: the heart, because it is concave, and DG-Cabochon and the Elongated Hexagonal Bipyramid, because they have no table for face-up light to enter.

Rankings

• Cut pages tag “Highest brilliance” and “Highest fire” on the gemstones with the highest rounded figure, and “Best overall” on the highest brilliance plus fire. Scintillation is left out there because it is the same in every gemstone. Ties tag every stone.
• Gemstone pages rank cuts by brilliance + fire + scintillation, rounded. Equal totals share a place (1, 2, 2, 4), and every cut placed tenth or better is shown, so a list can run to eleven. Brilliants and jewellery cuts are ranked separately.

What we do not model

• Polarisation. Fresnel losses are averaged over polarisation at each surface. No polarisation state is carried between bounces.
• Birefringence and pleochroism. Every stone is traced with a single index. Birefringence is listed, not rendered, so moissanite’s doubling does not appear.
• Full spectral dispersion. The simulator uses three wavelengths and the census two, both at the exit facet only. The Brightness & Tilt Analysis and the Cut Grading Tool’s brilliance use none.
• Partial internal reflection in every measure except the simulator. Internal total reflection is lossless everywhere.
• Scattering and surface roughness. Facets are perfectly flat and polished; the simulator’s polish grades act on the finished image.
• The observer. Apart from a head-shadow cone, no body, camera or setting blocks the light. The analytical measures view from infinitely far away.
• Real stones. Every figure describes the design. A cut stone’s figures will differ with its make, its material and its setting.

Check our work

The tools run in your browser, so their code is yours to read. The census is deterministic: the same seed gives the same score on every run, and our tests re-run it and require the stored scores to match exactly. If you find a formula that is wrong, a constant you would source differently, or a figure that does not follow from this page, write to us. We will correct it and say what changed.