The Development of the Geometric Concept of Relief Perspective

Abstract

The aim of this paper is to give an overview of the history of relief perspective from its origins in Palladio’s century so as to position the mathematical concept of relief perspective and its development in the context of intellectual history and the philosophy of science. While this concept emerged in art as bas-reliefs and in scenography, its mathematical formulation occurred only in the nineteenth century. Sixteenth-century scenographical perspective, in particular, planted an important seed for the later mathematical concept: here the example is Vicenza’s Teatro Olimpico by Andrea Palladio—a project that was actually built and is extant today. The main stages of the development of relief perspective are analyzed herein via the reconstruction of approaches taken by notable authors such as Breysig, Poudra, Staudigl, and Burmester. Studying this concept historically offers us a deeper comprehension of the intellectual history of projective geometry and its connection to scenography and architecture.

Introduction

The historical development of the geometric concept of relief perspective as the perspectival transformation of spatial objects reveals fascinating interrelationships among art, scenography, architecture, and mathematics. This article provides an overview of this history, tracing the foundations of relief perspective as a concept back to Palladio and the sixteenth century.

The initial aim of the invention of perspective was to represent space and spatial objects on a two-dimensional surface as they were seen by a viewer from a specific viewpoint. The descriptions of Alberti as well as the perspective machines of Brunelleschi (the mirror device) and Dürer were important first steps (Leopold 2014b: 226–228). Efforts in art to represent depth in perspective representations followed. The beginnings of relief perspective are found in bas-reliefs, where perspective images are formed with physical, material depth. Examples of bas-reliefs as three-dimensional transformations of perspective images include the “Gates of Paradise” by Lorenzo Ghiberti at the Battistero di San Giovanni in Firenze (1425–52). An architectural example of relief perspective is Donato Bramante’s Chiesa di Santa Maria presso San Satiro in Milan (1479–99). As there was not enough space to expand the old chapel of San Satiro without impacting a downtown street (Amoruso 2016), a perspectival illusion of the standard Latin cross plan for the church was realized instead within the actually truncated shape (Fig. 1).

Fig. 1

Plan and 3D-model of the truncated cross plan of Chiesa di Santa Maria presso San Satiro by student Marcello Felice Vietti, TU Kaiserslautern 2014

Another early architectural example, more like a bas-relief, is a façade cladding of the Scuola Grande di San Marco, today the Ospedale, in Venice, completed in 1495 by Mauro Condussi (Fig. 2a). Even before the Renaissance, during Gothic period, we can see similar illusions of depth, for example in the portals of the west façade of the cathedral at Amiens (Fig. 2b), which could be called a built perspective.

Fig. 2

a Façade of Scuola Grande di San Marco in Venice, 1495, photo by the author; b portal of the Cathedral in Amiens, 1220–1270, photo Henri le Secq 1852. Henri le Secq [Public domain], via Wikimedia Commons, https://commons.wikimedia.org/wiki/File:Cathedral_of_N%C3%B4tre_Dame,_Amiens.jpg

There are many other examples of relief perspectives that range from perspective image to bas-relief, for example the tabernacle in Venice’s Basilica San Marco and in the Chapel of Saint Anthony in the Padua Basilica.

Two opposing aims of relief perspective become apparent from these examples in art and architecture: to give a flat (or almost flat) perspective image greater depth and sense of space (as in the bas-reliefs), on the one hand, and to compress a three-dimensional object through perspectival shortening to reduce its size and depth (as in the architectural examples), on the other hand. These contrasting aims nevertheless share the fact that their simulated depth (either extended or contracted) corresponds with a viewer gazing from the point of view of a perspective image. The concept of representing spatial objects with perspective drawings was thus extended by spatial perspective representations. It is not coincidental that these techniques and the concepts behind them emerged after the invention of perspective and following new developments in theater stage design during the century of Andrea Palladio.

