By Rosemary Lee
Chapter 2 of Algorithm, Image, Art
Though it may be an unlikely place to look for insights into the topic of algorithmic media, an atlas from nearly two thousand years ago is a compelling example that bears comparison with modalities present in digital graphics today. Ingrid Hoelzl and Rémi Marie argue for thinking of images as “intrinsically merged with software” (2015, p. 7) in a sense that echoes the aesthetic, processual, and conceptual structuring of images through algorithms. They point out that the approach to mapmaking employed in Ptolemy’s Geography (c. 150 CE) offers meaningful insight into the formulation of images in terms of algorithmic sets of instructions through a relatively simple, analog example in comparison to current forms directly involving the use of software. This chapter looks into what implications the geometric description of real-world phenomena in analog instructional practices may have for algorithmic media, drawing from diverse examples in addition to Ptolemy’s Geography, including instructional approaches in anthropometric systems in drawing.
Contrary to what one might expect of an atlas, what can be found in the pages of Ptolemy’s Geography is not a collection of images portraying the terrain and characteristics of various regions of the earth. Instead, the atlas is composed primarily of text: indexes listing locations and associated numbers, with a few diagrams illustrating how these are to be interpreted geometrically. Geography exists as more of a database than a visual depiction: a structured collection of written sets of coordinates, accompanied by instructions for how to read and draw maps from that data. Employing an analog methodology for storing and visually articulating spatial information, textually, Geography is notable for the way it reveals that algorithmic modalities may exceed the specificity of a given medium, technology, or time period. Hoelzl and Marie note that this has wide-ranging implications for the algorithmic tendencies in recent forms of image-making, namely departing from the paradigm of photography in a turn toward images being thought of as based on data.
“As a program, the image, while still appearing as a geometrical projection on our screens, is inextricably mixed up with the data (physical and digital) and the continuous processing of these data.” (Hoelzl and Marie, 2015, p. 7) In the sense that Hoelzl and Marie discuss, relationships between software and images have not been isolated to such instances directly involving the use of digital computers. As Miguel Carvalhais argues, computational aesthetics “hinges on the discovery of what a computational system does and how it operates” (2022, p. 39) in a way that allows us to see how early aspects of the computational in media we might otherwise consider static. In this sense, aspects of Ptolemy’s atlas, including the geometrical projection of spatial imagery from alphanumeric information, foreshadow methods and theories that have come to have central importance in recent discussions of algorithmic visual media.
One of Ptolemy’s central innovations in Geography was the implementation of a coordinate system by which geographic locations may be plotted according to a grid. Demonstrating a method for geometrically modeling the world, this not only enabled proportional relationships to be maintained between geographic locations and features within a map, but it also permitted the transcription of images in textual form. An image may thus be created following written, algorithmic instructions or the reverse: starting from an image, one may transcribe written instructions for how to reproduce it. The Ptolemaic atlas thereby presents us with a rather elegant analog example of an algorithmic approach to image production, as well as touching upon the interrelation that may exist between images and texts.
Ptolemy’s Geography places emphasis on the execution of images according to clearly defined, sequential operations. Providing written data and instructions for how to decode it into a visual depiction helped to solve the problem presented by the laborious task of duplicating maps by sight alone. Formulating the visual content of the atlas in a way that it could be transcribed in written form also made it easier to faithfully reproduce a map, as the reproductions would be based on the same coordinates. This held significance for the storage and transmission of spatial information, enabling the maps to be re-plotted even centuries after the atlas was originally composed. Transcoding an image or other artifact into an alphanumeric sequence facilitates its storage and transmission in the form of data, which is especially relevant to current contexts involving digital media.
