Anamorphic Experiences in 3D Space: Shadows, Projections and Other Optical Illusions
The paper presents recent research on the reconstruction of Anamorphic effects and other optical illusions, shadows and projections, with the use of CAD systems. The first part of the paper is a bibliographical overview about the appearance of optical illusions in art, ranging from the work of Niceron to the extravagant sculptures of contemporary artists such as Markus Raetz. The second part of the paper reports on an educational approach that introduces anamorphic geometries into the teaching of digital methods of representation at Graz University of Technology. There is an overview of the experiments and methodology for constructing optical illusions in a CAD environment as well as examples drawn from student projects. The paper concludes with some observations and remarks relating to the aforementioned educational experience.
KeywordsProjections Optical illusions CAD Didactics Art Anamorphosis Digital representations
The understanding of space through diverse projections and the study of shadow has ordinarily formed part of the curriculum of architectural education. Courses of descriptive geometry traditionally taught with compass and straightedge, are currently enriched with digital media, introducing the ever-growing field of computational geometry. Based on this propaedia we are now moving a step further, utilizing a new vocabulary of anamorphic effects, optical illusions and anagrams which can result in fascinating spatial experiences. In this a game of perception a spectator in either physical or digital space might experience confusion or surprise walking through spaces that look different than what they really are! Nevertheless, the so-called “illusion” always comes down to simple geometric rules. All the aforementioned effects have a concrete geometric explanation. This paper presents research into examples of optical illusion and their digital reconstruction, and an educational process that attempts to understand, analyze, and construct optical illusions and visually ambiguous spaces.
Background and Motivation
Within the agenda of the course “Digital Methods of Representation” at Graz University of Technology, students were asked to employ their skills in descriptive geometry in new ways. MySpace was a course about digital representation held at the Institute of Architecture and Media during summer semester 2013. The course introduced architecture students to 3D modeling in Rhinoceros software. Students had some previous experience with CAD software and also with descriptive geometry, and in the aforementioned course they were asked to combine their previous knowledge and use 3D modeling in new ways. The aim of the course was not to offer a mere software tutorial but to challenge the students to use 3D modeling in new creative ways, challenging the notion of perception, understanding the space and reconstructing spatial uncertainties. The thematic area of three-dimensional optical illusions offered a very fertile ground for such explorations and generated a great variety of student projects while teaching about projections, shadows, Boolean operations and advanced surface editing. This paper reports on the methods developed during this course and the obtained results.
The semester-long course was structured in units of one or 2 weeks duration, conducting a specific sequence of exercises and tutorials, with intermediate submissions during the entire semester. The course was based on a long tradition of similar courses held at the Institute of Architecture and Media, always though exploring a different thematic area. This year the course MySpace was focusing on developing design strategies that challenge the viewer’s understanding of space, comprehending the links between geometry and perception.
A Brief Introduction to Optical Illusions
An optical illusion or visual illusion is characterized by visually perceived images that differ from objective reality. The most common ones are certain 2D configurations that trick the eye, giving a false impression about lengths, inclinations of lines, or tonality of colours. In 3D space, optical illusions usually refer to our ability to perceive perspective views, the understanding of light and shadow as well as our wrongly perceived distance from certain objects. More specifically, Anamorphosis is a distorted projection of an image or object, which only becomes clear when the observer’s point of view originates from the so-called vantage point or it is viewed as a reflection produced by special devices, such as a specific curved mirror surface. The word “anamorphosis” originates from the Greek compound word, whose prefix ana- means back or again, and the word -morphe, which means shape or form. There are two main types of anamorphosis: perspective or oblique, and mirror or catoptric from Greek catoptron, which means mirror.
Artistic References and Objectives
One of the objectives of this research was to study certain three-dimensional optical illusions in order to understand the relationship between geometry and perception. The students that took part in this course used the underlying principles of the examples presented to work on their own case studies as part of the course requirements. Among the aims of MySpace was to create experiences/environments of illusion and surprise through the design of geometrically complex objects and spaces with the use of digital media. Students were challenged to create ambiguous objects where 3D geometry would manifest different meanings when perceived from different points of view. Using 2D curves from numbers or letters as initial input, they would create and edit solids aiming at the design of an Ambigram (Hofstadter 1987). An Ambigram is a word, symbol or artistic form, consisting of one or more 3D elements; it can be understood or read in more than one way when viewed from a different direction, perspective or orientation. This first exercise introduced students to the geometry of projections, the perspective and orthographic views and the editing of solids.
