The Helmholtz legacy in color metrics: Schrödinger’s color theory

Abstract

This study is a continuation of the authors’ previous work entitled “Helmholtz and the geometry of color space: gestation and development of Helmholtz’s line element” (Peruzzi and Roberti in Arch Hist Exact Sci. https://doi.org/10.1007/s00407-023-00304-2, 2023), which provides an account of the first metrically significant model of color space proposed by the German polymath Hermann von Helmholtz in 1891–1892. Helmholtz’s Riemannian line element for three-dimensional color space laid the foundation for all subsequent studies in the field of color metrics, although it was largely forgotten for almost three decades from the time of its first publication. The rediscovery of Helmholtz’s masterful work was due to one of the founders of quantum mechanics, Erwin Schrödinger. He established his color metric in three extended papers submitted in 1920 to the Annalen der Physik. Two memoirs were devoted to the so-called lower color metric, which laid the basis for the development of his higher color metric, exposed in the last paper. Schrödinger’s approach to the geometry of color space has been taken as a starting point for future elaborations of color metrics and allows a close examination of the current assumptions about the analysis of color-matching data. This paper presents an overall picture of Schrödinger’s works on color. His color theory developed a tradition first inaugurated by Newton and Young, and which acquired strong scientific ground with Grassmann’s, Maxwell’s, and Helmholtz’s contributions in the 1850s. Special focus will be given to Schrödinger’s account of color metric, which responded directly to Helmholtz’s hypothesis of a Riemannian line element for color space.

Full text

Archive for History of Exact Sciences (2023) 77:615–635
https://doi.org/10.1007/s00407-023-00317-x
The Helmholtz legacy in color metrics: Schrödinger’s color
theory
Valentina Roberti1
·Giulio Peruzzi1
Received: 26 June 2023 / Accepted: 7 August 2023 / Published online: 29 August 2023
© The Author(s) 2023
Abstract
This study is a continuation of the authors’ previous work entitled “Helmholtz and the
geometry of color space: gestation and development of Helmholtz’s line element”
(Peruzzi and Roberti in Arch Hist Exact Sci. https://doi.org/10.1007/s00407-023-
00304-2, 2023), which provides an account of the first metrically significant model of
color space proposed by the German polymath Hermann von Helmholtz in 1891–1892.
Helmholtz’s Riemannian line element for three-dimensional color space laid the foun-
dation for all subsequent studies in the field of color metrics, although it was largely
forgotten for almost three decades from the time of its first publication. The rediscovery
of Helmholtz’s masterful work was due to one of the founders of quantum mechanics,
Erwin Schrödinger. He established his color metric in three extended papers submitted
in 1920 to the Annalen der Physik. Two memoirs were devoted to the so-called lower
color metric, which laid the basis for the development of his higher color metric,
exposed in the last paper. Schrödinger’s approach to the geometry of color space has
been taken as a starting point for future elaborations of color metrics and allows a close
examination of the current assumptions about the analysis of color-matching data. This
paper presents an overall picture of Schrödinger’s works on color. His color theory
developed a tradition first inaugurated by Newton and Young, and which acquired
strong scientific ground with Grassmann’s, Maxwell’s, and Helmholtz’s contributions
in the 1850s. Special focus will be given to Schrödinger’s account of color metric,
which responded directly to Helmholtz’s hypothesis of a Riemannian line element for
color space.
Communicated by Jed Buchwald.
BValentina Roberti
valentina.roberti@unipd.it
Giulio Peruzzi
giulio.peruzzi@pd.infn.it
1Department of Physics and Astronomy, University of Padua, Via Marzolo, 8, 35131 Padua, Italy
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616 V. Roberti, G. Peruzzi
1 Introduction
The scientific value of Erwin Schrödinger’s contributions cannot be overstated. His
fame resulted in particular from his contribution to quantum mechanics, for which
he was awarded the Nobel Prize in 1933 jointly with Paul Dirac. In addition to this,
Schrödinger explored a wide variety of research topics, and, in particular, from 1920
to 1926, he wrote extensively on color. This paper investigates Schrödinger’s break-
throughs in the field of color theory, with a special focus on his implementation of
Helmholtz’s line element for color space. In Sect. 2, Schrödinger’s Viennese back-
ground will be discussed. During the years spent at the University of Vienna, he
became aware of the latest advances in colorimetry thanks to the studies of Exner
and Kohlrausch. While Exner and Kohlrausch limited their research interests to the
domain of lower color metrics, Schrödinger went further extending the mathematical
treatment of color to the field of advanced color metric. This led him to the redis-
covery of Helmholtz’s contributions of 1891 and 1892, in which he found crucial
insights for the development of his own color metric. Schrödinger’s seminal works
of 1920 will be exposed in paragraph 3. In Sect. 4, the influence of both Helmholtz’s
and Schrödinger’s works on the subsequent developments of color metrics will be
analyzed. The main improvements related to this new branch of color science will
be outlined, starting with the experimental results obtained by MacAdam in 1942. In
the last paragraph, Schrödinger’s other papers on color—both popular and scientific
publications—written between 1923 and 1926 will be discussed.
2 Schrödinger’s Viennese background
In the winter semester 1906–1907, Erwin Schrödinger enrolled in the University of
Vienna. During this time, he came under the influence of Friedrich Hasenhörl, profes-
sor of theoretical physics, and Franz Serafin Exner, professor of experimental physics.
Both Hasenhörl and Exner can be considered as Schrödinger’s “scientific fathers”
since they were able to leave a strong impact on his future scientific career. Franz
Exner, in particular, initiated Schrödinger into the study of color science.
When Schrödinger matriculated at the University of Vienna, Exner was the director
of the “Zweites Physikalisches Institut” (“Second Physics Institute”), the new name
of the “Physikalisch-chemisches Institut” (“Physico-Chemical Institute”) since 1905.
Exner devoted his career to a wide range of physical research topics, in which he
obtained remarkable results, to the extent that he was recognized as one of the leaders
of the Viennese physics community. These included electrochemistry, atmospheric
electricity, meteorology, radioactivity, crystal physics, spectroscopy, and color the-
ory. From 1902 to nearly until his death, Exner investigated the consequences of the
Young–Maxwell–Helmholtz color theory: his main aim was to further establish the
experimental foundation of the trichromatic theory, defending it against attacks from
scholars supporting Hering’s theory of color. At that time, indeed, one of the most
heated controversies in the field of color was the Helmholtz–Hering dispute. Until the
1870s, the Young–Maxwell–Helmholtz trichromatic theory imposed itself as the only
fulfilling theory of color vision and attracted important supporters among the scientific
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The Helmholtz legacy in color metrics: Schrödinger’s color theory 617
community. Between 1872 and 1874, the German physiologist Ewald Hering provided
an alternative theory in his “Zur Lehre vom Lichtsinne” (“Theory of the Light Sense”),
published as a whole in 1878 (Hering 1878). This work developed an experimental cri-
tique of Helmholtz’s psychophysical theory of simultaneous contrast and contained the
formulation of Hering’s theory of color vision, in which black–white, blue–yellow,
and green–red were interpreted as antagonistic psychophysical processes. Hering’s
theory was developed and strongly defended by his school against criticism from the
supporters of Young–Maxwell–Helmholtz’s theory for the subsequent half-century.1
Besides the topic just mentioned, Exner’s work on color includes studies on the
Purkinje effect, i.e., a shift of the peak sensitivity of the eye toward the violet/blue
end of the spectrum in twilight, the fundamental colors and their sensitivity curves,
and the function of rods and cones. He also provided a quantitative determination of
brightness for different colors, which was a much-debated topic at the time.2Exner
served as a mentor with strong enthusiasm, sharing his research interests with his
pupils so that his research group in Vienna started to be named after him, “Exner
Kreis”, Exner Circle.3Schrödinger was also powerfully influenced by his professor.
