Abstract
Diamond is said to sparkle because its refractive index is high. What this paper shows is that there is a window of pavilion angles at which light entering from straight above is returned upward by two total reflections, and that this window opens at refractive index $\sqrt2$, is widest at $2.304765$, and narrows beyond. No new theorem or law is claimed.
First, five conditions fix the window——total reflection at the right face, travelling towards the left face, total reflection at the left face, leaving through the top, and not tilting too far to the other side on the way out (Section 2).
Second, the window opens at refractive index $\sqrt2$——below a critical angle of $45$ degrees, refractive index $\sqrt2=\mathbf{1.414214}$, there is no window. Water ($1.333$) has none; glass ($1.5$) has $\mathbf{4.2529}$ degrees and quartz ($1.544$) $6.1786$ degrees (Section 3).
Third, and this is the core. The window is widest, $\mathbf{12.857143}$ degrees, at refractive index $\mathbf{2.304765}$ (critical angle $180/7$ degrees), and beyond that narrows to half the critical angle——diamond’s $\mathbf{12.2199}$ degrees is slightly narrower than cubic zirconia’s $\mathbf{12.6902}$ degrees (Section 3).
Fourth, tracing rays one by one, the window’s edges agree with the closed form to within $0.0005$ degrees——but the closed form first written left out the fifth condition, and diamond’s upper edge was off by $0.7435$ degrees; checking by two routes found it (Section 4).
Fifth, Tolkowsky’s $40.75$ degrees lies in diamond’s window and not in glass’s——the same angle returns light or not according to the refractive index (Section 4).
Sixth, the separator is which condition fixes the edges of the window——at low index the total-reflection conditions fix the edges, at high index the conditions for leaving through the top, and where they exchange the window is widest (Section 5).
Keywords: gem cutting, total internal reflection, critical angle, refractive index, diamond.
Scope (what is not claimed)
No new theorem or law is claimed——total reflection, the critical angle, and Tolkowsky’s way of following rays in a cross-section are all standard. Only light entering from straight above is considered——oblique light, refraction at the crown facets, the variation of index with colour (fire) and absorption are not treated. Rays are followed by direction only, in a two-dimensional cross-section——which face a ray meets is decided by its direction; face sizes and points of incidence are not included. Refractive indices are representative. No gem is assessed and no cut is recommended. Relation to earlier papers: Paper 474 showed that on nights when distant sounds carry, the air above is warm and sound bends back down——the total reflection that returns sound in air and the total reflection that returns light in a stone come from the same critical-angle formula $\sin\theta_c=1/n$; as two total reflections met in separate places, they share a root in the sense of Paper 300. Paper 300 showed that sharing a root is decidable——the three cases of the window width here are cases of one formula, and are not called sharing a root. What is added is writing the window with five conditions and its width in closed form over three ranges of index, finding the index $\sqrt2$ at which it opens and $2.304765$ at which it is widest, checking the closed form by ray tracing and finding what the first form left out, and placing the separator on which condition fixes the edges.
1. Notation and the Question
Setting: take a cross-section of a gem with a horizontal top face (the table) and two lower faces (the pavilion) tilted at angle $\theta$. Light entering from straight above reaches the viewer if it reflects at the right face, then at the left face, and leaves through the top. With critical angle $\theta_c$ ($\sin\theta_c=1/n$), the conditions are
$$\theta>\theta_c,\qquad \theta>30^\circ,\qquad 180^\circ-3\theta>\theta_c,\qquad 180^\circ-4\theta<\theta_c,\qquad 4\theta-180^\circ<\theta_c$$(total reflection at the right face, heading for the left face, total reflection at the left face, leaving through the top, not tilting too far to the other side).
Observation 1.1 diamond is said to sparkle because its refractive index is high. The higher the index, the smaller the critical angle and the easier total reflection becomes——whether a higher index always returns light more readily is another question.
Question: how does the window of pavilion angles that return light upward change with refractive index?