Scenography and Palladio’s Century

One essential source for the development of relief perspective lies in scenography. To build a theater—and, especially, to design a stage—gave early moderns the opportunity to reflect upon the relationship between stage and audience. With the revival of ancient theatrical literature during the age of the Renaissance, new theater stage concepts had to be invented. Turning away from medieval stage design, which permitted simultaneous performance, Renaissance stage design sought to present a precisely formulated pictorial thought (Beyer 1987: pp. 15–17). In this way, the perspective image and its spatial variant became important. Likely early in the sixteenth century, designers began using perspectival decorations on theater stages and what had been the medieval simultaneous stage became the perspective stage. Several art historians report that the 1508 performance of Ludovico Ariosto’s La Cassaria in Ferrara employed one of the first perspective stages (Hass 2005: 178); later ones have been reported from performances at Urbino (1513) and Rome (1515) and are described as natural extensions of the scenically represented space on stage. Baldassarre Peruzzi (ca. 1515) set a perspective stage inside the Vatican palace for the comedy La Calandria. Hara (2016) refers, in her study on this stage, to the script by Cardinal Bernardo Dovizi da Bibbiena and to Giorgio Vasari’s 1568 Lives of the Artists, which she quotes as follows:

Baldassarre made two scenes that were marvellous and that opened the way for those made in our times. Nor could one imagine how he, in such a narrow place, made room for so many streets, palaces, and various temples, balconies and cornices so well made that they seemed not imitations but very real, and the piazza not painted and little but real and very large (quoted in Hara 2016: 589).

Some drawings by Peruzzi are preserved in the Uffizi Gallery, but no preparatory drawing for the set of La Calandria is extant. Vasari’s words are very similar to a statement about the aim of a perspective theater stage by Sebastiano Serlio. In the second book of his Tutte l’opera d’Architettura et Prospettiva, first published 1545 in Paris, he described:

… una scena, dove si vede in piccol spatio fatto dall’arte della Prospettiva, superbi palazzi, ampliffimi tempij, diversi casamenti, & da presso, & da lontano spatiose piazze ornate di varij edificij, drittissime & lunghe strade incrociate da altre vie, archi trionfali, altissime colonne, piramide, obelischi, mille altre cose belle… (Serlio 1584: 48r).

…. a scene, where we see in a small space created by the art of perspective, superb palaces, very large temples, various houses, and from close by and far away spacious squares adorned with various buildings, very straight and long streets crossed by other roads, triumphal arches, very high columns, pyramids, obelisks, a thousand other beautiful things… (my trans.).

For theater stages, a question arose regarding how these spatial impressions of depth could be achieved in a small room. Serlio thus described how to create theater stage designs such that a viewer would be presented with the sense of spatial depth but not its physical reality. His model posits a bottom line D and an eye level A (Fig. 3b): the most realistic perspectival illusion would therefore be seen from the first row of seats G. From stage height C a platform rises slightly from B to A. The screen wall P, on which a perspective drawing appears, is installed at a small distance from the rear wall M. The horizon O is at the eye level of an actor standing on the stage. The line between O and L intersects the screen wall at a point that serves as the vanishing point for the perspectival representations depicted on the screen. At the same time, the line B-O is used to align the perspectivally altered floor pattern (see Fig. 3b). This drawing demonstrates that Serlio’s stage concept considers the relationship between stage and audience to be essential.

Fig. 3

a Stage designs by Sebastiano Serlio, Della Scena Comica; b stage concept. Images: (Serlio 1584: 47–49), letters enlarged by the author

These early perspective stage concepts indicate that the idea of transforming space according to a single viewpoint was driven by the practical needs of scenography to show large buildings, palaces and streets in a small room, as described by Vasari and Serlio. The invention of perspective drawings found its way during the 1500s to spatial perspective stage design. There were additional perspective stage designs in this period, especially several attributed to Andrea Palladio. Massimiliano Ciammaichella (2018) has reconstructed these from drawings and frescoes and suggests that they acted as models for Palladio to test his idea of theater. Palladio analyzed and drew many Roman theaters in plan and in section. The drawings reveal his interest in the spatial relationship between stage and audience. It was he who established a middle axis and conceived of theater as a site of visual relations, an idea he implemented in the Teatro Olimpico.