After obtaining his doctorate in 1910, he became an assistant to Exner in experimental
physics. From 1911 to 1914, the year of his promotion as Privatdozent, Schrödinger
contributed to various topics of both experimental and theoretical physics, such as,
dielectric properties of matter, atmospheric electricity, and X-ray diffraction in lattices,
whereas from 1918 to 1920, his most important research was in the field of color theory,
in the footsteps of Exner.
Schrödinger shared his interest in color theory with his colleague and closest friend
Karl Wilhelm Friedrich Kohlrausch. Kohlrausch, three years older than him, was assis-
tant to Exner from 1908 to 1920, and in those years, he inevitably got acquainted with
both the established laws of color mixing and the most debated topics in the field of
color perception. From 1917 to 1920, indeed, he held a lectureship on subtractive color
theory (pigment mixing) in Vienna at the Kunstgewerbeschule, the School of Applied
Arts. In particular, he investigated the artist’s color pigments in terms of the three vari-
ables, hue, shade, and tint, first introduced by the mathematician Hermann Günther
Grassmannin 1853 (see Sect. 3). In 1920, Kohlrausch published three contributions,
which included, inter alia, a demonstration of the additivity law for brightness, i.e.,
1For more details on the controversy, see the contribution of Turner, “In the Eye’s Mind, Vision and the
Helmholtz-Hering Controversy” (Turner 1994).
2A collection of four of Exner’s contributions to color theory can be found in the volume “Vorlesungen
über die Physikalischen Grundlagen der Naturwissenschaften” (“Lectures on the Foundations of the Natural
Sciences”), first published in 1919 and extended in a second edition in 1922: Chapter 82. “Gegensatz von
Aussen- und Innenwelt. Korrelation derselben in Bezug auf das Licht. Farbentheorie. Grassmanns Regeln.
Newtons Prinzip und das Farbendreieck” (“Contrast between the outer and inner world. Its correlation with
respect to the light. Color theory. Grassmann’s rules. Newton’sprinciple and the color triangle”); Chapter 83.
“Sättigung der Farben. Begriff des Weiss. Young-Helmholtzschen Theorie” (“Saturation of Colors. The
concept of white. The Young-Helmholtz theory”); Chapter 84. “Bestimmung der drei Grundempfindungen.
Festlegung von Farben. Die Funktion der Stäbchen und Zapfen im Auge” (“Determination of the three
fundamental sensations. The Definition of colors. The function of rods and cones in the eye”); Chapter 85
“Herings und Goethes Theorie der Farben” (“Hering’s and Goethe’s color theory”). See Exner (1922,
pp. 627–656).
3On Franz Serafin Exner and his circle in Vienna, see the contribution of Karlik and Schmid, “Franz Serafin
Exner und sein Kreis—Ein Beitrag zur Geschichte der Physik in Österreich” (Karlik and Schmidt 1982).
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618 V. Roberti, G. Peruzzi
he proved that the quantity of brightness could be expressed as a linear combina-
tion of the fundamental colors (Kohlrausch 1920b). In the third contribution entitled
“Bemerkungen zur Oswaldschen Theorie” (“Observations on the Ostwald’s theory”),
Kohlraush expressed strong criticism toward Ostwald’s theory of color (Kohlrausch
1920c), which supported Hering’s opponent-color theory. The German chemist Wil-
helm Ostwald was a keen amateur painter. He used his knowledge of chemistry to
study pigments and, inspired by the works of his contemporaries, to develop his own
color classification system. This was extensively exposed in the monumental work
“Farbenlehre” published in 1918 (Ostwald 1918). Using the Maxwellian color spin-
ning top method (Clerk Maxwell 1855), Ostwald was able to detect hundreds of color
tones that he arranged on a disk, sorting them according to the perceived equal dif-
ferences and compensating each tone with the one diametrically opposed. In his color
atlas, however, he introduced only twenty-four full colors, Vollfarben, by selecting the
ones most easily available at that time, and he used filters to measure their intensity.
These were arranged on a circle, which formed the boundary between an upper and a
lower cone so that his final model assumed the form of a double cone. According to
him, this form was naturally based on the choice of three parameters: full color, white
and black. By recognizing blackness and whiteness as the attributes that contribute
to human color perception, Ostwald openly manifested his support for Hering’s color
theory.
In 1920, Schrödinger provided a preliminary study on color theory in his paper
“Theorie der Pigmente grösster Leuchtkraft” (“A Theory of Pigments of Maximum
Luminous Efficiency”), published in the Annalen der Physik (Schrödinger 1920a). In
this first contribution, he studied the maximum intensity of light that may be manifested
by a pigment for a particular point of the color diagram and for an arbitrary illumination
condition. Schrödinger introduced the concept of “ideal colors”, i.e., colors for which
the spectral reflectance has a value of either 0 or 1. Although his model was purely
theoretical—in practice, indeed, the reflectance function does not change abruptly from
0 to 1—some actual pigments approximate this behavior quite closely. Schrödinger
called these pigments “optimal colors” because, when mixed, the resulting color will
have the maximum brightness obtainable for that particular pigment. He formalized
the theory of subtractive color mixing by providing a two-dimensional manifold for
optimal pigment with fixed intensity (he chose the intensity of sunlight). Although
Schrödinger departed from Ostwald’s color theory, his results furnished a theoretical
understanding of some empirical rules obtained by Ostwald a few years earlier.
Schrödinger’s first memoir testified to his profound knowledge of the subject under
investigation, he cited, among others, the works of Newton, Grassmann, Helmholtz,
Helmholtz’s pupils, and Ostwald’s experimental results. This memoir was an important
contribution to color theory and the concept of “ideal colors” has since been used in
the development of color models and for the analysis of color pigments.
3 Schrödinger’s “Grundlinien”
« […] a computational error by Helmholtz, the discovery of which further weakens the
connection to experience, is probably the reason why his intellectually interesting idea
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The Helmholtz legacy in color metrics: Schrödinger’s color theory 619
of a Riemann geometry of color has not been appreciated, as far as I know. The relevant
sections have even been deleted from the latest edition of the Helmholtz Handbook of
Physiological Optics» (Schrödinger 1920b, p. 401).
After having displayed the concepts of ideal color pigments, Schrödinger turned to
a more ambitious project, i.e., the definition of a metric in color space, inspired by the
works published by Helmholtz in 1891 and 1892.
His studies on color metrics were published in 1920 in the Annalen der Physik
and appeared as three separate contributions to the organic work entitled “Grundlin-
ien einer Theorie der Farbenmetrik im Tagessehen” (“Outline of a Theory of Color
Measurement for Daylight vision”). “Grundlinien” Part I, Part II, and Part III rep-
resent Schrödinger’s main achievement in the field of color theory and made him
be honored by the prestigious Haitinger Prize of the Austrian Academy of Sciences
“for fundamentals of color theory”, which he received in the same year of their pub-
lication. In the introductory paragraph of Part I, Schrödinger coined the dichotomy
lower-higher color metric and clarified the distinction between these two theoretical
domains: “It emerges through experience that all efforts to quantify color fall into two
fundamental categories. Their distinction lies in the criterion used for the adjustment
of two adjoining color fields on an experimental device. Either a criterion of complete
identity is applied exclusively (indistinguishability), or other criteria are applied (such
as just-noticeable difference, maximum similarity, or eventual maximum contrast).
Measurement results of the first kind do form a unified, internally consistent system
whose simple axiomatic rules have been formalized by Grassmann, which rules have
been experimentally verified by König. I believe that this system of rules—frequently
known as the laws of light mixture—may reasonably be called lower color metric,
to distinguish it from higher color metric, whose laws are much more complicated
and much less well understood” (Schrödinger 1920b, p. 399, translated from German
to English by the authors4). According to Schrödinger’s definition, the pioneering
color-matching experiments carried out by James Clerk Maxwell5in the 1850s felt
in the domain of the lower color metric, while Helmholtz’s work on the line element
implied the definition of just noticeable color differences (i.e., the smallest perceptual
4In this section, all the subsequent quotations from Schrödinger’s original work have been taken from the
English translation provided by Niall (2017). However, in order to allow a more accurate reproduction of
the original, we have translated the terms “niedere Farbenmetrik” and “höheren Farbenmetrik” as “lower
color metric” and “higher color metric”, instead of “basic colorimetry” and “advanced colorimetry”.