2. Five Conditions
Observation 2.1 two of the five conditions fix the edges of the window. The lower edge is the later of “total reflection at the right face” and “leaving through the top”; the upper edge is the earlier of “total reflection at the left face” and “not tilting too far”——the $30$-degree condition never binds within the window.
3. How the Window Opens
Computation 3.1: the window for six materials, from the closed form and from ray tracing in steps of $0.0005$ degrees.
| material | index | critical angle | window | width |
|---|---|---|---|---|
| diamond | $2.417$ | $24.4397$ deg | $38.8901$--$51.1099$ deg | $\mathbf{12.2199}$ deg |
| cubic zirconia | $2.15$ | $27.7177$ deg | $38.0706$--$50.7608$ deg | $\mathbf{12.6902}$ deg |
| sapphire | $1.77$ | $34.4003$ deg | $36.3999$--$48.5332$ deg | $12.1333$ deg |
| quartz | $1.544$ | $40.3660$ deg | $40.3660$--$46.5447$ deg | $6.1786$ deg |
| glass | $1.5$ | $41.8103$ deg | $41.8103$--$46.0632$ deg | $\mathbf{4.2529}$ deg |
| water | $1.333$ | $48.6066$ deg | none | $0$ |
The width has a closed form in three ranges of index: $(180-4\theta_c)/3$ for $\sqrt2<n<1.701302$ (critical angle $36$ to $45$ degrees), $(180-\theta_c)/12$ for $1.701302<n<2.304765$ (critical angle $180/7$ to $36$ degrees), and $\theta_c/2$ for $n>2.304765$. At index $1.40$ and $1.41$ the width is $0$, at $1.42$ it is $0.3107$ degrees and at $1.45$ $1.8629$ degrees.
Observation 3.2 the window opens at refractive index $\sqrt2=\mathbf{1.414214}$. With a critical angle above $45$ degrees, an angle that gives total reflection at the right face cannot give it at the left——inside water, light cannot be returned upward by two reflections.
Observation 3.3 this is the core. The window is widest, $\mathbf{12.857143}$ degrees, at refractive index $\mathbf{2.304765}$ (critical angle $180/7=25.714286$ degrees), and beyond that narrows to half the critical angle——with too high an index, light returning upward after two reflections tilts too far to the other side at the table and is totally reflected there. Diamond’s $12.2199$ degrees is slightly narrower than cubic zirconia’s $12.6902$ degrees.
4. Tracing Rays
Computation 4.1: as a route separate from the closed form, the direction of light was reflected face by face, and the window’s edges read in steps of $0.0005$ degrees. For all six materials the traced edges agreed with the closed form to within $0.0005$ degrees.
Observation 4.2 the closed form first written left out the fifth condition. Setting the upper edge by “total reflection at the left face” alone, $(180-\theta_c)/3=51.8534$ degrees, put diamond’s upper edge $0.7435$ degrees above the traced $51.1099$ degrees——the condition $4\theta-180<\theta_c$, that light after two reflections must not tilt too far to the other side to leave, was missing. Without checking by two routes, diamond’s window would have been written $0.7435$ degrees too wide.
Observation 4.3 Tolkowsky’s $40.75$ degrees lies in diamond’s window and not in glass’s. Glass’s window begins at $41.8103$ degrees, so at $40.75$ degrees there is no total reflection at the right face——the same angle does not return light at a different index. At $45$ degrees the returned light leaves straight up (tilt $0$ degrees).
Observation 4.4 a high index gathers incoming light close to the vertical. Light entering from air is tilted inside by no more than the critical angle: $24.4397$ degrees in diamond, $41.8103$ degrees in glass——this paper looked only at light from straight above, but in diamond oblique light too is bent close to the vertical.
5. The Separator
Observation 5.1 the separator is which condition fixes the edges of the window. At low index the lower edge is fixed by total reflection at the right face and the upper by total reflection at the left; at high index both edges are fixed by the conditions for leaving through the top——where they exchange (index $2.304765$) the window is widest. “A higher index returns light more readily” holds only up to there.
