Cut registry › T-series
The standard · Diamond Design, 1919
What Marcel Tolkowsky actually calculated.
One ray, traced by hand, and a set of angles that the trade still ships a century later. Here are his numbers, what he was solving for, and what each cut in the T-series changes about them.
The short version
Tolkowsky was barely twenty, from an Antwerp cutting family, studying engineering in London. In Diamond Design he did something nobody had done: he treated the round brilliant as an optics problem and solved it with trigonometry rather than tradition.
He asked one question. Take a ray entering the table, bounce it off one pavilion facet, then the other, and send it back out through the crown. What pavilion angle reflects it twice instead of leaking it out of the bottom, and what crown angle lets it escape where an eye can see it? Diamond’s refractive index of 2.417 fixes the answer. He got 40.75° below the girdle and 34.5° above it, with a 53% table.
Those three numbers are the whole specification, and they are what every cut on this site is measured against. What he did not solve for is just as important, and it is where the T-series lives.
His numbers
Tolkowsky published proportions, not a facet map. The right-hand column is the geometry we build from them and measure as T57, so you can see where the model sits against the paper.
| Term | Diamond Design, 1919 | T57 as modelled | Note |
|---|---|---|---|
| Table | 53.0% | 53.0% | The width of the flat top, as a share of the girdle diameter. |
| Crown angle | 34.5° | 34.5° | Bezel facets. Sets where returned light exits. |
| Pavilion angle | 40.75° | 40.75° | The eight mains. The critical-angle result, and the number he is remembered for. |
| Crown height | 16.2% | 16.2% | Follows from the crown angle and table. |
| Pavilion depth | 43.1% | 43.1% | Follows from the pavilion angle. |
| Total depth | 59.3% | 59.2% | A tenth apart: the model resolves the knife edge to a finite point. |
| Girdle | Knife edge | Knife edge | He assumed a vanishing edge. No real stone has one, which is what T73 tests. |
| Culet | Point | Point | No culet facet, so 57 modelled facets rather than his 58. |
What he solved for
His model followed one ray and asked whether it came back. That makes it a calculation about white light return: how much of what goes in comes back out through the crown.
It says nothing about dispersion, because a single monochromatic ray has no colour to split. It says nothing about how the stone behaves when you tilt it, because the ray enters straight down. And it says nothing about how many separate flashes the stone produces, because that is a question about facet count, and he was solving for angles.
What that leaves
Fire, the dispersion of returned light into colour. Tilt stability, how much of the return survives when the hand moves. Scintillation, the density of distinct flashes, which on this engine is a direct function of facet count.
He got the first term close to right and left the other three on the table. A century of cutting has mostly been an argument about those three, and every cut in the T-series is a different answer to it.
The T-series
Every cut below carries Tolkowsky’s angles unchanged: 53% table, 34.5° crown, 40.75° pavilion. Only the facet pattern differs. Figures are Global in diamond, measured on one protocol, with the standard at 88.61.
| Cut | Facets | What changed | Global | vs T57 | Where the margin comes from |
|---|---|---|---|---|---|
| T57 standard | 57 | Nothing. The stone as published: 8 stars, 8 bezels, 16 upper girdles, 8 mains, 16 lower girdles, table. | 88.61 | 0.00 | The benchmark. |
| T73 | 73 | The knife edge becomes a real girdle: 16 vertical facets. | 89.85 | +1.24 | +0.85 scintillation, +0.29 tilt, +0.20 brilliance, −0.10 fire. |
| BL-Vitruvio | 185 | The same girdle band, taken to 128 facets. | 91.65 | +3.04 | +2.66 scintillation. Everything else nets to +0.38. |
| T105 | 105 | The only one that touches light-bearing surfaces: 8 new crown facets at 37°, and 40 new pavilion facets at 41.3°, 42.6° and 43.5°. | 90.55 | +1.94 | +1.85 scintillation and +1.00 brilliance, against −0.50 tilt and −0.45 fire. |
Each cut has its own page: T57, T73, T105 and BL-Vitruvio.
Reading the series
T73 and BL-Vitruvio add facets that cannot bend light. A girdle facet stands edge-on to the viewer. It is not a surface light returns through, so adding 16 of them, or 128, cannot change where the light goes. Run the census and that is exactly what you see: across T57, T73 and BL-Vitruvio brilliance moves 0.6 of a point, fire 0.5 and leak 0.3.
What does move is scintillation, from 71.8 to 80.3 to 98.4, because the engine counts flashes and flashes track facet count. Of BL-Vitruvio’s 3.04-point margin, 2.66 is that one term. It is the cleanest demonstration on this site of how much of a cut’s score can be bought without improving the optics at all.