5In his masterful papers “Experiments on Colour, as perceived by the Eye, with Remarks on Colour-
Blindness” and “On the theory of compound colors, and the relations of the colors of the spectrum”
published in 1855 and 1860, respectively, James Clerk Maxwell laid down the foundation for the con-
struction of chromaticity diagrams, derived from his color matching experiments. Maxwell’s color diagram
is a 2-dimensional representation of color space, which assumes the shape of an equilateral triangle in
the Euclidean plane with the chosen primary colors as vertices, V (vermillion), U (ultramarine), and EG
(emerald green), and white (W) as the orthocenter. The two variables defining the color diagram are hue
(wavelength) and saturation, whereas the brightness is maintained constant. For this reason, all saturated
colors (wavelengths) are located on the perimeter of the triangle. Any other color inside the triangle can be
expressed in terms of the “amount” of light of the three primary colors involved in the mixture and iden-
tified in terms of its distance from white. Once the position of a given color inside the triangle is detected
graphically, it is possible to identify both its wavelength and the wavelength of its complementary color by
extending the straight line passing through it and W (white) to a side of the triangle (Clerk Maxwell 1855,
1860).
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620 V. Roberti, G. Peruzzi
differences) between two color stimuli and, therefore, belonged to the domain of the
higher color metric. Schrödinger maintained a strong distinction between lower and
higher color metrics. Indeed, as we will show in this section, the first two memoirs
of the “Grundlinien” are entirely devoted to the analysis of the lower color metric,
while in Part III Schrödinger entered the domain of higher color metric, proposing his
implemented version of Helmholtz’s line element for perceptual color space.
In the “Grundlinien”, Part I, Schrödinger introduced the nomenclature related to
light and color, which he coherently adopted for the definition of the general laws of
light mixture. In particular, he introduced the definition of the term “color”, in the
wavelength range from λ=400 nm to λ=800 nm, as follows: “For quantitative ends,
let us designate the set of identically-appearing lights as a color” (Niall 2017, p. 20,
original in Schrödinger 1920b, p. 420). Starting point for the whole treatment were
Grassmann’s laws of colorimetry, which, therefore, require further elucidation. In his
seminal paper—his sole contribution to color theory—published in 1853 and entitled
“Zur Theorie der Farbenmischung” (“On the theory of compound colors”), Grass-
mann proposed the formal equipment of affine geometry (or projective geometry) to
the problem of characterizing color space by introducing his four laws of additive color
mixture, later known as “Grassmann’s laws”. The first law concerns the dimension of
color space; Grassmann considered the nature of human color vision as dependent on
three variables, i.e., tint,shade, and hue. Grassmann’s first rule was experimentally
verified by Maxwell in 1855 (Clerk Maxwell 1855). Although Maxwell considered
red, blue, and green as the three independent color variables, he showed that these two
methods (his own and Grassmann’s) of considering color could be easily deduced one
from the other via coordinate transformations. The second law can be expressed by
the following statement: “In the second place, we assume that if one of two mingling
lights be continuously altered (while the other remains unchanged), the impression of
the mixed light also is continuously changed” (Grassmann 1854, p. 256). Grassmann’s
second law, also known as “continuity law”, reflected the property of color singled out
by Riemann in his 1854 Habilitation dissertation, and later by Helmholtz in 1868: col-
ors form a continuously extended manifoldness. The first two laws imply the existence
of a complementary color for each color of the spectrum. However, Helmholtz’s exper-
imental findings of 1855 contradicted this proposition (see Peruzzi and Roberti 2023).
By analyzing pairs of complementary spectral lights, indeed, Helmholtz was not able
to find a simple complementary color for green, which had to be mixed with purple,
i.e., a mixture of violet and red, to furnish white (von Helmholtz 1855). Grassmann’s
third law states that there are lights with different spectral power distributions that
appear identical (the color match is said to be metameric), while Grassmann’s fourth
law concerns the additivity of brightness: the total intensity of an additive mixture of
colors is the sum of the intensities of the mixed colors.
Schrödinger performed a synthesis of the most important achievements in the field
of colorimetry in a set of axioms, which incorporated, extended, and refined Grass-
mann’s laws. In Sect. 2, Schrödinger discussed the composition of lights and colors
(“Addition von Lichtern und Farben”), which he interpreted as the superposition of
colored light vectors, following Grassmann’s intuition. In this paragraph, Schrödinger
elucidated his first axiom by rephrasing Grassmann’s third law as follows: “the uncon-
ditional and complete equivalence of lights which we have defined as the same in color,
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The Helmholtz legacy in color metrics: Schrödinger’s color theory 621
is retained when they are mixed” (Niall 2017, p. 21, original in Schrödinger 1920b,
p. 411). Section 3is devoted to the operation of scalar multiplication (“Multiplicaktion
mit einer Zahl), which allowed Schrödinger to introduce his second axiom. He refined
Grassmann’s second law, replacing it with the following proposition: “There are no
two lights that become equal and later unequal in a periodic manner as the intensities
of these lights increase proportionally” (Niall 2017, p. 27, original in Schrödinger
1920b, p. 418). The third proposition is introduced in Sect. 4“Die Dimensionstat-
sache”, which deals with the dimensionality of color space. Schrödinger extended
Grassmann’s first law to the case of dichromatic and monochromatic color vision.
While in trichromatic color space, there are linearly independent triplets of colors,
and in dichromatic color space, there are linearly independent pairs of colors, i.e.,
color space for normal trichromats is three-dimensional and color space for dichro-
mats is two-dimensional. In monochromatic color space, on the other hand, any two
colors are linearly dependent, i.e., color space for subjects affected by achromatopsia
is one-dimensional. It is worth pointing out, that Grassmann’s fourth law of colorime-
try was not considered in this first memoir. A deeper analysis was reserved to this law,
known as the additivity law of brightness, in Part III of the “Grundlinien”, devoted to
advanced color metric.
As shown above, Schrödinger’s account responded directly to Grassmann’s formal-
ization of color theory as a vector space: “it turns out that basic colorimetry provides
a complete model for the affine geometry of a pencil of vectors, taken to represent the
color manifold” (Niall 2017, p.13, original in Schrödinger 1920b, p. 420). He envis-
aged the color domain to have a bijective correspondence with a pencil of vectors in
space, namely vectors that radiate from a point. Schrödinger recognized, however, that
the laws of color mixture deviate from the axioms of affine geometry on one point:
this concerns the application of the operation of subtraction and the lack of physical
significance related to negative colors, as well as the operation of multiplication by
negative numbers. For any vector representing a color, the vector of the opposite sign
cannot be used, since there are no lights that produce darkness when superimposed.