Ambigrams can fall into several categories, ranging from natural, to rotational, to 3-dimensional. One of the most famous ambigrams is the one created by Douglas Hofstadter, American professor of cognitive science, who also coined the term (Polster 2000). For the cover of his book Gödel, Escher, Bach: an Eternal Golden Braid (Hofstadter 2000), where each of the 3 projections to the Cartesian planes recreate the silhouette of the three initial letters, “G”, “E” and “B” representing visually the connections and reciprocities among the works of Gödel, Escher and Bach. Following the same lines and going into mathematics and computation, the sqriancle (SQuare, tRIANgle and cirCLE, according to Sela and Elber 2007) is an object that resembles a square, a triangle and a circle when viewed from three different angles. Understanding the underlying geometrical principles led to the construction of ambigrams for the first set of projects to be undertaken during the MySpace course at Graz University of Technology. The methodology followed will be explained in detail in the following section of the paper.
Similar to Quinn, the work of Felice Varini, which was presented and analyzed by a recent Nexus Network article (Di Paola et al. 2015), also explores the issue of transformation and is closely linked to an architectural framework. It is the spectator who moves within the architecture to find the point of “encounter with the work”, where the components are assembled to represent a precise figure. Varini’s work is related to the surrounding architecture, as he paints on buildings, urban spaces, vertical and horizontal surfaces. The characteristic of his paintings is the unique Vantage Point from which the viewer can see the complete work of art, which usually consists of simple geometric shapes. If not standing at the vantage point, the viewer will only perceive fragmented shapes. The construction process of the works of Quinn and Varini follows the same principles of projection onto irregular surfaces, i.e. on surfaces that are either connected or not, located in variable distances from the Vantage Point.
Just as the work of Quinn and Varini is an interpretation and extrapolation of perspective or oblique anamorphosis in 3D space, so is the work of British artist Jonty Hurwitz translating the mirror or catoptric anamorphosis in actual sculptures. The artist utilizes 3D scans of bodies or animals, which are subsequently distorted according to Pi. The reflection of the sculptures on a cylindrical mirror recreates the model as it would look if none of the transformations had taken place. Cylindrical Mirror Anamorphosis has been studied in depth by Čučaković and Paunović, and their work is documented in their article included in the 17th volume of the Nexus Network Journal (Čučaković and Paunović 2015). They study urban installations and describe their workflow, which ranges from experiments to digital methods undertaken in AutoCAD 3D.
Methodology and Digital Workflow
Before setting up the teaching methodology for the aforementioned course, we studied some precedents in computational design and optical illusions, including some recent work presented in conferences such as Advances in Architectural Geometry, Sigraph and Nexus Network Journal. The course involved the analysis of famous case studies and the implementation of the knowledge gained in the projects undertaken during the semester. The project “Escher for Real” supervised by Professor Elber (Elber 2010) is a study on the “impossible drawings” of Escher, where tangible 3D models are created so that from a certain viewing direction they appear to be identical to the original 2D drawings, while from any other direction they are revealed to be valid yet deformed 3D geometric shapes. In their paper “Shadow Art” Mitra and Pauly utilize outline shadows as a source to try and synthesize 3D objects (Mitra and Pauly 2009). Since multiple shadow images often contradict each other, the authors have worked on a geometric optimization algorithm that computes a 3D shadow volume whose shadows best approximate the provided input images.
Departing from Hofstadter’s iconic ambigrams, students studied possible ways to create solids that would generate a different shape according to the point of view. The method was based on the visual hull, introduced by Laurentini (1994) as the closest approximation of a 3D object that can be obtained from 2D silhouettes (the curves obtained from a parallel or perspective projection on a plane) with volume intersection approach. It is understood that an object cannot be reconstructed in precision based only on its 2D silhouettes. For the same reason a set of 3 2D silhouettes can correspond to several different 3D objects. Within this space of solutions, the students studied different extrusions and projections of a set of curves that were given as input. All students were assigned a different set of curves according to their matriculation number; the curves were generated from the outlines of the numbers. For the 3D experiments, the curves were constructed with a minimum number of control points, so that the resulting surfaces would be easy to manipulate through control point editing. The students implemented tools of geometrical Boolean operations (Union, Subtraction, Intersection), keeping or discarding parts of the solid that would correspond to the intended silhouette.
The planar curves generate 3 different solids through straight extrusion to define the potential space for the development of the Ambigram.
The solids are combined (through solid editing in Rhino) in one closed polysurface that can be viewed from different angles recreating the silhouette of the initial number-curves
Further operations of solid editing and reduction of material.
This second step might include several related processes; the geometrical problem is not to merely find the Boolean intersection of two or more solids. The understanding of topological relationships is of crucial importance, as the combination of topologically different solids would result in a hard-to-solve problem. However this task can be achieved through several different approaches, concerning the combination of shapes, hence the students would need to adopt solutions that would optimize the representational criteria. An added parameter to the Ambigram exercise was the intention to utilize as little ‘material’ as possible, thus reducing the volume where necessary without compromising the resulting silhouette. In that sense, the students would first obtain the visual hull and subsequently perform additional operations, to subtract volumes. “The visual hull is the maximal shape that projects consistently into a set of silhouettes, and is obtained by intersecting visual cones from the corresponding calibrated viewpoints” (Sinha and Pollefeys 2005, p. 2). Thus the perspective view of the Ambigram usually did not reveal the triple interpretation of the shape, but a rotation around it would make the whole process clear and visible to the spectator.
