The Geometry of Vision : Hermann Maertens ’ Optical Scale for a Deterministic Architecture

In 1870s the German architect Hermann Eduard Maertens grounded his Optical Scale research on Hermann Helmholtz and Franciscus Donders’ works about the physiology of vision and engrafted it in the tradition of Renaissance perspective and proportion theory’s applications to architecture and urban planning. This article describes the scientific core of his approach, in the context of a general revision of aesthetic enjoyment of artworks and a deterministic reorientation of human knowledge toward the industrial production; his elaboration of a triad of visual angles to determine size and organization of space according to visual targets; the diffusion of visual cones as a graphic tool to include perceptual values in the project; the immediate success of his formula among architects and urban planners but, at the same time, the critical reception of his static concept of urban perception; the means of transmission of his ideas in the XX century and their often unaware long-term influence on some postwar years researches.


Introduction
In the second half of the XIX century, art history, aesthetic reflection and psychology of perception often crossed issues and methods, implicitly seeking for a mutual legitimacy and a developmental orientation.In the fundamental Der Stil (1860-63), Gottfried Semper had linked the three spatial moments of aesthetic perception to the human body: height, width and depth were synonymous with symmetry, proportion and direction.A few years later, Robert Vischer had exposed his idea of Einfühlung or Theory of Empathy in Über das optische Formgefühl: and Beitrag zur Ästhetik (1873), correlating the perceived form with the subject, his mood and his personal predisposition to grasp the meanings and sensations.
The work of the German architect Hermann Eduard Maertens 2 (1823-1898), and in particular his research on the Optical Scale or Optische-Maassstab in 1870s and 1880s, lies in a common area with the various scientific and artistic disciplines that were evolving rapidly in those years.Thus, his work can be analyzed from both the point of view of visual physiology and cognitive psychology, as well as in an architectural and urban planning key.This enquiry focuses particularly on the scientific base of his studies and the reception of his studies on the correct visual distance in the field of urban studies and design, often by unaware scholars and designers of the XX century.

Visual perception at the end of the XIX century
The picture of scientific studies at the end of the XIX century is particularly lively thanks to the engagement of Hermann Helmholtz (1821Helmholtz ( -1894)).His research on the physiology of the eye and vision was part of the activities of the Berlin Physical Society, whose members were concerned with the measurement and representation of space and time.They elaborated, used and refined tools such as "Telegraphy, imaging devices, electromagnetic devices for time measurement, and graphic display of temporal processes connected with light, sound, or neurophysiological phenomena" 3 and, through the pages of the journal Die Fortschritte der Physik, they undertook to find application fields, for example, for their methods of measuring small time intervals through graphical devices capable of enhancing shifts or imperceptible transformations 4 .
In his treatise entitled Handbuch der Physiologischen Optik (1856-1867), Helmholtz not only had pointed out that the human eye sees in detail only a small fraction of the visual field but he also established how this fraction could be measured.Thus, he determined the unit of measurement in the evaluation of the vision, establishing that the size of the figures represented should be subtracted to one-minute arc size at the conventional reading distance.
The work of the Dutch Franciscus Donders (1818Donders ( -1889)), the most accredited founder of the science and art of vision measurement, with significant relapse on ophthalmology, has been engaged properly on this fundamental observation.
Already before his lucky On the Anomalies of Accommodation and Refraction of the Eye (1864) translated into French, German and Italian, Donders had coined the term "visual acuity" to measure and describe the sharpness of vision through the relationship between the result of subject and the average result of the popu-

FABIO COLONNESE
The Geometry of Vision: Hermann Maertens' Optical Scale for a Deterministic Architecture La Geometría de la Visión: la Escala Óptica de Hermann Maertens para una Arquitectura Determinista 62 lation.The most famous result of his studies consists of the so-called optotypes, a series of graphic symbols on a 5x5 square matrix that his colleague surgeon Hermann Snellen (1834Snellen ( -1908) ) translated into letters, perfected and published in the form of the homonymous table that still today is on the wall of every ophthalmologists' room.On the basis of that table, Snellen defined normal the human ability to recognize an optotype when it underwent 5 minutes of arc and then to discriminate a single stroke length of 1 minute arc 5 .
Even today the definition of the limits of the "representable", which is generally