Vicenza’s Teatro Olimpico, designed in 1580 by Andrea Palladio and completed in 1585 by Vicenzo Scamozzi, realized Palladio’s theater concept. The first free-standing roofed theater since Antiquity (Beyer 1987: 7), it is an impressive building that remains accessible as an example of the scenographic idea of presenting spatial depth on stage and that had, therefore, a significant impact on stage design and, later, on mathematical relief perspective. Palladio was commissioned by the Academia Olimpica to design the theater; it was his last great work. Palladio had already illustrated Daniele Barbaro’s Italian translation of Vitruvius’s De architectura (Barbaro 1556) which included floor plans for Roman theaters and an elevation for the scaenae frons, the permanent architectural background of a theatre stage. Figure 4 shows perspective street views similar to those built for the Teatro Olimpico. Palladio’s son Silla took over the construction of the theater after Palladio’s death, while Scamozzi took care of the illusionistic scenes for the first theater performance scheduled by the Accademia, the Athenian tragedy Oedipus (Amoruso et al. 2019: 428).

Fig. 4

Scaenae frons of the Roman theater, I dieci libri dell’architettura di M. Vitruvio tradutti et commentati da Monsignor Barbaro, Libro Quinto. Image: (Barbaro 1556: 156), drawing by Andrea Palladio. http://doi.org/10.3931/e-rara-7582

The on-stage space of the Teatro Olimpico offers seven viewing directions with perspectivally transformed corridor spaces, as shown by this floor plan and longitudinal section (Fig. 5). It is interesting to see that, given how the stage is related to the audience, the corridors’ different angles allow different views for the audience from different seats. Figure 6a shows the mise-en-scène of the Teatro Olimpico with visitors on stage. The drawing makes clear that actors played on the horizontal proscenium stage in front of the stage wall, which contains niches with statues of figures, a central barrel vault opening to streets with buildings, and, symmetrically on each side, two rectangular openings with views into streets with buildings (Fig. 6b).

Fig. 5

Image: Ingberg Image Archive: https://www.flickr.com/photos/144598141@N07/albums/72157675822931166

a Floor plan of Teatro Olimpico in Vicenza; b longitudinal section, drawings by Ottavio Bertotti Scamozzi, 1776

Fig. 6

a Drawing of the stage in Teatro Olimpico around 1750, unknown artist. https://commons.wikimedia.org/wiki/File:Print,_Prospetto_Interno_del_Teatro_Olimpico_Nella_Citta_di_Vincenza_(View_of_the_Stage_of_the_Teatro_Olimpico_at_Vicenza),_ca._1750_(CH_18433643).jpg); b photo of Teatro Olimpico today by the author

Scamozzi built other versions of theater stages with perspectival transformations, in particular the theater in Sabbioneta of 1590 (see Baglioni and Salvatore 2017), and similar concepts emulating Serlio, Palladio, and Scamozzi were employed for theater construction in other countries. Joseph Furttenbach’s theater stage designs published ca. 1640 in Ulm, Germany, are one such example, though they are much simpler, with a focus upon one perspectivally transformed stage. Furttenbach did, however, attempt to refine the concept with prisms for changeable stages (Fig. 7), though while we have books with descriptions and drawings by Furttenbach (1640, 1663), no information about any stages actually built could be found for this study.

Fig. 7

Two excerpts from Furttenbach’s Stage Design (1640: 219, 222)

Returning to Italy, we must also note Guidobaldo del Monte, who described in De Scenis, Perspectivae libri sex (1600) how to design a stage (Fig. 8), as well as Andrea Pozzo’s scenographical innovations in his Perspectiva Pictorum et Architectorum of 1707 (Fig. 9).

Fig. 8

Guidobaldo del Monte, De Scenis. Perspectivae libri six (del Monte 1600: 283; 292)

Fig. 9

Andrea Pozzo, Perspectiva Pictorum at Architectorum. De Teatris Scenicis (Pozzo 1707: Fig. 72)

The necessity of transforming the theater stage according to the various viewpoints of the audience drove the development of perspectival transformations of spatial objects such as streets and buildings. The concept of perspectival theater stages as it arose in Palladio’s century thus had an important impact on later stage designs, especially through preserved examples like the Teatro Olimpico. The scenographical approach, therefore, constitutes an important foundation for the much later mathematical formulation of relief perspective in the context of projective geometry. Theater stage design in fact remained the primary application of relief perspective in books by nineteenth-century mathematicians.

The Mathematical Concept of Projective Geometry

The systematically worked out concept of relief perspective in geometry was stimulated by built perspectival scenography, but could be developed mathematically only after the establishment of projective geometry. This field had been prepared by the introduction of vanishing points and lines as well as by the mathematical concept of perspective as a transformation of figures, which were introduced in the eighteenth century and refined in the nineteenth, the history of which is traced in brief below. As so often occurs, artistic preceded scientific developments.