After this work of axiomatization, the next aim was that of representing geometri-
cally the color manifold. “Grundlinien”, Part II, is entirely focused on this topic. To
construct the tridimensional color space, Schrödinger chose three objective colors as
calibration colors, between which there is no linear relation. These calibration colors
are assigned to three arbitrarily chosen noncoplanar (i.e., linearly independent) vec-
tors of the pencil. Once indicated as A,B, and Cthe calibration colors, i.e., the basis
vectors, any other color Fcan be expressed by the following relation:
F=x1A+x2B+x3C,(1)
where xiindicates the color coordinates of F, namely its trichromatic coefficients,
relating to A,B, and C. Any color, therefore, can be specified by its components in
the direction of each basis vector. By representing each color vector in this way, the
color manifold of vectors can be easily built up: it assumes the form of a regular
convex cone, which Schrödinger called the “spectral cone”, embedded in a real vector
space and whose apex coincides with the origin (see Fig. 1). Any given spectral
distribution could be represented by a curve on the surface of the cone. The heads of
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622 V. Roberti, G. Peruzzi
Fig. 1 Schrödinger’s cone of perceived colors. A Spektralkurve, spectral curve, and its intersection with an
arbitrarily inclined plane (Schnitt mit einer Ebene) are shown. The planar surfaces ROG, ROV, and VOI are
also indicated (Schrödinger 1920c, p. 428)
the vectors indeed trace a specific curve—whose shape depends on the spectral energy
distribution—around the surface of the spectral cone. This curve represents the spectral
curve and its position in the color manifold is determined by the brightness. The higher
the brightness, the farther the spectral curve will be from the origin (Schrödinger 1920c,
p. 428). It is worth noticing that the intersection between the color cone and an arbitrary
inclined plane (Fig. 1) gives rise to a plane section whose shape closely approximates
that of Helmholtz’s two-dimensional color diagram (Fig. 2).
The most peculiar property of Schrödinger’s cone is that it is “not so tightly curved”:
the envelope of the cone has, indeed, three plane sections, ROG (red, origin, green),
ROV (red, origin, violet), and VOI (violet, origin, indigo), as shown in Fig. 1., which
represent the spectral intervals detected by one of Helmholtz’s pupils, Arthur König:
from λ=655 nm to λ=630 nm; from λ=630 nm to λ=475 nm and from λ=
475 nm to λ=430 nm, respectively (König and Dieterici 1892). According to the
experimental data, the preferred basis vectors would be those which lie on these three
Fig. 2 Helmholtz’s color diagram at constant brightness, which was experimentally derived in 1855 (von
Helmholtz 1855, Tafel I, Fig. 5)
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The Helmholtz legacy in color metrics: Schrödinger’s color theory 623
plane sections, in the directions of the three spectral primaries detected by König:
far-red end of the spectrum, R; far violet, V; and green, G.6The two sections ROG
and VOI form the borders of the envelope, and between ROG and VOI the spectral
cone has a convex curvature. Moreover, in the interior of the envelope, there is a vector
direction that corresponds to white. The sheaf of planes passing through the direction
of white separates the totality of colors into groups: “The groups which lie on the same
half-plane of the sheaf of planes are related to the same spectral color by ‘dilution with
white’” (Niall 2017, p. 37, original in Schrödinger 1920c, p. 430). Pairs of colors that
lie on the same plane enclosing white are defined as complementary colors.
Once he displayed the construction of his color cone and elucidated its geometrical
properties, Schrödinger provided in the final section of the paper the derivation of
color coordinates for an arbitrary function of wavelength and explored the relation
between two-dimensional and three-dimensional color space, i.e., between the color
space for dichromats and trichromats.
Schrödinger’s discussion of lower color metrics contained in these first papers
laid the foundation for his work on higher color metrics, “Grundlinien”, Part III. As
discussed above, in the domain of lower color metrics, an affine space sufficed to
describe the laws of color mixture. On the other hand, in the domain of higher color
metrics, a Riemannian space was demonstrated to be necessary to translate color
differences into color distances in the three-dimensional and two-dimensional color
spaces. Therefore, in higher color metrics distances between colors are defined by a
Riemannian metric. This is, as Moore noticed, the same kind of geometry used by Ein-
stein in his general theory of relativity, although the color space is three-dimensional
and not four-dimensional as space–time (Moore 2015). It is worthwhile to point out
that Schrödinger had previously familiarized himself with Riemann geometry while
working on problems of general relativity (see Schrödinger 1918a,b). Certainly, his
consolidated knowledge of the Riemannian formalism allowed him to envisage its
application to color theory.
In “Grundlinien”, Part III, Schrödinger acknowledged, on the one hand, Helmholtz’s
intellectually interesting idea of a Riemann geometry of color, whereas, on the other
hand, he recognized some critical aspects that had escaped Helmholtz’s attention in
the definition of the line element. Schrödinger argued that the identification of these
shortcomings by Helmholtz’s pupils might have led to the suppression of the key
passages related to higher color metrics in the third edition of the “Handbuch” (Von
Helmholtz 1909–1911), condemning his pioneering work to oblivion.
Helmholtz’s original line element formulation (Helmholtz 1892) is reported below:
dE2=dx
x+a2
+dy
y+b2
+dz
z+c2
,(2)
where x,y, and zrefer to the physiologische Urfarben (physiological primary colors)
and a,b, and crepresent the self-light constants. The physiological primary colors
were expressed as linear homogeneous equations of R(red), G(green), andV(violet),
6See Sect. 9 of Schrödinger’s paper entitled “Irreelle Eichfarben. Die Young-Helmholtzsche Theorie”
(“Virtual-Valued Calibration Colors. The Young-Helmholtz Theory”).
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624 V. Roberti, G. Peruzzi
i.e., the Elementarfarben, elementary colors, found by Helmholtz’s assistants Arthur
König and Conrad Dieterici (König and Dieterici 1886).
By analyzing Helmholtz’s mathematical treatment, Schrödinger first noticed that
the new primaries chosen by Helmholtz as a linear combination of König’ and
Dieterici’s fundamentals had to be computed ad hoc to make the line element for
just distinguishable color pairs approximately constant. Moreover, Schrödinger rec-
ognized that the brightness function (V(λ) in modern terms) obtained from his model
was in contradiction to experience. Two pronounced maxima resulted from the spec-
tral sensitivity of the primary sensations: “this brightness function is quite absurd,
since e.g., it produces a hideous dromedary-like curve with two pronounced maxima
as a brightness distribution for the spectrum of sunlight” (Niall 2017, p. 71, original
in Schrödinger 1920d, p. 493).
But for a still more general reason this brightness function, and therefore
Helmholtz’s line element, was unacceptable to Schrödinger. Experiments carried out
by William de Wiveleslie Abney,7Edward Robert Festing (see Abney and Festing
1886), and subsequently by Exner and Kohlrausch (Exner 1918,1920; Kohlrausch
1920b), proved that brightness, at least to a high degree of approximation, was an addi-
tive property of colors, confirming Grassmann’s fourth law of colorimetry (Grassmann
1854). Geometrically speaking, this means that the surfaces of constant brightness
should be portions of planes in the 3-dimensional Euclidean space in which the color
space is embedded. The surfaces of constant brightness derived from Helmholtz’s
metric were not planar, i.e., brightness was not an additivity property in Helmholtz’s
model.
Between the nineteenth and the twentieth centuries the modern concept of lumi-
nance saw birth, thanks to more accurate and elaborate studies in the field of
photometry. It has to be underlined, however, that Helmholtz had already conducted
refined experiments on brightness comparison using a modified version of the color
wheel used by Maxwell for mixing colors. Schrödinger was aware of this study and
argued that Helmholtz had misinterpreted his results, which only apparently led to a
non-additive brightness function. Since that time, Helmholtz had always doubted the
additivity of brightness.
Another significant aspect related to the nature of color space was highlighted
by Schrödinger. Helmholtz’s model was Riemannian in the sense that any distance
between two neighboring points was given by a differential expression of coordinates,
but the resulting color space turned out to be isometric to Euclidean space. Indeed, by
neglecting the self-light constants a,b, and cin Eq. (2), Helmholtz’s color space is
Euclidean on a logarithmic basis (lnx,lny,lnz):
dE2=(dlnx)2+(dlny)2+(dlnz)2.(3)
7William Abney was a photographic chemist. He succeeded Arthur König as the leading international
pioneer in precision colorimetry. Abney formulated the now-called “Abney’s law”, or brightness additivity
law, named after him. Abney’s Law of additivity states that the total brightness of a mixture of wavelengths
is equal to the sum of the brightness of its monochromatic components, something already postulated by
Grassmann.