The parameters defined by a given algorithmic system may afford degrees of variation, such as changes of scale while maintaining stable proportional relationships between components, granting images greater potential to be stored, printed, duplicated, or transmitted. Tying images to algorithmic procedure and to data — as opposed to a particular physical instantiation — renders them highly mutable, abundant, and transient. Algorithmic images may be understood as exceeding the singular execution as a physically individuated object. And, in some respects, they may also be integrative of the data and instructions these instantiations derive from. The possibility to have multiple readings of a given set of instructions endows algorithmic images with a mutability more commonly associated with computational media than with analog processes of image-making. For example, digital media can simulate other forms of media or be read or expressed in more than one way across a variety of different media. The instructions for a given algorithmic image may be performed manually, but they also have a great deal of potential to be re-interpreted, misinterpreted, or iterated upon.
In a sense anticipating the seriality and iterability afforded to images by algorithmic approaches currently, Geography was not revolutionary for being an exclusively original work, but rather a reinterpretation from the work of another cartographer. Geography is primarily based on the maps and writings of Marinus of Tyre with only a few updates to select regions (Johnston, 1999). Consequently, Ptolemy’s main contribution with this atlas was not the composition of the maps, themselves, but the methods he employed and explicated in its pages. The etymology of the word geography may be argued to echo this sentiment, by breaking the word down into the Greek “geo” and “graphein.” In connecting the word to these origins, Johnston (1999) underlines the importance of the methodological approach employed, as well as the interrelationship between text and image. Following this logic, we can think of geography as a form of “earth writing” or “earth description”, emphasizing geography not merely as a visualization of the world, in itself, but a way of visualizing the world.
Projections made using Ptolemy’s methods resulted in maps that were aesthetically different from their predecessors, with the added functional implications that arose from creating maps that were more compatible with measurements of the territory represented. When the Ptolemaic atlas was translated from Greek to Latin in 1407, it conflicted with the then-current methods of medieval cartography, which at the time based the relative size of countries on power relations. As a result of Ptolemy’s methods of mapmaking, the map becomes a closer reflection of real-world measurements than the mostly symbolic representations that were prevalent up to that point. Beyond the aesthetic effect this had on the resulting maps, it also gave rise to a greater degree of accuracy in mapmaking. This is important not only for the relationship between maps and the world they are intended to represent but also for how they function as tools, affording particular kinds of relationships with that world. By changing the way that an image acts as a visual representation related to the world, a shift occurs in how that image, consequently, acts on it.
Hoelzl and Marie contend, in contrast to Ptolemy’s own description of the atlas, that what he actually created with Geography was “not ‘a representation in pictures of the whole known world’ but a dataset that allowed Renaissance cartographers to draw what came to be known as the world’s first accurate world maps” (p. 99). Rigorous attention to geometrical relationships and real-world data, as seen in the Ptolemaic maps, indicates a shift toward the development of a view of the world based on mathematic and scientific principles. In contrast to the more ideologically based and symbolic forms of representation that had been in place up to that point, the maps in Geography have not only internal geometric consistency between their compositional elements but these depictions are also intended to correspond to the relative size of objects in the real world.
In Geography, an analog algorithm acts as a structuring mechanism in the visual compositions of maps, allowing the internal proportional relationships between elements to be informed by geometric constraints. Hoelzl and Marie point out that analog geometric compositional strategies were also employed in Ancient Egyptian and Roman art. Ancient Egyptian canons of representation involved systematic rules relevant to contemporary algorithmic media. In such cases, systematic constraints were used to standardize proportional relations internal to visual compositions. According to such an anthropometric system, the relative size of depicted figures and objects was made in accordance with pre-defined rules and units of measure that structured various motifs in relation to a grid.
Adherence to a geometric canon of representation allowed visual elements’ height and width to be specified as multiples of units, enabling a degree of consistency to be maintained in the construction of visual compositions. For example, in Ancient Egyptian canonical representation, the proportions of a human figure would be defined as a set number of units in height and width, with each feature relative to those number of units. Constraining the execution of an image to a standard unit of measure, in this case, the cubit, allowed widespread unity in visual depictions. The cubit was a standard unit of measure in the Ancient world, since c. 3000 BCE. It was based on the length of the human forearm, measured from the elbow to the tip of the fingers. As it was based on a bodily measurement, the size of a cubit varied according to whose body was used as a reference. But as long as a depiction was produced according to the formula, its proportions would be the same as every other iteration.