A triad of visual angles
Maertens' approach to the problem of the visual relationship between the façade of a building and the space before it is expressed by a triad of visual angles that set distances and thresholds for three different ways of contemplating architecture.He fixed these angles in a series of profile illustrations: 18° is the angle of field in which an artwork or a building appears to be part of the surrounding context around it in a whole image; when it is seen under a visual angle of 27°, it appears in its integrity and completeness; under a visual angle of 45° or more, the observer's attention is conquered by the details [Fig.3].
To provide historical and operational validity to his conjectures, Maertens worked on two parallel tracks: on the one hand, he elaborated a number of perceptual analyzes of Italian, French and German squares and monumental historical complexes; on the other, he proposed a series of typical design situations to demonstrate how his Optical Scale would optimize the placement and dimensioning of artworks, signs, screens and pathways within a museum room, a street section, a garden or a palace [Fig. 4].His analyzes naturally needed to demonstrate the validity of the approach and to make the system more appealing especially to designers who pursued the inscrutable canons of space harmony of so many historical sites.
While the numerical data after the analyzes resulted into complex oversized tables for the benefit of the most resolute historians, the meta-design devices are illustrated by a series of drawings in section and, rarely, in plan.For example, the com- Fuente: Elaboration by the author.
[Fig.4] The 18, 27, and 45 degrees visual angles in the contemplation of pictures in a gallery, of a monument in a garden, of the facade of a building, and in designing a street section.
Fuente: Hermann Maertens, Der Optische-Maassstab, 1884, figg. 8, 9, 10, 15. bination of plan and section of a garden organized around a monument shows the Maertens proposed investigations and integration of optical correction devices, which were also an heritage of an historical architectural practice at risk of extinction.A separate discourse would deserve the architectural detail.He analyzed it through the same readability parameters that indicate the correct size of the letters on a road sign to be seen in the distance.He had established that, apart from some correction coefficients due to brightness and chromatic contrast effects, the body of the typographical character should be the 3.450th part of the maximum reading distance 8 .He then devoted himself to the elements of classical moldings and their almost imperceptible formal variations in the highest parts of a temple or a church.In the size of the triglyphs or thin dentils, Maertens felt he had found the equivalent of Donders' optotypes, the limit case in which the visual acuity of the observer was to be tested.
Although Maertens measured the visual cone with angular values, it is important pointing out that his angular triad refers to the fundamental ratios of 1: 3, 1: 2 and 1: 1, the divisions resulting maliciously approximated at 18, 27 and 45. 45 is actually the sum of 18 and 27 and the three numbers are in the ratio of 2: 3: 5 among themselves, having 9 as a common denominator [Fig.6] 9 .This arithmetical relationship suggests that Maertens, aspiring to discover and disseminate a law with a universal value, felt it was necessary to design it with some elementary geometric relationship, in order to dress it with some absolute or divine attribute or just to make it easy to remember and apply.9 There is also a fourth ratio, equal to 1:6, approximated to 72°, which would mark the threshold of the panoramic view but it was quite secondary to Maertens' main interests.Of course, Maertens adopted other small licenses.For example, he almost ignored the width of the buildings -their extension on a horizontal plane -to focus on their height, the most important feature to influence human behavior.He also ignored the lower portion of the visual field below the horizon as if it does not affect the distance to look at a building from 12 .Moreover, he assumed that human gaze is kept steadily along a horizontal axis while it instinctively moves to look for the optical focus and the visual balance point of the building, tilting -also due to the differences in the ground -and altering the behavioral pattern assumption.
Human space and the Optical Scale 12 If one considers also the part of the view angle from the horizon plan to the base of the building, the angular triad would become 36, 54 and 90 degrees.In particular, the angle of 54° corresponds to the amplitude that has been measured observing people staring at medium and large objects at a distance of vision corresponding to the extension of the object itself.In particular, the dotted line cones Maertens had drawn in plans and sections to envision the look of pedestrians or tourists marked the imagination of architects and urban planners who aimed at turning the visualizations of their insights into scientific demonstrations.The visual cones, hitherto limited almost exclusively to the design of fortified and theatrical architecture, were adopted throughout the XX century as a visual tool to illustrate a number of different situations.The historian August Choisy adopted them in the analysis of the Athenian Acropolis while the architect Eugène Hénard used them to evaluate the Parisian vehicular traffic.Alexander Klein adopted them in his diagrams to demonstrate the inefficiency of traditional buildings while Le Corbusier used them to suggest a panoramic relationship between architecture and landscape, like in the projects for Algiers.Luigi Moretti used the cones to visualize the optical control of spatial sequences in the rhetorical apparatus of Fascist architecture 16   suggested a ratio between 1:2 and 1:4 between height of enclosing walls and a space dimension is most "comfortable" with the sense of "enclosure", which is lost when the ratio is beyond 1:4 and if less than 1:1, the space being like a "pitch or trench" 20 .