The first mathematical step towards projective geometry was the concept of vanishing points. Although Guidobaldo del Monte had introduced the punctum concursus ca. 1600 (Fig. 10), it took some time for vanishing points and lines to be articulated by Brook Taylor as a clear mathematical concept ca. 1715. Taylor defined the vanishing point as the point where the visual ray, which is parallel to any original line, cuts the plane of projection. The vanishing line is defined, accordingly, as the intersection line of a plane through a point of view that is parallel to an original plane. As seen in Fig. 11, Taylor used the general case with an oblique plane of projection for his definitions.

Fig. 10

Introduction of punctum concursus by del Monte (1600: 42)

Fig. 11

Introduction of vanishing point and line by Taylor (1715: Plate 1, Fig. 2)

The final formulation of projective geometry with the introduction of points and lines at infinity can be found in Karl Georg Christian von Staudt’s Geometrie der Lage (Staudt 1847), which posited that points and lines at infinity are projected to vanishing points and lines, while in the reverse direction the vanishing points and lines have points and lines at infinity as their preimages: “Zwei Geraden, welche in einerlei Ebene liegen, haben entweder einen Punkt gemeinsam oder einerlei Richtung. Zwei Ebenen haben entweder eine Gerade gemein oder einerlei Stellung." (Two straight lines, laying in one plane, have either one common point or a common direction. Two different planes have either a common straight line or a common position) (Staudt 1847: 23). A more abstract mathematical step had been achieved, one that was necessary for the development of projective geometry. Staudt also described the projective relationship between figures and spatial systems, which had been formulated before, by Jean-Victor Poncelet in particular.

The second mathematical step towards projective geometry was the transformational concept of plane figures, then expanded to spatial figures. Johann Heinrich Lambert’s “Perspectograph” (Fig. 12) was able to transform a plane figure (here a garden plan) into a perspective figure. The figures are set in relation to each other; the transformations are described by their invariants and are translated by machine movements.

Fig. 12

Perspectograph by Lambert (1752: Tafel IX)

After Lambert’s invention of the Perspectograph, Poncelet articulated an understanding of projective transformations by studying the characteristics of projective figures in his Traité des propriétés projectives des figures (1822). In the Supplement sur les propriétés projective des figures dans l’espace, he included a section entitled “Des figures homologiques dans l’espace, ou de la perspective relief; application au tracé des bas-reliefs” (Homological figures in space, or relief perspective; application to the drawing of bas-relief) (Poncelet 1822: 357ff). Poncelet concluded that the homological figures must be relief projections of one another; his concept of transformation remained theoretical, without drawings or practical applications.

Several other mathematicians worked on the development of projective geometry during this period by looking for the projective relationship between geometric figures. Jacob Steiner, for instance, introduced the principle of duality (Steiner 1832). In his Traité de perspective linéaire of 1859, engineer and mathematician Jules de la Gournerie connected theoretical studies on homologic transformation (or collinear transformation, the usual German term) with applications in reliefs and theater stages. Book IX of this work, entitled “Bas-reliefs” (de la Gournerie 1859: 225ff), studies the theory and effects of relief perspective using the term perspective relief.

The First Book on Relief Perspective

As described in the previous section, the idea of relief perspective had been prepared by the conceptualization of projective transformations and their application to spatial figures undertaken by various mathematicians. Johann Adam Breysig, a professor of art in Magdeburg and Gdansk who had been educated as an architect and scene painter, invented the notion Reliefperspektive and wrote a 134-page, 11-table volume with drawings on the topic: Versuch einer Erläuterung der Reliefperspektive zugleich für Mahler eingerichtet (Breysig 1798). Although the book was written for painters, Breysig claimed that the volume formulated the mathematical rules for relief perspective. He introduces the term “relief perspective” early on, explaining that he first wanted to call it Bildner-Perspektive (sculptor perspective), but decided that neither this nor the French haut-relief were the correct terms, given that the method is applicable more widely than to sculpture alone and that not all haut-reliefs are related to perspective. Breysig describes relief perspective drawing methods step by step and introduces important related notions; the volume ends with an index. His annotated drawings apply the rules of relief perspective for example to a basic room and to a more complex room with angular walls and two doors (Fig. 13) as well as to three-dimensional shapes such as a pyramid or a cylinder (Fig. 14). The rooms and shapes as well as the construction of relief perspectives are shown in top and side views.

