Art in Shadows of the Six-dimensional Cube
Year: 2011 Authors: László Vörös
Core claim
A 6-cube 3-model can produce stable shadow images and reconstructive unit mosaics that connect higher-dimensional geometry with artworks by Vasarely and Farkas.
Topics
hypercube projections, tessellation, impossible forms, spatial reconstruction, op art
Domains
higher-dimensional geometry, orthogonal projection, regular polygons, Archimedean solids, optical art, mosaic design, frieze and relief forms, visual perception
Methods
geometric modeling, orthogonal projection, spatial reconstruction, tessellation analysis
Media
AutoCAD, AutoLISP, photographs of artworks, shadow images
Paper text
The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.
Bridges 2011: Mathematics, Music, Art, Architecture, Culture
Art in Shadows of the Six-dimensional Cube
László Vörös Institute of Architecture Pollack Mihály Faculty of Engineering University of Pécs 7624 Pécs, Boszorkány utca 2. Hungary E-mail: vorosl@pmmk.pte.hu
Abstract
The three-dimensional model of more-dimensional cubes can be constructed on a rotational axis and on the joining central point in symmetrical form, based on a regular polygon. An orthogonal projection of this kind of model of the six-dimensional cube shows an image, like the projection of the cube in the direction of its diagonal, perpendicularly to the plane of the image. The projection of any derived (6 > j > 2)-dimensional solid fits to the network of triangles joined by their sides in this method. The hull of the 6-cube’s 3-model may be the Archimedean truncated octahedron as well and the top view of the 3-model of a derivative 3-cube shows a special shadow casted by parallel beam of light. Based on all this, a reconstruction maintaining the topology of the forms made up of cubes, like hinted by the pictures for instance of V. Vasarely and T. F. Farkas, is possible. These hold latent unit mosaics of tessellations and in this manner may inspire to construct geometrical structures of further creations.
Three-dimensional Models of More-dimensional Cubes and Shadows of Derived Cubes
The projection of the cube in the direction of its diagonal, perpendicularly to the plane of the image, creates a special shadow. The images of the edges give the face and half-diagonals of a regular hexagon. This kind of projection of normal cubes joined at their faces fits onto a network of regular triangles joined at their sides (Figure 1).
In general: lifting the vertices of a -sided regular polygon from their plane, perpendicularly by the same height, and joining with the center of the polygon, we get the edges of the -dimensional cube (-cube) modeled in three-dimensional space (3-model). From these the 3-models or their polyhedral surface (Figures 1 and 2: top, elevation and general views) can be generated by the well known procedure of moving the lower-dimensional elements along edges parallel with the direction of the next dimension [9, 13, 17]. We may think simply about the moved vertex, edge and face of a cube and we gain the 3-model of the 4-cube by sliding the cube along new parallel model edges joining the vertices.
A shadow of this kind of model of the 6-cube shows an image, like the projection of the normal cube described in the first paragraph, independently from the angle between edges and the basic plane of the construction. (Based on the latter, one of the derived three-dimensional elements could be congruent with a simple cube, but the image of this remains unchanged.) The projection of any derived (6 > j > 2)-dimensional solid fits to the network of triangles joined by their sides in this method (Figures 4-5).
Former proofs show that by the parallel sliding of edges in case of three Archimedean solids, we obtain special 3- models of the 6-, 9- and 15-cubes inside these solids [12]. The hull of the 6-cube’s 3-model is the Archimedean truncated octahedron. The top view of a derivative 3-cube’s 3-model shows a special image. It is the same one like a shadow of a normal cube casted by parallel beam of light onto the plane of a face of the cube (Figure 3).
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Vörös
Figures 1-2
Figures 3-5
Connection to Arts
The wired shadows of a cube can be interpreted in two ways on behalf of the inner vertices’ visibility (middle upper part of Figure 1). The lacking eighth of a cube shown in regular hexagonal shadow as a solid, is perceived either as a negative or a positive form. These phenomena are known in psychology through the experiments of Necker and Koffka [7,16]. Numerous works of art were created with this perception and interpretation phenomenon and the above discussed geometrical base structure. Two-dimensional images often show “impossible” forms, seen as non interpretable in 3 dimensions. We even tend to perceive objects that can definitely be reconstructed as “impossible”, in cases of more complex geometrical structures, more so if the applied light effects and “distractional” portrayal tools encourage us to do so. Naturally, this is not always and not only the sole purpose of the work. Based on all this, a reconstruction maintaining the topology of the forms made up of cubes, like hinted by the pictures, is possible. From the transformations of the placement in space, new images can evolve. Further modifications and additional sources of light could help to perfect the illusion but our first aim is to show some examples by spatial reconstructions of Vasarely’s and Farkas’s pictures. Some of these or their parts are latent unit mosaics of tessellations and in this manner may inspire to construct geometrical structures of further creations. Our 3-models of the 6-cube can be the generalized base of the discussed interrelations.
Art in Shadows of the Six-Dimensional Cube
Three Examples Based on Victor Vasarely’s Pictures
The first picture indicates two different oriented, alternating spatial interpretations (Figure 6). Thus we think first of an impossible three-dimensional shape. The first reconstruction is based on models of two derived 3- and 4-dimensional parts of the 3-model of the 6-cube. With our solution, we can create a tessellation as well (Figures 7-9). After further investigation, it turned out, we can have a reconstruction with the same parallelepiped or simple cubic form (Figure 10). It can be the model of the derived 3-dimensional element of the above model of the 6-cube. We can see an example for tessellation with the gained spatial unit in figures 10-12. Other possible arrangements with the same unit can result in statically better solutions to create real two sided frieze- or relief-like products from stones detached by faces.
