THE OPHTHAMOLOGIST WHO USED MATHEMATICS TO KNOW AND UNDERSTAND ABOUT ASTIGMATISM
Allvar Gullstrand’s theoretical studies on the optics of the human eye, which he learned mathematics on his own, were awarded the Nobel Prize in medicine in 1911.
If we told an ophthalmologist, rather than an optician, that differential geometry has been critical to understanding image formation in the human eye, he might look at us with skepticism. But a mathematician who is not an expert in differential geometry would be equally skeptical if we told him that Allvar Gullstrand, a Swedish ophthalmologist and Nobel Prize laureate in medicine, made relevant discoveries in differential geometry while attempting to understand astigmatism geometrically.
Gullstrand was born in Landskrona, Sweden, on June 5, 1862, and died in Stockholm on July 28, 1930. As a high school student, he showed a strong interest in mathematics, particularly differential geometry, which he had self-taught and which would become his lifelong passion. Despite considering a career in engineering, he eventually decided to study medicine after being influenced by his father. Following graduation, he specialized in visual optics, a field that encompassed optics, optometry, and ophthalmology. This is a branch of knowledge that studies the eye as an optical instrument, or how the optical elements of the eye -cornea and crystalline — form images in the retina -the photographic film of the eye-.
The mathematical model of image formation in the eye proposed by Alhacén in the tenth century was one of the pillars of geometric optics. This stated that observing an object within the eye formed a visual image that corresponded to the intended purpose point-by-point. Thus, vision could be modeled as a set of point relationships, with each point P of the real object matched by another point, the image point P’. This mathematical correspondence occurred because a main ray emanated from P and passed through the pupil’s center to reach P’.
This model was expanded over the next nine centuries by examining not only what happened with this main ray, but also what happened with other rays that emanated from the same point source and entered the eye. Geometric optics researchers discovered as early as the nineteenth century that if a few rays are chosen around one of these main rays and what happens in the image is studied, several phenomena can occur.
First, all rays must converge at the same point in the retina to form a perfect point image. Second, the point of convergence appears in front of the retina, resulting in myopia. Finally, make it behind the retina, which corresponds to nearsightedness. The retina produces a punctual, circular blurred image in the last two cases. However, it is possible that the rays do not converge in a single point and that the blurring is elliptical rather than circular. In this case, the major axis of the ellipse, also known as the axis of astigmatism, marks the preferred direction of blurring.
Gullstrand thoroughly investigated the geometric properties of these rays near the main beam. He focused on the wave front, which is a surface associated with those rays and perpendicular to all of them. Gullstrand discovered that when astigmatism is zero — which in differential geometry is equivalent to saying the wave front has an umbilical point — there is an abrupt change — which we call mathematical singularity — in the region where the rays converge known as the focal surface.
Gullstrand also analyzed and classified the various types of mathematical singularities that appear. He discovered, in particular, a method of distinguishing between different types of umbilical points, which had previously only been addressed, with less success, by the mathematician Jean G. Darboux. These findings made significant contributions to differential geometry.
His contributions to the mathematical theory of image formation within the eye, along with others, earned him the Nobel Prize in medicine in 1911. Despite this recognition, Gulltrand’s work had little impact on the scientific community at the time. Two factors contribute to this: on the one hand, his work was mostly published in Swedish; on the other hand, working between two seemingly disparate fields led to his misinterpretation. Ophthalmologists did not understand the mathematics used in their research, and mathematicians did not take an ophthalmologist’s mathematical work seriously.
Gullstrand’s findings, on the other hand, demonstrate the importance of interdisciplinarity as a source of new knowledge in modern science. Another example is a recent work on progressive lenses that combines geometry and optics, which we will discuss in a subsequent article in this section.