Similar systematic use of geometric proportions to those employed in Ancient Egypt can be found in what is referred to as the Vitruvian Man. An anthropometric system of proportions, the Vitruvian Man was outlined by the Roman architect Vitruvius in De Architectura c. 27 BCE. It is effectively a formula, according to which the male human body is drawn according to proportional relationships:
For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. (p. 72)
Following such an algorithmic description, the depiction of the human body is thereby guided by formulaic instructions, which constrain its dimensions. While such image-making methods may employ mathematical constraints on the proportions of the depiction of human bodies, they are more concerned with notions of compositional harmony than with representational precision in the sense of producing realistic or accurate depictions. Though based to some extent on direct measurements of actual human bodies, they are generalized, using average sizes of bodily features to approximate a norm from multiple bodies. In addition to working from direct measurements of human bodily proportions, Vitruvius’s anthropometric system also sought to invoke what were thought of as divine relationships in geometry:
if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centred at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square. (1914, p. 73)
Circles, squares, the Fibonacci sequence, and the golden ratio were seen as having special properties and they held connotations of perfection, balance, symmetry, and unity. Using these as guiding principles of visual aesthetics, such as in the Vitruvian Man, sought to draw on conceptual associations through their structuring of spatial relationships. The best-known example of the Vitruvian Man is that drawn by Leonardo da Vinci, where the body of a man is aligned within the proportions of a square and circle.
Echoes of these ancient compositional strategies can also be found in modernist design and architecture. For example, Le Corbusier’s Modulor Man, (1945) was intended as a design tool that would facilitate particular aesthetic relations in design and architecture, relative to the proportions of the human body. Aimed at achieving a degree of aesthetic unity, Le Corbusier defined a systematic approach to the relative measurements of designed objects and spaces.
Various measures are now in use:
The inch and foot by the British (it kept their architecture related to human proportions in spite of the machine age). The meter, derived from the meridian of the globe, is an artificial and arbitrary measure that has nothing to do with human proportions and which, as a result, has led to a certain disintegration in the architecture of those countries which used it.
In view of the immense task of manufacture and prefabrication to be completed, a unified scale of measurement based on the human body had to be created, a highly significant mathematical expression capable of giving innumerable combinations that are really satisfactory and above all harmonious.
Le Corbusier’s development of the Modulor man with the intention of informing the design of functional objects and spaces shows how the same principles used in two-dimensional visual compositions and images may also be applied to three-dimensional contexts such as in architecture. This dimensional application of geometric proportions draws an interesting parallel with the cubit, which was employed both in visual compositions on flat surfaces and in built constructions. Its specific emphasis on geometrical harmony also recalls the same quality in Vitruvius’s system.
Bringing the structuring principles of algorithmic compositional strategies into the design of functional objects and spaces adds a dimension of interactivity that is relevant to our later discussion of how algorithmic forms of visual media may be understood as not just shaping visual aesthetics, but also acting on the world. It’s noteworthy that in the Vitruvian Man, the human figure is normalized as a European adult male, which is described in terms of its presumed universality. The Modulor man is even more specific, tailored to the bodily proportions of an idealized average French man. Positioning such a particular demographic as if it is neutral raises similar issues to those that arise in the tendency of machine learning systems to reiterate and even amplify existing patterns and biases. This is illustrative of how specific assumptions and value judgments may be embedded in an algorithmic system before it is even employed. Recalling Hoelzl and Marie’s description of algorithmic methodologies as not producing visualizations, but acting as a mode of visualization, Luciana Parisi argues in her book Contagious Architecture (2013, pp. 102–7) that parametricism imposes a particular logic upon not only on design but also on interaction with its constructions. According to this perspective, algorithms may be understood to inform not only the outputs of the processes in which they are applied but also the relationships such designed objects and structures have with the world around them.