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The Helmholtz legacy in color metrics: Schrödinger’s color theory 625
If the mathematical treatment is reduced to a two-dimensional color vision sys-
tem, i.e., color space for subjects suffering from dichromatism, the orthogonality
requirement between lines of constant brightness (isophotes) and lines of constant
hue (isohues) leads to a contradiction to Abney’s additivity rule. One of the conse-
quences of Helmholtz’s line element was, thus, that isophotes and isohues were not
mutually orthogonal (see Vos 2006).
The disagreement with Abney’s law of additivity led Schrödinger to undertake a
re-examination of the entire metrical theory of color perception, and his proposal of a
new metric compatible with both Abney’s and Weber-Fechner’s results took shape.
Schrödinger used the additivity law of brightness as a guide for his mathematical
treatment:
h=a1x1+a2x2+a3x3,(4)
where hstands for “Helligkeit” (brightness), aiindicate constants, derived experimen-
tally by Exner (a1=43,33;a2=32,76;a3=1) and corresponding approximately
to the brightness of the three fundamental processes x1,x2,x3, as experimentally
determined by König and Dieterici in 1892 (König and Dieterici 1892). To agree with
the existing experimental data, therefore, the line element must produce a family of
planes of constant brightness:
a1x1+a2x2+a3x3=const.(5)
Planes of this family should be everywhere perpendicular to those of constant hue.
By performing a coordinate transformation, Schrödinger obtained the Euclidean line
element, which turned out to satisfy the orthogonality requirement but, on the other
hand, did not satisfy the Weber–Fechner law. The incompatibility with the law of
psychophysics was first pointed out by Wolfgang Pauli, whose key contribution was
openly acknowledged by Schrödinger in the original memoir. Thus, by selecting a
factor for reciprocal luminance of the following form:
1
a1x1+a2x2+a3x3
.(6)
Schrödinger obtained the final expression for his line element, which satisfied both
Abney and Weber-Fechner laws:
ds2=1
a1x1+a2x2+a3x3a1dx2
1
x1
+a2dx2
2
x2
+a3dx2
3
x3.(7)
In Schrödinger’s metric, furthermore, color space has a curvature different from
zero, i.e., it is not isometric to Euclidean space.
A highly significant application of the line element reported in this paper was the
determination of the lines of constant hue (wavelength) inside the triangular color
diagram. Indeed, using his line element, Schrödinger was able to depict in the color
diagram containing all colors with equal brightness the geodesic lines connecting the
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626 V. Roberti, G. Peruzzi
Fig. 3 Theoretical constant-hue lines in Maxwell’s color triangle (Rot–Blau–Grün, Red–Blue–Green) show-
ing the spectrum locus determined by König and Dieterici (thicker black line with the indication of
wavelength in nm) (Schrödinger 1920a,b,c,d, p. 515)
white point with the spectral color points on the perimetry. These were in general
curved lines, as shown in Fig. 3, except for the shortest lines to the three fundamental
colors, which were straight. Colors of constant hue were those that lay on the geodesics
between the fully saturated color and white. Such geodesics were loci of constant
hue,orisochromes.Theisochromes diverged from yellow, magenta, and cyan and
converged toward the primaries red, green, and blue. The derivation of geodesic lines
of constant hue provided an important clue for the non-Euclidean interpretation of the
geometry of color space and constituted a pivotal contribution to the advancement of
color science.
4 The reception of Schrödinger’s line element
Since Schrödinger’s theoretical work, other mathematicians and physicists have devel-
oped line element theory further, so that higher color metric started constituting a
proper branch of color theory, as Turner pointed out (Turner 1994), almost inaccessi-
ble to other specialists working in the field.
Whereas Helmholtz’s line element had a firm structural basis in the Weber–Fechner
law, Schrödinger’s line element was obtained by a purely mathematical construction.
It gave, however, a satisfactory description of the experimental color discrimination
data available at the time.
The first significant implementation of line element theory was provided by the
American physicist and color scientist David MacAdam, who experimentally derived
his ellipses for color stimuli in 1942 (MacAdam 1942). He was well aware of
Schrödinger’s work, as testified by the publication of an abridged translation of
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The Helmholtz legacy in color metrics: Schrödinger’s color theory 627
Schrödinger’s papers of 1920 in his “Sources of Color Science” (MacAdam 1970).
MacAdam had the opportunity to perform precise measurements at the Kodak firm in
Rochester, New York. These aimed to create a connection between the perception of
the chromatic difference between two colors and a geometric chromatic distance in the
CIE 1931 color space (the first color space created by the International Commission
on Illumination) of constant luminance. He carried out color-matching experiments
and determined the spread of color matches for a set of 25 test colors. MacAdam used
a bipartite comparison field of view, exploiting a method first developed by Maxwell
in the 1850s; one color served as a test, and the other color was subjected to small
variations in chromaticity by the observer until it matched the test color. He found out
that colors, within a certain small region in the diagram, which assumed the shape of an
ellipse, appeared identical to a human eye. Figure 4shows MacAdam’s discrimination
ellipses, containing all the colors that are indistinguishable to the human eye from the
color located at the center of the ellipses, and the perimeter of each ellipse represents
the just noticeable difference in chromaticity; therefore, the smaller the perimeter, the
better the discrimination. MacAdam’s ellipses rapidly became a key set of data, used
as a reference for line elements and color difference formulae.
Fig. 4 MacAdam’s ellipses, represented ten times their true size on the CIE (x, y) chromaticity diagram.
Each ellipse on the diagram contains all the colors that are indistinguishable to the human eye from the
color at the center of the ellipse. The perimeter of the ellipses represents the just noticeable difference in
chromaticity. Therefore, the smaller the perimeter, the better the discrimination of chromaticity. The reason
for the large variability in size, shape, and orientation of the ellipses is expressed in Helmholtz’s papers of
1891 and 1892 (MacAdam 1942, p. 271)
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628 V. Roberti, G. Peruzzi
Another metric for color space was provided by the English physicist and math-
ematician Walter S. Stiles in 1946. Stiles, relying on his skills as a mathematical
physicist, proposed a version of the line element based on his experimental investiga-
tions. He found some weaknesses in both Helmholtz’s and Schrödinger’s treatments.
In the intermediate luminance range, both scientists assumed that the Weber fractions
were equal for all three response mechanisms. By conducting his sensitivity exper-
iments, however, Stiles recognized that different Weber fractions had to be applied
to the three cone systems rather than the common fraction used by Helmholtz and
Schrödinger. Stiles suggested, thus, a modification of the line element introducing the
different Weber fractions for each fundamental process:
dE2=dx
0.78x2
+dy
1.0y2
+dz
4.46z2
,(8)
where x,y, and zrepresent the three fundamental processes and 0.78, 1.0, and 4.46
indicate the quantities, obtained experimentally, that are proportional to the limiting
Fechner fractions for red, green, and blue, respectively. By rewriting Eq. (8)asa
logarithmic expression, the color space can be converted to a uniform Euclidean space,
in line with Helmholtz’s proposal, but with different scales along the three axes.
An exhaustive collection of the main achievements in the field of color metrics
up to 1971 can be found in the Proceeding of the Memorial Symposium on Color
Metrics entitled to Helmholtz on the occasion of the 150th anniversary of his birth.
Among the 64 participants, the names of MacAdam and Stiles stood out. On that
occasion, a whole range of line elements and color difference formulas passed the
review. One of the most advanced attempts to define a line element in color space
was that proposed by J. J. Vos and P. L. Walraven of the Institute for Perception
TNO of Soesterberg. It consists of a theoretical model, the so-called “zone-theory”
compromise, which contemplates the validity of both Young-Maxwell-Helmholtz’s
theory and Hering’s opponent process theory, as we will see in the following section.