Fig. 13

a, b Drawings by Breysig, constructing relief perspectives of two rooms in top and side views (Breysig 1798: 88, 90)

Fig. 14

a, b Drawings by Breysig, constructing relief perspectives of a pyramid and a cylinder (Breysig 1798: 94, 96)

After the breakthrough of vanishing elements and projective transformation, several mathematicians with theoretical and applied backgrounds wrote important works on relief perspective. Noël-Germinal Poudra’s Traité de perspective-relief (1860) consists of both text and an atlas with drawings. The book opens with a 50-page report on Poudra’s work that Michel Chasles presented to the Académie des Sciences in 1853. Chasles explains that Poudra understands perspective relief to be the representation of a three-dimensional solid by means of another figure that is also three-dimensional, depending on certain geometric rules similar to the rules of perspective on a plane, thus presenting the eye with a faithful imitation. After reflecting upon earlier works on bas-reliefs, for example by Abraham Bosse, Chasles concludes, confirming my own research, that these authors’ rules had been incomplete and did not form a theory of bas-reliefs:

Le premier ouvrage dans lequel, à notre connaissance, la question ait été envisagée sous un point de vue géométrique, quoique encore exclusivement pratique, date de la fin du siècle dernier. Cet ouvrage, écrit en Allemand, a pour titre: Essai sur la perspective des reliefs, par Breysig, professeur à l’Ecole des Beaux-Arts de Magdebourg (Poudra 1860: 28).

The first work in which, to our knowledge, the question has been considered from a geometric point of view, although still exclusively practical, dates from the end of the last century. This work, written in German, is entitled: Essay on the perspective of reliefs, by Breysig, professor at the School of Fine Arts Magdeburg (my trans.).

In his text, Poudra explained how to derive relief perspectives from an axonometric drawing. He had a systematic view of relief perspective that was based on the work of Poncelet. In the first part of his book he explains the methods; in the second part he discusses the applications. Besides architecture, sculpture, and theater stages, he found applications to garden design. He describes several applications and suggests drawing a rectangular room as an auxiliary construction (Fig. 15a) using a perspective scale (Poudra 1860: 113ff). On Plates 16 and 17 (Fig. 15b), he modifies the plans of Paris’s Notre Dame Cathedral so that the resulting building offers, for the observer entering through the large portal indicated with V, the same appearance though with less actual depth. His figures 25 and 26 shows the floor; and figures 27 and 28 the simplified façade as well as the results of their respective relief perspectives.

Fig. 15

a Relief perspective construction of a rectangular room as an auxiliary construction; b relief perspective construction of simplified Notre Dame in Paris (Poudra 1860: Planche 15–18)

The Theoretical Embedding of Relief Perspective

Several mathematical works with applications on relief perspective, in particular in German, followed during the late nineteenth century. The work of the Austrian Staudigl, Grundzüge der Reliefperspektive (1868), explained the theoretical background and formulated the concept of collinear or homological transformation. The relief perspective and its original are described as two perspective spatial systems that are collinear related spatial structures. Staudigl pointed to the importance of relief perspective in relation to a systematic approach to perspective transformation:

Abgesehen von dem Werthe, welchen derartige Untersuchungen für den Bildhauer haben, dürften dieselben auch für diejenigen, welche sich dem Studium der darstellenden Geometrie widmen, insofern von Interesse sein, als die Reliefperspektive die allgemeinste Projektionsmethode ist, aus der sich die orthogonale, die schiefe und die perspektivische Projektion als spezielle Fälle ergeben (Staudigl 1868: II–III).

Apart from the values which such investigations have for the sculptor, they should also be of interest to those who devote themselves to the study of descriptive geometry, because the relief perspective is the most general method of projection, from which the orthogonal, the oblique and the perspective projection arise as special cases (my trans.).