In each of the cases discussed in this chapter, a set of proportional rules establishes particular mathematical relationships between compositional elements and the human body. These strategies inform the production of images, both in terms of the processes involved, but also guiding the geometric relationships within compositions. In this sense, anthropometric systems such as Ancient Egyptian canonical representation, the Vitruvian Man, and the Modulor system demonstrate how algorithmic relationships can be used to inform visual media, processually, and aesthetically. They also invite comparison with strategies that are currently employed in art and design contexts through comparatively simple methods. It is possible to produce images according to the analog methods described here manually, using only simple tools, whereas more complex, automated image-making systems can be relatively opaque to human understanding.
Looking at analog instances in which systematic implementation of mathematical relationships and algorithmic sets of instructions are used to determine visual outputs enables us to recognize similar compositional strategies across vastly different applications and time periods. Establishing median values for the representation of a human figure based on multiple real-world measurements, as we saw in the Vitruvian Man, bears resemblance to the development of statistical models in machine learning, which are based on the analysis of numerous examples. In parametric design and architecture, complex mathematical relationships inform the design of objects and urban spaces. This draws a parallel with approximation, which in mathematics describes the use of a simpler function in place of another to achieve a more accurate result specific to a given context. Describing real-world phenomena mathematically through the development of models facilitates understanding them in new ways, a theme that cuts across these fairly diverse examples.
Hoelzl and Marie (2015) argue that Geography demonstrates a turn towards “the world as database” in which “software […] is also part of the structure of the image” (p. 83) As they describe, images, now, are often taken as visual representations of the world based on data. According to this perspective, data-based images are not merely visual depictions of the world, but rather, visual depictions derived from data about the world, representing “a world that does not exist in and for itself, but only insofar as its natural and cultural resources and its inhabitants are transformed into the common language of symbols and numbers that allow their universal commensurability” (p. 100)
This is a significant philosophical distinction that lies at the heart of how technologically mediated imaging — including, but not exclusive to algorithmic media — may meaningfully depart from or connect to other image paradigms. While the idea of images as or based on data is one we are quite familiar with today, it is important to recognize that it has not always been a pervasive view of visual media. It is a construct that has been built up over time with quite specific built-in value judgments, cutting across diverse media and time periods. Treating images as interchangeable with, representative of, or the product of data affords certain technical modalities, such as those explored in this chapter or more complex processes covered later in this book, but it also imposes a particular logic on the outcomes of such processes.
As we will further examine in later chapters, the idea of the image as a database offers a useful perspective on images created using machine learning, which have a relatively direct relationship to visual data in comparison to other forms of image-making. This may be thought of in several ways, for example, that training a machine learning model requires many images that ultimately inform the production of generated images. We may also think of machine learning-generated images as a form of data, although this comes with the caveat that machine learning has a demonstrated tendency towards bias and error that undermines the presumptions of objectivity typically associated with data-based processes.
The following chapter expands on the idea that the transcription of images in terms of data or algorithmic sets of instructions, as in the examples covered here, affords images particular qualities that extend beyond attempts at merely capturing or replicating visual appearances. This leads us to discuss the implications of transcodability, seriality, latency, and iterability in image-making, which are important to our later consideration of generative strategies and highly automated forms of visual media. These aspects tie into discussions in later chapters of this book concerning instruction-based practices and the operative image.
Bibliography
Carvalhais, Miguel. Art and Computation. Rotterdam: V2_ Publishing, 2022.
Hoelzl, Ingrid, and Rémi Marie. Softimage: Towards a New Theory of the Digital Image. Bristol: Intellect, 2015.
Johnston, Ron. “Geography.” In Brittanica, 1999.
Le Corbusier. Modulor. 1945. System of anthropometric proportions.
Parisi, Luciana. “Parametricism or Deep Relationality.” In Contagious Architecture: Computation, Aesthetics, and Space, 102–7. Cambridge, London: MIT Press, 2013.
Vitruvius. Vitruvius: The Ten Books on Architecture. Translated by Morris Hicky Morgan. Cambridge: Harvard University Press, 1914.