Based on the knowledge gained about projections, and inspired by the artwork of Thomas Quinn, Felice Varini, Marte Haverkamp and by the previously presented research undertaken by the author, the students carried out a second exercise which required that they digitally construct the shadow that an object casts on a given surface. The shadow was cast on multiple surfaces that would contain at least one change of plane.
Selection of the angle of the sun for the given model
Orientation of the camera parallel to the light beam
Creation of a Construction Plane in Rhino perpendicular to the light beam
Setting the camera as axonometric (parallel view in Rhino, CPlane top) as opposed to the default perspective, so that the projected image on the background surface coincides with the silhouette of the object
Extracting the silhouette of the object for the given camera view
Projecting the silhouette onto the “built environment” where the shadow would be cast according to the given Construction Plane.
As a last step before rendering the results, the students would give thickness to their models, through offsetting to solid or extrusion of a surface, thus, based on the shadow, they would create a complete new object that may or may not resemble the initial model.
Both processes described above relate to a design approach where 2D representation is transferred and transformed into 3D representation and back to 2D when projected on a surface. During these transformations, the 3D objects either retain or lose some of their characteristics, for example maintaining their silhouettes but losing their volumetric information, thus leading to completely new objects that trick our perception of space, resulting in different illusions according to the point of view, as the visually perceived images are different from the objective reality.
The use of geometrical curiosities to motivate learning has been also used in a simpler form for younger students, in secondary education. A recent project, the Geometry Playground, a traveling exhibition, initially presented at the Exploratorium Science Museum in San Francisco, encouraged visitors to use spatial reasoning, a kind of thinking where you make mental pictures of shapes and spaces. The students that visited the exhibition and actively explored different geometric experiences, such as anamorphic effects, moving geometries and tiling patterns, improved attitudes towards the focal topic (in this case geometry). According to the scientific reports of the Geometry Playground, studying the response of a number of students, the experience with hands-on geometry experiments helped “to bridge the gap between museum and school, perhaps ameliorating some negative associations students have with school geometry and potentially enhancing the educational effectiveness” (Dancu et al. 2009, p. 3). In the above report it is also discussed that student attitudes toward mathematics and geometry are “often negative, and those negative attitudes can have damaging effects on academic success and later career choices” (Dancu et al. 2009, p. 5). Geometry and computation is often a subject that receives negative preconception from students. There is a strong psychological component that highly influences the learning experience. Introducing a more playful yet challenging syllabus can aid students to discard this negative image and creatively engage in the mysteries of computational geometry. Evaluating the research and teaching experiences described in this paper, we can summarize and conclude with some basic remarks. During the course the students developed their abilities for spatial reasoning while improving their attitude towards geometry and mathematics in general. Such geometry experiments trigger students to think “outside the box” and instigate memorability; students acquire experience in problem solving and gain confidence with complex geometric issues. It was observed, and also verified through the evaluation of an exam on 3D modeling skills, that after the semester-long course, the students responded better to 3-dimensional problems and acquired a skillset that was utilized and further enriched during architectural studio courses.
From the obtained results, it was understood that the thematic area of shadows, anamorphic effects and optical illusions offered great opportunities for formal experimentation and geometry research. Students reacted very creatively to this rather abstract brief, very often exceeding the expectations and the academic requirements for this course. The participants of MySpace were equipped with a whole new skill-set about dealing with projections, different perspectives and shadow casting, while learning about precise 3D modeling and advanced surface editing. The design brief motivated students to think out of the box and create projects that are playful and challenging. The in-depth study of the work of the aforementioned artists opened the path for both students and educators to discover new methods and techniques; this experience highly influenced the way we perceive space. Just as the motto of the exhibition, ‘nothing is as it seems’, young architects should always question the relationship between the visually perceived image and the reality. Especially for contemporary architects, as the digital media very often generates ambiguities among design objects, it is important to be able to distinguish the difference between those designs that could be constructed and those that can only exist on the computer screen. Our encounter with digital objects will always generate certain ambiguities as well as learning opportunities. As developmental psychologist Edith Ackerman explains in her interview “Learning is all about moving in and out of focus, shifting perspective, and coming to see anew […] it is like the art of living itself, as it is about navigating uncertainties rather than controlling what we cannot predict” (Ackermann and Hirschberg 2013, p. 78). Based on the results of the course, but also on the evolution of the ideas during the semester, we can say that the learning experience described above allowed students to learn, engage their minds and enrich their spatial cognition. As Ackermann affirms, it is important to establish a balance among the experience that the learners project and the intelligence gained by interaction with innovative tools.
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