Between 1955 and 1965, important breakthroughs in the study of vision came from
various direct techniques, which established the existence of three photopigments in
the retinal cones. Those were demonstrated to provide neural inputs to cells located
higher in the visual tract, which in turn displayed chromatic opponency in a manner
very similar to that predicted by Hering.8
Up to the present time, several research studies have been performed aimed at
fulfilling what Helmholtz and Schrödinger first began, i.e., an ever more accurate
mathematical description of color by the application of differential geometry to the
perceptual color space, in step with the modern achievements in the fields of both
geometry, psychophysics, and visual neuroscience.
8In particular, the 1950s-1960s contributions of the American psychologists Dorothea Jameson, Leo M.
Hurvich, Russell L. De Valois and Arthur E. Jones made the scientific community begin reconsidering
Hering’s opponent process theory (see, e.g., Jameson and Hurvich 1955, De Valois and Jones 1961).
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The Helmholtz legacy in color metrics: Schrödinger’s color theory 629
5 Schrödinger’s other papers on color
After his masterly work on color metrics, Schrödinger continued publishing occasional
papers on color theory between 1923 and 1926. In this paragraph, we present some
of these largely unexplored contributions, which testify to the breadth and depth of
Schrödinger’s research interests.
On February 3, 1923, Schrödinger’s short popular article entitled “Ton und Farbe”
(“Sound and Color”) appeared in the Swiss daily newspaper Neue Zürcher Zeitung
(Fig. 5). Schrödinger offered a comparison between the two sensations of sound and
color. He stated that sounds and colors were both periodic phenomena, but their esthetic
effects through the respective senses of hearing and sight were quite different in modal-
ity. According to him, music depends upon the temporal sequence of sounds, whereas
painting depends on the spatial arrangement of colors: “wahrgenommene Farben
bilden räumliche Formen; die Klänge bilden nur zeitliche Formen, im besonderen
Melodien” (“perceived colors form spatial forms; sounds form only temporal forms,
especially melodies”). This Fuilletton reveals Schrödinger’s inclination for philosoph-
ical investigations and also testifies to his interest in both music and art. Concerning
the latter, he wondered about the qualities that made a painting satisfactory. In partic-
ular, he did not believe that an abstract arrangement of colors could constitute an art
piece because it did not furnish an adequate depiction of reality.
In the following year, Schrödinger submitted to Die Naturwissenschaften asci-
entific paper entitled “Über den Ursprung der Empfindlichkeitskurven des Auges”
(“On the Origin of the Eye’s Sensitivity Curves”). This memoir contains key argu-
ments in favor of a phylogenetic evolution of the human visual system and testifies to
Schrödinger’s awareness of the latest achievements in the field of biology.9He con-
sidered the spectral brightness sensitivity function of the ordinary human eye in terms
of the availability of natural sources of illumination over time. Schrödinger specu-
lated that the peak brightness sensitivities of cones and rods arose in phylogenetic
development, with the cone system being relatively recent in appearance compared to
the rod system. In that period, the comparative physiology of vision offered support
to these conjectures, based on differences in illumination for terrestrial and aquatic
animals. For the development of this study, he embraced the duplicity theory formu-
lated in 1894 by the German physiological psychologist Johannes von Kries. This
theory concerns the comparisons and interactions between the cone and rod systems
in the visual pathways, with the assumption that twilight vision is a function of the
retinal rods and daylight vision is a function of the retinal cones. In 1866, the Ger-
man anatomist Max Schultze supposed the rods to be more primitive organs than the
cones, i.e., achromatic twilight vision to be more primitive than chromatic daylight
vision (Schultze 1866). Most of the subsequent intervening discussions of rod vision
9Schrödinger extended his interest toward biology while he was a Senior Professor at the School of
Theoretical Physics of the Dublin Institute for Advanced Studies. In 1943, Schrödinger gave a series of
lectures at Trinity College in which he explored areas of biology. These lectures were focused on two
main topics: the nature of heredity and the thermodynamics of living systems. In particular, Schrödinger
investigated to what extent quantum physics could be used to explain the genetic structure of living beings.
These lectures were collected in a volume entitled “What is life? The Physical Aspect of the Living Cell”
(Schrödinger 1944), published in 1944. “What is life?” was widely read and became one of the most
influential science books of the twentieth century.
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630 V. Roberti, G. Peruzzi
Fig. 5 The original text of Schrödinger’s “Ton und Farbe”. Courtesy of the Österreichische Zentralbibliothek
für Physik, University of Wien
and cone vision implied an evolutionary distinction between them. In 1892, e.g., the
American psychologist, logician, and mathematician Christine Ladd-Franklin made
this evolutionary distinction focal.10
Whereas the sensitivity curves for the perception of the fundamental colors seemed
to be easily explained according to the Young–Maxwell–Helmholtz theory by refer-
ring to the spectral distribution of the solar light, Schrödinger raised questions about
the sensitivity curve of the rod mechanism. This appeared to shift toward the blue
end of the spectrum. He investigated the possible reasons for this displacement of the
rod curve toward shorter wavelengths. One plausible explanation could reside in the
fact that the rod apparatus originated under the influence of another type of illumina-
tion light with a different energy distribution curve from that of the cone apparatus.
However, as Schrödinger pointed out, this light had more or less the same energy
distribution as daylight. The most probable explanation laid, according to him, in the
green–blue color exhibited by water in thicker layers: “for an aquatic animal that lives
under the surface at some depth, the composition of sunlight must be transformed in
just about the way we need for our explanation. Such an animal would have particular
need of robust adaptation to different levels of brightness, too, if it explored changes
in depth under the water’s surface. Under this hypothesis, the rod system would be
an older system of vision which emerged during the age of aquatic life” (Niall 2017,
p. 137, original in Schrödinger 1924, p. 927). According to this explanation, color-
blind vision could be regarded as a pure return to an atavistic state. Although many
of the details of rod and cone evolution are still uncertain, recent studies have demon-
strated that our distant aquatic ancestor, living in the ocean more than 500 million
years ago, had already evolved two classes of cones, which were sensitive to short and
long wavelengths. In line with Schrödinger’s intuition, the ancestral cones provided
10 In 1892, Ladd-Franklin presented her first findings in London at the Second International Congress
of Psychology with her contribution “A new theory of Light Sensation” (Ladd-Franklin 1892). It is worth
highlighting that Ladd-Franklin traveled to Europe between 1891 and 1892. In particular, while in Germany,
she devoted herself to research in color vision, working, inter alia, at Helmholtz’s laboratory in Berlin under
the supervision of Arthur König.
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The Helmholtz legacy in color metrics: Schrödinger’s color theory 631
dichromatic color vision in daylight. From these two classes of cones, new photore-
ceptors had developed, resembling modern-day photoreceptors, which endowed the
organism with a major survival advantage: the ancestral rods, which enabled vision at
very low light levels, at night and in the deep ocean, and three new classes of cones,
which provided trichromatic vision in daylight lighting levels.11
In 1925, another contribution, entitled “Über Farbenmessung” (“On Color Measure-
ment”), was published in the Physikalische Zeitschrift. Here Schrödinger expressed
strong criticism toward Ostwald’s theory of color, in line with Kohlrausch’s arguments
on the subject (Kohlrausch 1920c).