The girdle is not entirely inert, and T73 shows why. Adding a band physically separates crown from pavilion, taking total depth from 59.2% to 61.7%. That small change is worth +0.29 of tilt stability and +0.20 of brilliance, and costs 0.10 of fire. The knife edge Tolkowsky assumed is slightly worse than a girdle a cutter can actually finish, which is a pleasing result for a man who never had to polish one.
T105 is the only member that changes the optics, because it is the only one that adds crown and pavilion facets at new angles. It carries Gabi Tolkowsky’s 105-facet pattern, whose own proportions were never published, laid over the family’s 1919 angles. It returns 95.2% of light usefully and leaks 1.38%: the brightest and cleanest pair of figures in the series.
It pays for that in movement. Those extra pavilion families sit steeper than the mains, at up to 43.5°, and steep pavilions lose light sooner as the stone tilts. T105 holds 87.3 under a 20° tilt against the standard’s 90.4, the weakest tilt figure in the series. More light dead-on, less light as the hand turns.
Facet count supplies +1.85 of its margin and brilliance +1.00, with fire and tilt handing back nearly a point between them. It is still the only cut in the series where any of the result is something Tolkowsky would have called an improvement to the design rather than to the finish.
Read together, the four cuts price two different things. T73 and BL-Vitruvio price a facet count. T105 prices an angle change. Only the second one is a design decision in the sense Tolkowsky would have recognised.
Why the line is here
Every benchmark on a maker’s own website has the same problem: the maker chose it. T57 is the exception we can point at. Tolkowsky published it, the trade adopted it, and we did not get to pick the numbers.
The cost is that the bar is not high. T57 carries the fewest facets of anything we publish, so almost any modern cut clears it, and a good deal of what clears it does so on scintillation alone. That is why every margin on this site is published term by term rather than as a single score. A cut at 145 or 201 facets banks roughly 2.4 to 2.7 points against this standard before its optics are assessed at all. Subtract that, and what is left is the part the cutter actually designed.
The test
Diamond Design derives the pavilion from the material: the refractive index gives the critical angle, the critical angle gives total internal reflection, and the two-bounce path gives the angle. That is a strong claim, and it is testable. We swept the pavilion angle across three refractive indices on his own architecture, then held the index fixed and changed the facet layout instead.
Sweeping the pavilion on his own 57-facet architecture: cubic zirconia at n 2.16 wants 41.25°, diamond at 2.417 wants 40.25°, moissanite at 2.65 wants 38.25°. Monotonic, in the direction the book predicts, and worth 3.0° across the range. The index sets the angle, as he says it does.
In diamond the plateau runs 40.00° to 40.75°, all of it inside the engine’s resolution of the maximum. His 40.75° sits at the top of that plateau, 0.05 of a point off the best value the sweep can find. Solved by hand in 1919, and a Monte-Carlo sweep a century later cannot beat it on his own stone.
Hold the index at 2.417 and take the pavilion from 57 facets to 145. The plateau moves to 41.00° to 41.50°. That is about 1°, roughly a third of what the index is worth. The critical angle itself never moves: 24.44° in diamond, and every cut in this registry sits 16 to 17° above it.
The index decides whether each bounce reflects. The two-bounce path decides where the ray goes afterwards, and that path assumes a pavilion of opposed mains meeting at a culet. Replace it with several tiers and the second bounce lands on a differently inclined facet. Measured, the leak penalty for steepening roughly halves, and that is what buys the extra degree.
Global holds its value across 0.75° of pavilion but across 2.5° of crown, from 33.0° to 35.5°. The pavilion answers to the critical angle; the crown sits downstream of whatever the pavilion has already done with the light. That asymmetry is why the modern cuts here keep his pavilion and rewrite his crown.
On pure light return this engine puts the 57-facet diamond optimum at 39.25°, not 40.75°. His figure lands on our Global plateau, which also rewards fire and tilt, terms a single monochromatic ray cannot see. We reproduce his answer by a different route, not his calculation. Sweeps run at four times the standard ray count.
Method
Monte-Carlo on exact facet geometry: 3,200 rays face-up plus 2,200 at 20° tilt, cosine-weighted from a 7° near-vertical cone. Brilliance is the light that exits through the crown within 64° of vertical.
Counts here come from the inclination census of each cut’s own plane set, grouped at 0.45°. T57 reads 6 families, T73 seven, BL-Vitruvio seven, T105 ten.
Proportions as published in Diamond Design: A Study of the Reflection and Refraction of Light in a Diamond, Marcel Tolkowsky, 1919.
Engine estimates, ± 2 points, not lab-certified grades. Symmetry reads 100 for every cut because each is modelled from exact geometry, so it cancels out of every comparison on this site.