A Glass Galaxy
Recently, a project led me down an interesting path. I applied linear Principal Component Analysis (PCA) to the index profiles of glasses in the SCHOTT glass catalog at 20°C. The Sellmeier dispersion equation for these glasses has six parameters, and accordingly, there were six principal components. However, the first three principal components captured most of the index-variation information. This, along with the fact that I cannot visualize more than three dimensions, led me to focus on the first three principal components for this visualization.
Shown below is the 3D distribution of glasses in the space defined by the first three principal components. Note that I had to scale the second dimension by a factor of 10 and the third dimension by a factor of 50 to make the distribution visually comprehensible. At the center of this plot, the X and Y axes are shown as red and green lines, respectively, corresponding to the first two principal components. The Z axis follows the right-hand rule. In this plot, each “ball” represents a glass. The color of each ball indicates the Abbe number, with dispersion increasing from red to blue. Simply put, the overall index increases along the X axis. Note that two glasses are missing from this visualization: P-SF68 and SF6G05.
The grouping of glasses looks more like “filaments” or “threads” than distinct clusters. Five such groups are clearly visible, while the remaining glasses are more sparsely distributed. In the XY view, two large filaments can be seen: one for crown glasses and one for flint glasses. They appear to meet at a point in the low-index region, where BK7 is located. Looking in the [1, 1, 0] direction, the SF and N-SF glasses form distinct filaments. PK and FK glasses have very low dispersion and stand out as a separate filament. They are used for high-performance chromatic correction.
To round off, open the Interactive Abbe Diagram from SCHOTT in another tab and explore.