The Visual Planning
The main field of application of the Optische-Maassstab was certainly the urban planning, especially those experiences that critically recovered the picturesque tradition, giving rise to the concept of Townscape and Visual Planning 21 .Eventually, Maertens had the ambition to find a scientific justification for the compositional principles practiced for centuries even if according to several different aims [Fig.9].Although the perspective figurative revolution had taken place in Florence at the beginning of the XIV century, there are only few written notes about the perceptual opportunities offered by the squares, like Leon Battista Alberti's in the epigraph of this article.Only in the second half of XVI century, when Michelangelo's innovative square at the Campidoglio was still under construction, Andrea Palladio defined the squares like those "wide places left in the cities" in order to see "the appearance of some beautiful fabrica and mostly of some Temple" 22 .
The XVII century records a new awareness about the relationship between the right distance and the visual effect of architecture.Jules Hardouin-Mansart questioned on the value of the incidence of the visual angle on the perception of his urban projects23 .Writing about the new axis of the Chapel of the Sorbonne, in a letter to Cardinal Richelieu, it is noted that "without the opening of this avenue and the square, the church would not emerge the half of what emerges now"24 .
Gian Lorenzo Bernini is possibly an international exporter of this new sensitiveness.In order to "see perfectly"25 the facade of his Louvre, he designed a square that was equal to one and a half the height of the palace, a ratio corresponding  The visual approach of Maertens deeply binds the shape and position of the architectural body to its immediate spatial surroundings, making them indissolubly bound.If it is necessary to preserve a specific free distance for the best enjoyment of the monument and volume of the church, then the surrounding visibility area becomes implicitly a part of the building itself, a sort of visual servitude.
Quite the same could be claimed for the path route considered propaedeutic to the visual discovery and exploration of the building.
While this observation contributed to a mature concept of the city as an interconnected system of spaces and volumes, on the other hand it could legitimate also the idea of the city as a collection of monuments.Describing the Cologne Cathedral, for example, Maertens asserted "it is therefore justified the use of leaving space around the most valuable buildings (churches), although this can lead to demolitions" 39 .This and other similar statements may have had more than one consequence on the historical-operational conception of Gustavo Giovannoni.In 1913, he elaborated the famous and notorious "Teoria del diradamento edilizio" that was occasionally applied in the historic center of Rome to reestablish hygienic conditions and visibility to the monuments, at the expense of not only the medieval fabric but also of important buildings.Renaissance-perspective static vision, with an attention to proportion that will be recovered by few modern architects, like Le Corbusier.
The dissemination of Maertens' results was limited by the complexity of his treatises as well the lacking of an organic translation of them.It gradually exhausted in the years between the two world wars and the results of his researches were transmitted partially or unconsciously, thanks to the reproduction of a few of his illuminated diagrams or some rough written synthesis.The Optical Scale did not strike deep into the architect's practice, confined in small appendixes of the most diffused design manuals or graphic devices applied on a plan to remind the presence of a vantage point or to attribute at least a semblance of science to architectural processes and products.From this point of view, his story confirms the congenital difficulty, in some ways ever present, of transforming the design practice into a properly scientific process, as if the designer felt his creative autonomy threatened by principles, methods and tables.But even if an almost indirect way, Martens' ideas fertilized a ground that is common to a number of interdisciplinary studies that deeply marked the development of urban and architectural culture from the 1960s.Even today, the idea that there is a proper way to look at a building, with a precise distance according to the finality and the definition of the detail, is something that could be very useful in the production and use of visual interfaces and images generated by the computer often with no awareness of the physiological behavior of the human eye.
adopted to determine what is to be drawn and what to be excluded at the different scales of architectural representation, is basically the result of the observations of those scholars.Quite appropriately, since the first edition of the Hermann Maertens' treatise, the first illustration included in the text is a pair of Gothic characters over the 5x5 grid of Donders and Snellen: the letters "u" and "i" are there to indicate how all his monumental work is based on the ability of ocular resolution and visual acuity to distinguish, respectively, a single line and to two lines divided by a narrow blank space [Fig.1].

