On the left, an image as an eye without visual defects would see.
On the right, a simulation of how that same image would see someone with astigmatism.Nancy Zambrano
If we were to tell an ophthalmologist, not a specialist in optics, that differential geometry has been crucial to understanding the formation of images in the human eye, they might look at us with disbelief. But no less skeptical would be the attitude of a mathematician, not an expert in differential geometry, if we were to tell you that a Swedish ophthalmologist and Nobel Prize in Medicine, Allvar Gullstrand, made relevant discoveries in differential geometry trying to understand, geometrically, astigmatism.
Gullstrand was born on June 5, 1862 in Landskrona, Sweden, and passed away on July 28, 1930 in Stockholm. As a high school student, he showed a great interest in mathematics and, in particular, in differential geometry, which he had learned in a self-taught way and which would be his great passion throughout his life. However, although he thought of pursuing a career in engineering, eventually, influenced by his father, he chose to study Medicine. Upon graduation, he specialized in
visual optics
, an area that straddles optics, optometry, and ophthalmology. This is a branch of knowledge that studies the eye as an optical instrument, that is, it seeks to understand how the optical elements of the eye –cornea and crystalline– form images on the retina –the photographic film of the eye–.
One of the pillars of geometric optics was the mathematical model of the formation of images in the eye proposed by Alhacén in the 10th century.
One of the pillars of geometric optics was the mathematical model of the formation of images in the eye proposed by Alhacén in the 10th century. This stated that when observing an object within the eye, a visual image was formed that corresponded, point to point, with the object contemplated.
Therefore, vision could be modeled as a relationship between sets of points;
He made each point
P
of the real object correspond to another, the image point
P '
.
This mathematical correspondence occurred, since
a
main ray
emanated
from
P
which, passing through the center of the pupil, reached
P '.
During the next nine centuries, an attempt was made to expand this model by analyzing, not only what was happening with this main ray, but also with other rays that also emanated from the same point source and entered the eye. As early as the 19th century, researchers in geometric optics observed 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.
In the first place, that all the rays converge in the same point located in the retina; then a perfect spot image is formed. Second, that the convergence point is in front of the retina, and then myopia appears. Finally, that it is behind the retina, which corresponds to hyperopia. In the last two cases, a dotted image is generated in the retina in a circular blur. But, in addition, it can also happen that the rays do not converge in a single point, and that the blurring does not have a circular but elliptical shape. In this situation, the major axis of the ellipse marks the preferred direction of blurring, which is called the
axis of astigmatism
.
Gullstrand investigated in detail the geometric properties of these rays close to the main ray.
Specifically, he studied the wave front, a surface associated with these rays and perpendicular to all of them.
Gullstrand investigated in detail the geometric properties of these rays close to the main ray.
Specifically, he studied the
wavefront,
a surface associated with these rays and perpendicular to all of them.
Gullstrand realized that when astigmatism is zero - which in differential geometry is equivalent to saying that the wavefront has an
umbilical point
- then there is an abrupt change - which we call a
mathematical
singularity -
in the region where the rays converge called
focal area
.
In addition, Gullstrand analyzed and classified the different types of mathematical singularities that appear. Specifically, he discovered a particular way to differentiate between different types of umbilical points, something that had previously only been approached, with less success, by the mathematician Jean G. Darboux. These results were contributions of great importance in differential geometry.
These and other notable contributions to the mathematical theory of imaging within the eye earned him the Nobel Prize in medicine in 1911. Despite this recognition, Gulltrand's work did not, at the time, have a great impact on the scientific community. . Two reasons point to this: on the one hand, his work was published mainly in Swedish; on the other, when working between two seemingly disparate fields, he was misunderstood. Neither the ophthalmologists understood the mathematics that incorporated their investigations; not even mathematicians took the mathematical work of an ophthalmologist very seriously.
However, Gullstrand's discoveries exemplify the importance for modern science of interdisciplinarity as a generator of new knowledge.
Another example of this is a recent work that combines geometry and optics, on progressive lenses, which we will talk about in a future article in this section.
Sergio Barbero
is a researcher at the Optics Institute of the Higher Council for Scientific Research
María del Mar González
is a researcher at the Autonomous University of Madrid and a member of the ICMAT
Ágata Timón G-Longoria
is coordinator of the Mathematical Culture Unit of the
ICMAT
and editor and coordinator of this section
Café y Teoremas
is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of encounter between mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems.
The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems."
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