He illustrates this understanding with different representations of a cube W_o, where the relationship between the full plastic model W_p at smaller scale, relief perspective W_r, and perspective drawing W_z are viewed from the fixed position of the viewer’s eye A_o (Fig. 16). Staudigl explains that all three representations give the same impression of the original cube from the viewpoint A_o because the original and their image points lie along the same visual rays. Figure 16 shows these relationships with an axonometric drawing that includes projection on ground plane RS for better spatial understanding. Projected elements are always marked with a prime symbol′—for example, A_o′ is the projection of the viewpoint or eye A_o on the ground plane. The original cube W_o is standing on this horizontal plane RS (he labels the planes in a manner unusual today: with two Latin capital letters, the first upper left, the second downward right of the boundary lines of the plane) with one side parallel to image plane HG₁, where the result of the perspective drawing W_z is visible. Plane PQ, which is parallel to image plane HG₁, is the starting definition for relief perspective W_r of the cube, in which the perspective image of the backward square of the cube is drawn. The proposition for the relief perspective, then, is that the image of the front square of the cube is not drawn in the same plane PQ but on a plane closer to the eye. The image of point a_o is then a_r. The position of a_r determines the depth of the relief perspective. Staudigl described that relief perspective is the general case whereas the full plastic model and the perspective drawing are special cases.

Fig. 16

Relationship between original cube W_o, full plastic model W_p, relief perspective W_r, and perspective drawing W_z from the viewpoint A_o by Staudigl (1868: 3)

With the concept of relief perspective clearly defined, the spatial relationship of relief perspective transformation can be now expressed in a mathematical way: the half infinite space behind a defined front plane is transformed in a spatial layer between the front plane and the vanishing plane, which is called the relief depth. Figure 17 shows the parameters of the relief perspective and their interdependencies. Relief depth substantially determines relief perspective: relief model depth is derived from set relief depth by fixing the position of the vanishing plane; desired relief depth determines the position of the vanishing plane, also called relief depth. If the relief depth is zero, we get the usual perspective. This indicates that there are two methods for defining a relief perspective with a defined point of view: deciding the relief model depth, as Staudigl did it in his drawing in Fig. 14, or defining the position of the vanishing plane that determines the relief depth. The first method is more like an artist’s approach, considering the depth of the model and the needed material. The second method corresponds better with a mathematical understanding of transformation, where the infinite half space is transformed to the spatial layer. Fixing the position of the front plane, which defines the half space, as well as the position of the viewpoint are necessary in both procedures.

Fig. 17

Concept and parameters of relief perspective, transforming half space in a space layer (Leopold 2014b: 235)

After these fundamental works on relief perspective, the topic came to be included in descriptive geometry textbooks. For example, in his descriptive geometry textbook Die Darstellende Geometrie (Fiedler 1871), Wilhelm Fiedler, a professor in Zürich, included an entire section entitled “Die centrische Collineation räumlicher Systeme als Theorie der Modellierungs-Methoden” (The centric collineation of spatial systems as a theory of modelling methods). He mentioned that the construction of centrically collinear spatial systems would be called Relief-Perspective and lists some applications. Fiedler’s book remains mainly theoretical, with few drawings. Both volumes of the 1884 descriptive geometry book, Lehrbuch der Darstellenden Geometrie, by Christian Wiener, a professor in Karlsruhe, included chapters on relief perspective. The slide on relief perspective shown in Fig. 18a, impressively illustrates the relief perspective concept. The author of the undated image is not recorded. The assumption is obvious that it could be by Christian Wiener, because there is a similar drawing (Fig. 18b) in his book (Wiener 1884: 472), both in frontal axonometries.

Fig. 18

a Relief perspective of a body “Reliefperspective. Abbildung eines Körpers mittels einer räumlichen Centralkollineation”, model 1333 of Göttingen Collection of Mathematical Models and Instruments (http://modelcollection.uni-goettingen.de/index.php?lang=en&r=1&sr=42&m=1333); b drawing by Wiener (1884: 472)

The works on relief perspective by Burmester (1883, 1884), a professor of Descriptive Geometry at the Technical University of Dresden, combine the mathematical background with applications to architecture and theater stages. Burmester published detailed construction drawings (Fig. 19) and photographs of realized geometric and architectural examples (Fig. 20); these models for practical application stand out among works on relief perspective, and fulfill his stated goal to bridge the gap between theory and practical design. His architectural examples, in particular, show transformed spaces, buildings and objects in relation to a viewer. Unfortunately, Burmester’s relief perspective models by Burmester have been lost, though Lordick (2005) has rebuilt them using 3D modelling and 3D printing.