Figure 6: Victor Vasarely: Gestalt bleu, 1969 Photo: Xavier Zimbardo [2]
Figure 7-8
Figure 9: Tessellation
Fig. 10: Spatial unit
Fig. 11: Tessellation in frontal view
Fig. 12: Tessellation in backside view
The 3-dimensional reconstruction of the next picture is constructed from our first model of the 6-cube and from derived elements of this (Figure 13). Taking a wider frame, the gained centrally symmetric mosaic may be repeated periodically by mirrors on the border lines (Figures 14-15).
The hull of the 6-cube’s 3-model can be the Archimedean truncated octahedron as well. The top view of the 3-model of a derivative 3-cube is a special shadow described above (Fig. 3). We can build the spatial reconstruction of the third work of Victor Vasarely with this base element (Fig. 16).
Vörös
Fig. 13: V. Vasarely, Duo-2, 1967
Fig. 14: Selected elements and the reconstruction as unit mosaic
Fig 15: Some repetitions
Fig. 16: V. Vasarely, Felhoe, 1989
Fig. 17: Tessellation
Two Examples Based on Tamás F. Farkas’s pictures
Fig. 18-19: T. F. Farkas, Picture from a cover art [5] - Spatial reconstruction
Art in Shadows of the Six-Dimensional Cube
Figure 20: Spatial transformations of the reconstruction
We can build the spatial reconstruction of the first picture with two derived 3-dimensional elements of the 6-cube’s 3-model based on the former principles. We may gain six different pictures out of the eight ones by mirroring this form to the planes of the rectangular coordinate system (Figures 18-20).
Fig. 21-22: T. F. Farkas, Tibet I. 1997, (cut) [14] - Spatial reconstruction and tessellation
Vörös
The base of the second picture can be reconstructed from a unit mosaic built with only one derived three-dimensional element of the 6-cube’s 3-model and we may continue the tessellation (Figures 21-22). The backside view shows naturally a new picture of this periodical tiling. Other possible arrangements with the same basic element can result in statically better solutions built from stones detached by faces.
Remarks
This paper describes some specialties of a wide topic that naturally could not be detailed due to the necessary limit of size. More references listed below can be reached on the internet and may help in studying the foreground by related references as well. The creation of the constructions and figures was aided by the AutoCAD program and AutoLISP routines developed by the author.
References
[1] H. S. M. Coxeter, Regular Polytopes (3rd ed.) Dover, 1973 [2] Fondation Vasarely, La Fondation Vasarely – de l’op art a la cité polychrome du bonheur, Images En Manoeuvres Éditions, ISBN: 978-2-8499-5171-2, in English, French, Hungarian, 2010 [3] R. L. Gregory – E. H. Gombrich, Illusion in Nature and Art, Duckworth, London, 1973, in Hungarian: Illúzio a természetben és a művészetben, Gondolat, Budapest, 1982 [4] Gy. Darvas, The Art of Tamás F. Farkas, KoG Vol. 10. No. 10. p. 53, 2006 [5] T. F. Farkas, cover art, KoG Vol. 10. No. 10, 2006 [6] T. F. Farkas, Impossible Ornaments, Bridges Pécs – Mathematics, Music, Art, Architecture, Culture, Proceedings, pp. 513-514, 2010 [7] Lj. Radovic and S. Jablan, Vasarely’s Work – invitation to Mathematical and Combinatorial Visual Games, Bridges Pécs – Mathematics, Music, Art, Architecture, Culture, Proceedings, pp. 127-134, 2010 [8] L. Vörös, Reguläre Körper und mehrdimensionale Würfel, KoG Vol. 9. No. 9. pp. 21-27, 2005. [9] L. Vörös, Two- and Three-dimensional Tiling Based on a Model of the Six-dimensional Cube, KoG Vol. 10. No. 10. pp. 19-25, 2006 [10] L. Vörös, Specialties of Models of the 6-dimensional Cube, Bridges Pécs – Mathematics, Music, Art, Architecture, Culture, Proceedings, pp. 353-358, 2010 [11] L. Vörös, N-Zonotopes and their Images: from Hypercube to Art in Geometry, Proceedings: Library of Congress Cataloguing-in-Publication Data : Weiss, Gunter (editor on chief) and International Society for Geometry and Graphics ISGG = Proceedings of the 13th International Conference on Geometry and Graphics Dresden, August 4-8, 2008 ISBN 978-3-86780-042-6 electronic book / International Society for Geometry and Graphics ISGG (ed.) - Gunter Weiss (ed.). - Dresden: ERZSCHLAG GbR, Aue, 2008 [12] L. Vörös, Regular and semi regular solids related to the 3-dimensional models of the hypercube, 13th Scientific-Professional Colloquium on Geometry and Graphics, Poreč, 2008 http://www.grad.hr/sgorjanc/porec/abstracts.pdf [13] E. W. Weisstein, Hypercube, From MathWorld- A Wolfram Web Resource - http://mathworld.wolfram.com/Hypercube.html [14] Artportal: http://artportal.hu/lexikon/muveszek/f_farkas_tamas [15] KoG: Scientific and Professional Journal of Croatian Society for Geometry and Graphics http://hrcak.srce.hr/index.php?show=toc&id_broj=845 [16] http://www.mi.sanu.ac.rs/vismath/jadrbookhtml/part48.html [17] http://en.wikipedia.org/wiki/Hypercube [18] http://icai.voros.pmmf.hu