Observation 5.2 one root with Paper 474, in the sense of Paper 300. It showed that on nights when distant sounds carry, the air above is warm and sound bends back——the total reflection that returns sound in air and the one that returns light in a stone are two met in separate places, yet come from the same critical-angle formula $\sin\theta_c=1/n$.
Observation 5.3 what is not said. No gem is assessed and no cut recommended; oblique light, crown refraction, the variation of index with colour and absorption are not included——what is shown is that the window of angles that return light from above does not widen steadily with index, but is widest at a certain index.
6. Roll Call and Closing
- five conditions fix the window (Notation): two of them set its edges.
- the window opens at index $\sqrt2=\mathbf{1.414214}$ (Observation 3.2): water has none.
- glass has $\mathbf{4.2529}$ degrees (Computation 3.1): quartz $6.1786$.
- widest, $\mathbf{12.857143}$ degrees, at index $\mathbf{2.304765}$ (Observation 3.3): the core.
- diamond $\mathbf{12.2199}$ degrees, cubic zirconia $\mathbf{12.6902}$ degrees (Computation 3.1): diamond is narrower.
- the width in closed form over three ranges (Computation 3.1): $(180-4\theta_c)/3$, $(180-\theta_c)/12$, $\theta_c/2$.
- tracing and closed form agree to $0.0005$ degrees (Computation 4.1): two routes.
- the first form left out the fifth condition, off by $0.7435$ degrees (Observation 4.2): found by tracing.
- Tolkowsky’s $40.75$ degrees lies in diamond’s window (Observation 4.3): not in glass’s.
- the separator is which condition fixes the edges (Observation 5.1): sharing a root with the sound of Paper 474.
Closing: Diamond is said to sparkle because its refractive index is high. There is a window of pavilion angles that return light from straight above by two total reflections——it is fixed by five conditions: total reflection at the right face, heading for the left face, total reflection at the left face, leaving through the top, and not tilting too far. Below refractive index $\sqrt2=\mathbf{1.414214}$ there is no window, and in water light cannot be returned by two reflections; glass has a window of $\mathbf{4.2529}$ degrees. The window is widest, $\mathbf{12.857143}$ degrees, at index $\mathbf{2.304765}$, and narrows beyond——diamond’s $\mathbf{12.2199}$ degrees is slightly narrower than cubic zirconia’s $\mathbf{12.6902}$ degrees, because with too high an index the returning light tilts too far to the other side at the table and cannot leave. Tracing rays one by one, the edges agreed with the closed form to within $0.0005$ degrees, but the form first written left out this fifth condition and put diamond’s upper edge $0.7435$ degrees too high. The separator is which condition fixes the edges——the window is widest where the fixing conditions exchange, and “a higher index returns light more readily” holds only up to there. Placed among the earlier papers——the total reflection of sound in air in Paper 474 and the total reflection of light in a stone come from the same critical-angle formula; as two met in separate places, they share a root in the sense of Paper 300. To be honest——only light from straight above was followed, by direction, in a two-dimensional section, without oblique light, crown refraction, colour dependence or absorption. The indices are representative; no gem is assessed and no cut is recommended.
References
- Born, M., & Wolf, E. (1999). Principles of Optics (7th ed.). Cambridge University Press.
- Hecht, E. (2017). Optics (5th ed.). Pearson.
- Tolkowsky, M. (1919). Diamond Design: A Study of the Reflection and Refraction of Light in a Diamond. E. & F. N. Spon.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). On Nights When Distant Sounds Carry, the Sky Is Warm [Paper 474]. Zenodo. 10.5281/zenodo.22733583
Reference
Yamagishi, Y. (2026). The Window of Angles at Which a Gem Returns Light Upward Opens at Refractive Index $\sqrt2$ and Is Widest Near $2.3048$ [Paper 598]. YOI: https://www.yamagishiyuuki.com/yoi/598
DOI: 10.5281/zenodo.23014418
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 598. This paper is a preprint and has not been peer reviewed.