Fig. 19

Construction drawings of relief perspectives by Burmester (1883: Tafel I)

Fig. 20

Relief perspective models by Burmester (1883: Tafel IV)

The axonometric drawings in Fig. 19, upper row, show the spatial concept of relief perspective with indicated viewpoint O, image plane Σ, and vanishing plane Φ of the relief. Image plane Σ defines the half space. The points of the image plane, or as defined in Fig. 17, the front plane, are simultaneously original and relief points. Burmester then shows the relief construction of basic geometric elements like line, plane, and cuboid room. The upper right figure (his Fig. 3 in Fig. 19) scube with distance to the image plane. Burmester describes that the relief perspective can be also defined by the assumption that all edges perpendicular to the image plane should be shortened by half in the relief perspective. The more theoretical approach—defining the relief perspective through the position of the vanishing plane—can be replaced with a more practical approach—defining the relationship between the depth of the original figure and its relief perspective figure. Figure 19, lower figures, also illustrates the relief perspective construction of a cube with a cone in top, side, and front views as well as the relief perspective construction of an obelisk with edges in general position, not parallel to the image plane (Fig. 19). The construction drawings are finally followed by photographs of the realized models: No. 1 Typical Solids; No. 2 Arch Hall; and No. 3 Romanesque Basilica (Fig. 20).

Since the hype around relief perspective died down late in the nineteenth century, relief perspective has been more or less forgotten. There were only a few studies on relief perspective in the twentieth century and chapters about relief perspective in descriptive geometry books became shorter and shorter, disappearing by and by. In a descriptive geometry textbook published in the 1920s in Berlin, for example, relief perspective is still mentioned in the last chapter (Scheffers 1927: II, 424–430), but primarily in relation to its deficiencies, a potential reason for its disappearance. First there is the problem of shadow in sunlight. This problem could be only solved by artificial lighting that mimics the sunlight cast on the original object—a technical problem. A second problem regards theater perspective, because actors cannot be integrated into onstage relief perspective due to their fixed size. He summarizes:

Die Reliefperspektive sucht sowohl die Wünsche des Malers als auch die des Bildhauers zu befriedigen und scheitert daran, dass sie zwei Herren zugleich dienen will (Scheffers 1927: II, 429).

Relief perspective seeks to satisfy both the wishes of the painter and the sculptor and fails because it wants to serve two masters at the same time (my trans.).

Hermann Schaal, a geometry professor at the University Stuttgart was one of the first scholars to pick up relief perspective again. He reflected on the topic in a journal for teachers of mathematics (Schaal 1981), including a figure showing a combination of perspective drawing and a relief perspective model.

Conclusion

By studying the historical development of relief perspective, it becomes apparent that we must differentiate early approaches to perspectivally transformed spaces and objects in art, scenography, and architecture from the systematical working out of relief perspective as a mathematical concept based upon the foundations of projective geometry and the collineation of spatial figures. Nevertheless, stage design, bas-reliefs, and transformed architectural spaces prompted mathematicians to think about the mathematical fundamentals of this concept. The intellectual history of relief perspective brings to light important interrelations, in other words, among art, scenography, architecture and mathematics. This research emphasizes that the roots of relief perspective are firmly embedded in the scenographic inventions of Palladio’s century. The clear articulation of a projective transformational concept in mathematics was the theoretical result of these earlier theatre-related inventions. This conceptualization, in turn, now has the potential to provoke similar impulses in art and architecture that could potentially lead to a fruitful interaction between the disciplines. Reflection on the conditions and influences of the development of this concept supports a better understanding of representation methods across the interrelated disciplines of art, architecture, and mathematics.

A renewed interest in relief perspective could aim at reaching a more comprehensive understanding of the general concept of perspective transformation. Such a conception of relief perspective—as a transformation of solids related to a point of view—could influence architectural design processes, where spatial objects are related to their visual perception. Some experiments on design procedures with perspective transformations (Leopold 2017) have been carried out, for example in rendering the intended view of a designed building or in comparing interior designs that make the same impression from a fixed point of view. Further research connected with architectural design projects could yield yet more interesting applications.