In the same year, he presented before the Vienna Academy of Science his last paper
on color theory: “Über das Verhältnis der Vierfarben- zur Dreifarbentheorie” (“On the
Relation of the Four Color to the Three-Color Theories”). In this work, Schrödinger
revealed the formal relation between Young–Maxwell–Helmholtz’s theory and Her-
ing’s four-color theory by a simple geometric transformation on color space, i.e., by a
projective transformation, and proved that both theories are equally capable of repre-
senting color mixtures at the level of simple sensations, something that Helmholtz had
already tried to demonstrate in the second version of the Handbuch (von Helmholtz
1885–1896, pp. 376–382).12 Schrödinger also reported the pioneering attempts to
combine the two theories in the so-called “zone theory”. The zone theory compromise
was first postulated in 1881 by the Dutch ophthalmologist Franciscus Cornelis Don-
ders and later by von Kries (von Kries and Adolf 1882). It contemplated the validity
of both Young–Maxwell–Helmholtz’s theory and Hering’s opponent process theory,
although applied to different levels of the nervous system. For almost a century, the
two theories were considered mutually exclusive for the explanation of human visual
perception. As already mentioned, the debate between supporters of Helmholtz’s the-
ory, on the one hand, and Hering’s theory, on the other hand, constituted one of the
most persistent controversies of modern science relating to human color vision. At the
end of the nineteenth century, hybrid theories, “zone theories”, saw birth to unify the
two theories in a common interpretative framework. The name “zone-theory” suggests
that the mechanism of trichromatic color mixture is manifest at an “early stage” (first
zone) of the visual system, while the mechanism of the opponent process color mix-
ture comes into play at a “higher level” (second zone) of the visual system. Although
Schrödinger sympathized with von Kries’ proposal, his intent while writing this con-
tribution was not to provide support for the “zone theory”, but rather to demonstrate
the mathematical equivalence between the two theories of color vision: “This variant
theory provided by von Kries is highly credible, in my opinion. However, what I want
to show in the following is completely independent of the profound conception of the
physiological substrate of the visual process. It is simply a matter of establishing that,
from a formal point of view, the relationship between the two theories—the three-
and the four-color theory—is to be understood in a very simple manner, namely as a
mere transformation of the variables”) (Schrödinger 1925a,b, p. 472, translated from
German to English by the authors).
11 On the evolution of cone and rod photoreceptors, see the contributions of Lamb, “Why rods and cones?”
(Lamb 2016), and Morshedian and Fain, “The evolution of rod photoreceptors” (Morshedian et al. 2017).
12 The first English translation, enriched with commentaries, of the paragraph in question was provided by
Joseph David Everett in 1893 (Everett 1893).
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632 V. Roberti, G. Peruzzi
Schrödinger’s last contribution to color was a paragraph entitled “Die Gesicht-
sempfindungen” (“The visual sensations”), written for the eleventh edition of
Müller-Pouillets Lehrbuch der Physik13 and published in 1926 (Schrödinger 1926). In
1921, Schrödinger moved to the University of Breslau (now Wrocław) where he was
appointed full professor. Here he met the German physicists Otto Lummer, profes-
sor of experimental physics and a former assistant to Helmholtz in Berlin, and Ernst
Pringsheim, professor of theoretical physics. In Breslau, Schrödinger could work in
an institute, where research on optical phenomena and optical physics played a piv-
otal role. After the publication of his monumental work of 1920, “Grundlinien einer
Theorie der Farbenmetrik im Tagessehen”, he rapidly became recognized as one of
the world’s leading experts on color theory. Indeed, when Otto Lummer, together with
his colleagues Arnold Eucken and Erich Waetzmann, was in charge of planning the
eleventh edition of the “Lehrbuch”, he proposed to Schrödinger the writing of the
chapter related to visual sensations. Schrödinger masterfully fulfilled the assignment:
in more than a hundred pages, he exposed standard color theory, color sensations in
daylight and twilight, and higher color metrics.
6 Conclusion
During an interval of a few years, Erwin Schrödinger succeeded in laying the seeds for
future advancements in the fields of colorimetry and color metric, further developing
the 19th-century contributions of Maxwell, Grassmann, and Helmholtz. Thanks to
the flourishing scientific environment he encountered at the University of Vienna in
his first academic years, Schrödinger was able to address open problems in color
theory, inspired by the research carried out by his contemporaries, particularly by
his mentor Franz Exner and his colleague Karl Kohlrausch. His research led him
to rediscover Hermann von Helmholtz’s pioneering discussion on the line element,
which was deleted from the latest edition of the “Handbuch” falling into oblivion for
almost three decades. Although Helmholtz deserves credit for the first determination
of Riemannian metrics for color space, we have tried to emphasize in this paper the
importance of the work conducted by Schrödinger, who not only brought back to
light Helmholtz’s key insights on the nature of color space but also sought to carry
Helmholtz’s legacy forward.
Since Schrödinger’s formal development of the color metric, considerable effort has
been spent on the derivation of more refined line elements for color space, both from
theoretical considerations and experimental data. Nowadays, indeed, the definition
of an appropriate metric for color spaces has become increasingly important also for
technological applications.
Schrödinger’s papers on color show us the continuing importance of his color
theory, in that an analysis of his contributions permits a closer examination and a
better understanding of the main progress achieved in the field, and, in parallel, sheds
light on some critical aspects yet to be clarified today.
13 Some translated fragments can be found in MacAdam’s work “Sources of Color Science”, paragraph
“Thresholds of color differences” (MacAdam 1970, pp. 183–193).
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The Helmholtz legacy in color metrics: Schrödinger’s color theory 633
Funding Open access funding provided by Università degli Studi di Padova within the CRUI-CARE
Agreement. This work was supported by the DOR Peruzzi (Dotazione Ordinaria di Ricerca) funds of the
University of Padova.
Data availability statement The authors confirm that all data generated or analyzed during this study are
included in this article.
Declarations
Conflict of interest This work was supported by the DOR Peruzzi (Dotazione Ordinaria di Ricerca) funds
of the University of Padova. The authors have no competing interests to declare that are relevant to the
content of this article.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long
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References
Abney, William De Wiveleslie., and Edward Robert Festing. 1886. Color photometry. Philosophical Trans-
actions of the Royal Society 177: 423–456. https://doi.org/10.1098/rstl.1886.0013.
Cahan, David, ed. 1993. Hermann von Helmholtz and the foundations of nineteenth-century science.Berke-
ley: University of California Press.
Clerk Maxwell, James. 1855. Experiments on colour, as perceived by the eye, with remarks on colour-
blindness. Transactions of the Royal Society of Edinburgh 21: 275–298. https://doi.org/10.1017/
S0080456800032117.
Clerk Maxwell, James. 1860. On the theory of compound colours, and the relations of the colours of the
spectrum. Philosophical Transactions of the Royal Society of London 150: 57–84. https://doi.org/10.
1098/rstl.1860.0005.
De Valois, Russell L., and Jones, Arthur E. 1961. Single-cell analysis of the organization of the primate
color-vision system. Neurophysiologie und Psychophysik des Visuellen Systems/The Visual System:
Neurophysiology and Psychophysics, 178–191. Berlin: Springer. https://doi.org/10.1007/978-3-662-
22221-8_22.
Donders, Franciscus Cornelis. 1881. Ueber Farbensysteme. Albrecht Von Graefes Archiv Für Ophthalmolo-
gie 27 (1): 155–223.
Everett, Joseph David. 1893. Helmholtz on Hering’s theory of colour. Nature 47 (1216): 365–368.
Exner, Franz Serafin. 1918. Einige Versuche und Bemerkungen zur Farbenlehre. Sitzungsberichte Der
Kaiserliche Akademie Der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Klasse,
Abteilung 2a: Mathematik, Astronomie, Physik, Meteorologie, Und Mechanik 127: 1829–1864.
Exner, Franz Serafin. 1920. Zur Kenntnis der Grundempfindungen im Helmholtz’schen Farbensystem.
Sitzungsberichte Der Kaiserliche Akademie Der Wissenschaften in Wien, Mathematischnaturwis-
senschaftliche Klasse, Abteilung 2a: Mathematik, Astronomie, Physik, Meteorologie, Und Mechanik
129: 27–46.
Exner, Franz Serafin. 1922. Vorlesungen über die physikalischen Grundlagen der Naturwissenschaften.
Zweite, Vermehrte, Auflage. Leipzig und Wien: Franz Deuticke.
Grassmann, Hermann Günther. 1854. On the theory of compound colours. Philosophical Magazine 7:
254–264. https://doi.org/10.1080/14786445408647464.
Hering, Ewald. 1878. Zur Lehre vom Lichtsinne, sechs Mittheilungen an die Kaiserl. Akademie der Wis-
senschaften in Wien. Wien: Druck und Verlag von Carl Gerold’s Sohn.