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Fig. 3] Approaching Palazzo Farnese in Rome.The three pictures have been shot with a distance from the palace resulting from the application of 18, 27 and 45 degrees visual angles suggested by Maertens.
Maaastab's efficiency in determining the size and position of the protection fence, the crown of flowerbeds and the path of passage according to visual concentric circles derived by the application of the visual angles of 45, 27 and 18 degrees of the people walking or sitting down around the monument itself [Fig.5].
Maertens' work was a reference to the many designers interested in the aesthetic and narrative possibilities of Raumkunst: from Adolf Loos, who would be willing to size each single room according to the effect to exert on man [Fig.7], to Le Corbusier, who elaborated the idea of architectural promenade as a sequence of visual effects.It could appear as a happy combination of Vischer's idealistic aesthetics and Fechner's empiricism 13 and it gradually contributed to the elaboration of Gestaltungtheorie, bringing a substantial theoretical and operational contribution to an aesthetics of vision based on the physiology of the eye.Maertens's ta-10 Wilhelm Butte, Prolégomènes de l'arithmétique de la ie humaine contenant la classi ication g n rale des talens l chelle des ges de l homme et une ormule d aluation de toutes les situations géographique: d'après un même système (Paris-London-Landshut: Dentu, 1812), 125.11 Albert Eichhorn, Der akustische Maasstab (Berlin: Schuster & Bufleb Architectur-Buchhandlung, 1899).I thank Jesper Cepl for this reference.

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Fuente: Elaboration by the author.

For
example, Ildefonso Cerda analyzed the modern city around the key notions of rest and movement, comforted by a wealth of references to biology, anatomy and medicine, in his Teoría general de l'urbanización (1867).In his inaugural speech at the University of Leipzig in 1893, entitled Das Wesen der Architektonischen Schöpfung, or "the essence of architectural creation", August Schmarsow began to develop the perceptive empiricism in a foundational principle, by assuming the bodily movement through space as the very essence of architecture and condemned the stationary perception of form, as proposed in the same years by Wölfflin and indirectly by Maertens himself.Anyway, it took decades to evolve the Renaissance-born vision scientifically redefined by Maertens in something totally new.Quite in the same years in which Neufert was customizing Maertens' schemes for his manual, Herbert Bayer, another formerly student at the Bauhaus, was elaborating an innovative exhibition structure for a new dynamic and immersive perception, immortalized in 1935 in the famous Diagram of the Extended Field of Vision [Fig.11].Maertens was criticized not only for the implicit stillness of his visual analyses but also for the presupposed frontal vision of the work which prevails in his schemes.Stübben found that his formulas were likely to produce squares that look too dilated 37 .Brinckmann, who was author of analysis and schemes based on Maertens' work, tried to temper his stainless determination through a series of psychological considerations.He emphasized the role of optical illusions and that of the elements referable to the human scale in the optical measurement of building bodies.For example, he recalled the different effect of buildings more or less cut out, articulated or stylistically heterogeneous: a Gothic cathedral is articulated in a myriad of parts and may looks larger than a building of the same size in Doric or industrial style 38 .

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Fuente: Herbert Bayer, "Fundamentals of Exhibition Design", in PM: an intimate journal for advertising production managers, art directors and their associates (New York: PM Pub.Co., 1939-40), 25.

The title, together with the epigraph, is an explicit homage to Helmholtz's work as well as an attempt to share his discoveries about visual perception with artists. According to Maertens, the artists could no longer afford to ignore that an aesthetic judgment is determined by the first impression of a work and that this
In 1877 Hermann Maertens published the first edition of Der Optische-Maassstab or Die Theorie und Praxis des ästhetischen Sehens in den bildenden in the Auf Grund der Lehre der physiologischen Optik, that is "The Optical Scale or Theory and Practice of the Aesthetic Vision of the Arts on the Basis of Optical Physiology".(Bonn: Max Cohen & Sohn., 1884), Front page.