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

634 V. Roberti, G. Peruzzi
Jameson, Dorothea, and Leo M. Hurvich. 1955. Some quantitative aspects of an opponent-colors theory.
I. Chromatic responses and spectral saturation. Journal of the Optical Society of America 45 (7):
546–552. https://doi.org/10.1364/JOSA.45.000546.
Karlik, Berta, and Erich Schmid. 1982. Franz Serafin Exner und sein Kreis - Ein Beitrag zur Geschichte
der Physik in Österreich. Wien: Verlag der Österreichischen Akademie der Wissenschaften.
Kohlrausch, Karl Wilhelm Friedrich. 1920a. Beiträge zur Fabenlehre. I. Farbton und Sättigung der Pig-
mentfarben. Physikalische Zeitschrift 21: 396–403.
Kohlrausch, Karl Wilhelm Friedrich. 1920b. Beiträge zur Farbenlehre. II. Die Hellkigkeit Der Pigmentfar-
ben. Physikalische Zeitschrift 21: 423–440.
Kohlrausch, Karl Wilhelm Friedrich. 1920c. Beiträge zur Farbenlehre. III. BemerkungenZur Oswaldschen
Theorie. Physikalische Zeitschrift 21: 473–477.
König, Arthur Peter, and Conrad Heinrich Dieterici. 1886. The modern development of Thomas Young’s
theory of colour-vision. Report of the British Association for the Advancement of Science 56: 431–439.
König, A.P., C.H. Dieterici. 1892. Die Grundempfindungen in normalen und anomalen Farbensystemen
und ihre Intensitätsverteilung im Spektrum. Hamburg und Leipzig: Leopold Voss.
Ladd-Franklin, C. 1892. A new theory of light sensation. In Proceedings of international congress experi-
mental psychology, London; 1892.
Lamb, T.D. 2016. Why rods and cones? Eye 30: 179–185. https://doi.org/10.1038/eye.2015.236.
MacAdam, D.L. 1942. Visual sensitivities to color differences in daylight. Journal of the Optical Society
of America 32 (5): 247–274. https://doi.org/10.1364/JOSA.32.000247.
MacAdam, D.L. 1970. Sources of color science. Cambridge: MIT Press.
Moore, W. 2015. Schrödinger: Life and thought. Cambridge: Cambridge University Press.
Morshedian, A., and G.L. Fain. 2017. The evolution of rod photoreceptors. Philosophical Transactions of
the Royal Society B: Biological Sciences 372 (1717): 20160074. https://doi.org/10.1098/rstb.2016.
0074.
Niall, K.K. 2017. Erwin Schrödinger’s color theory: Translated with modern commentary.NewYork:
Springer International Publishing.
Ostwald, F.W. 1918. Die Farbenlehre, in fünf Büchern. Leipzig: Verlag Unesma.
Peruzzi, G., and V. Roberti. 2023. Helmholtz and the geometry of color space: Gestation and development
of Helmholtz’s line element. Archive for History of Exact Sciences.https://doi.org/10.1007/s00407-
023-00304-2.
Schrödinger, E. 1918a. Die Energiekomponenten des Gravitationsfeldes. Physikalische Zeitschrift 19: 4–7.
Schrödinger, E. 1918b. Über ein Losungssystem der allgemein kovarianten Gravitationsgleichungen.
Physikalische Zeitschrift 19: 20–22.
Schrödinger, E. 1920a. Theorie der Pigmente von grösster Leuchtkraft. Annalen Der Physik 4 (62): 603–622.
Schrödinger, E. 1920b. Grundlinien einer Theorie der Farbenmetrik im Tagessehen. Annalen Der Physik 4
(63): 397–426.
Schrödinger, E. 1920c. Grundlinien einer Theorie der Farbenmetrik im Tagessehen. Annalen Der Physik 4
(63): 427–456.
Schrödinger, E. 1920d. Grundlinien einer Theorie der Farbenmetrik im Tagessehen. Annalen Der Physik 4
(63): 481–520.
Schrödinger, E. 1924. Über den Ursprung der Empfindlichkeitskurven des Auges. Die Naturwissenschaften
12: 925–929.
Schrödinger, E. 1925a. Über Farbenmessung. Physikalische Zeitschrift 26: 349–352.
Schrödinger E. 1925b. Über das Verhältnis der Vierfarben- zur Dreifarbentheorie. Sitzungsberichte der
Akademie der Wissenschaften in Wien. Mathematisch-naturwissenschaftliche Klasse, Abteilung 2a
134, 471–90.
Schrödinger, E. 1926. Die Gesichtsempfindungen. Müller-Pouillets Lehrbuch der Physik, 2/1, 11, 456–560.
Auflage. Braunschweig: Vieweg.
Schrödinger, E. 1935. Les Prix Nobel en 1933 (Autobiographical Sketch), 86–8. Stockholm: Norstedt &
Söner.
Schrödinger, E. 1944. What is life? The physical aspect of the living cell. Cambridge: Cambridge University
Press.
Schultze, M.J.S. 1866. Zur Anatomie und Physiologie der Retina. Archiv Für Mikroskopische Anatomie 2:
175–286.
Stiles, W.S. 1946. A modified Helmholtz line-element in brightness-colour space. Proceedings of the Phys-
ical Society 58 (1): 41–65. https://doi.org/10.1088/0959-5309/58/1/305.
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

The Helmholtz legacy in color metrics: Schrödinger’s color theory 635
Stiles, W.S. 1972. The line element in colour theory: A historical review. Color Metrics 1–25.
Turner, R.S. 1994. In the eye’s mind, vision and the Helmholtz-Hering controversy. Princeton: Princeton
University Press.
Von Helmholtz, H. 1855. Ueber die Zusammensetzung von Spectralfarben. Annalen Der Physik 170 (1):
1–28. https://doi.org/10.1002/andp.18551700102.
Von Helmholtz, H. 1885–1896. Handbuch der physiologischen Optik. 2. umgearbeitete Auflage. Hamburg
und Leipzig: Leopold Voss; 1885–1896.
Von Helmholtz, H. 1909–1911. Handbuch der physiologischen Optik, Dritte Auflage. Hamburg und Leipzig:
Leopold Voss.
Von Helmholtz, H. 1891. Versuch einer erweiterten Anwendung des Fechnerschen Gesetzes im Farbensys-
tem. Zeitschrift Für Psychologie Und Physiologie Der Sinnesorgane 2: 1–30.
Von Helmholtz, H. 1892a. Versuch, das psychophysische Gesetz auf die Farbenunterschiede trichromatis-
cher Augen anzuwenden. Zeitschrift Für Psychologie Und Physiologie Der Sinnesorgane 3: 1–20.
Von Helmholtz, H. 1892b. Kürzeste Linien im Farbensystem: Auszug aus einer Abhandlung gleichen Titels
in Sitzgsber. der Akademie zu Berlin. 17. Dezember 1891. ZeitschriftFür Psychologie Und Physiologie
Der Sinnesorgane 3: 108–122.
Von Kries, J.A. 1882. Die Gesichts-Empfindungen und ihre Analyse. Leipzig: Verlag von Veit & Comp.
Vos, J.J. 2006. From lower to higher colour metrics: a historical account. Clinical and Experimental Optom-
etry 89: 348–360. https://doi.org/10.1111/j.1444-0938.2006.00091.x.
Vos, J.J. 1979. Line elements and physiological models of color vision. Color Research & Application 4:
208–216. https://doi.org/10.1002/col.5080040406.
Vos, J.J., Friele, L.F.C., and Walraven, P.L. 1972. Color metrics: Proceedings of the Helmholtz memorial
symposium on color metrics September 1–3, Driebergen 1971. Holland: AIC.
Vos, J.J., and P.L. Walraven. 1991. Back to Helmholtz. Color Research and Application 16: 355–359. https://
doi.org/10.1002/col.5080160604.
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