Advertisement

Nexus Network Journal

, Volume 20, Issue 1, pp 267–281 | Cite as

Double Orthogonal Projection of Four-Dimensional Objects onto Two Perpendicular Three-Dimensional Spaces

  • Michal Zamboj
Geometer’s Angle

Abstract

A method of visualization of four-dimensional objects using a double orthogonal parallel projection onto two orthogonal three-dimensional spaces is proposed. A synthetic approach, with the use of basic descriptive geometry tools of Monge’s projection in interactive 3D graphical software, is taken. Descriptive constructions of projections of points, lines, planes and three-dimensional spaces are shown. Methods of measuring true lengths of segments and rotating projections of objects into the modeling space are described. The focus lies on improving the spatial ability and understanding of the four-dimensional space. An example of a construction is carried out on a model of tesseract. All the constructions are analogies of constructions in Monge’s projection of a three-dimensional space. With advantages of interactive computer graphics, the constructions are easy to perform, and objects are easy to manipulate with.

Keywords

Four-dimensional visualization Multidimensional descriptive geometry Monge’s projection Tesseract 

Notes

Acknowledgement

The work was supported by the grant SVV 2017 No. 260454. I also wish to thank Lukas Krump (Charles University) for valuable comments on preliminary versions of the text.

Supplementary material

4_2017_368_MOESM1_ESM.ggb (43 kb)
Supplementary material 1 (GGB 44 kb)

References

  1. Abbott, Edwin Abbott. 2006. Flatland, a Romance of Many Dimensions. Edited with an Introduction and Notes by Rosemary Jann, Oxford University Press.Google Scholar
  2. Chilton, Bruce L. 1980. Principal Shadows of the 12 Pentagonal Regular 4-Dimensional Objects (Polytopes). Leonardo 13, 4: 288–294. DOI 10.2307/2688637Google Scholar
  3. Forsyth, Andrew Russell. 1930. Geometry of Four Dimensions. Cambridge: The University Press.Google Scholar
  4. Hanson, Andrew J. and Heng, Pheng Ann. 1991. Visualizing the Fourth Dimension Using Geometry and Light. In: Proceedings of the 2nd conference on Visualization ’91: 321–328 DOI 10.1109/VISUAL.1991.175821Google Scholar
  5. Hinton, Charles Howard. 1880. Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton. Edited by Rudolf v. B. Rucker, Dover Publications, Inc.Google Scholar
  6. Hinton, Charles Howard. 1904. The Fourth Dimension. London: Swan Sonnenschein & Co, Ltd., New York: John Lane.Google Scholar
  7. Hoffmann, Christopf M. and Zhou, Jianhua. 1991. Some techniques for visualizing surfaces in four-dimensional space. Computer-Aided Design 23, 1: 83–91. DOI 10.1016/0010-4485(91)90083-9Google Scholar
  8. Kageyama, Akira. (2016.) A Visualization Method of Four Dimensional Polytopes by Oval Display of Parallel Hyperplane Slices. Journal of Visualization 19, 3: 417–422. DOI 10.1007/s12650-015-0319-5Google Scholar
  9. Lindgren, Carlos Ernesto S. and Slaby, Steve M.. 1968. Four-Dimensional Descriptive Geometry. New York: McGraw-Hill.Google Scholar
  10. Manning, Henry Parker. 1956. Geometry of Four Dimensions. New York: Dover.Google Scholar
  11. Miyazaki, Koji. 1988. Visualization of a Die and Other Objects in Four-Dimensional Space. Leonardo 21, 1: 56–60. DOI 10.2307/1578417Google Scholar
  12. Monge, Gaspard. 1799. Géométrie descriptive. Paris: Baudouin.Google Scholar
  13. Séquin, Carlo Heinrich. 2002. 3D Visualization Models of the Regular Polytopes in Four and Higher Dimensions. In: Proceedings of Bridges: Mathematical Connections in Art, Music, and Science: 37–48.Google Scholar
  14. Şerbănoiu, Bogdan V., and Şerbănoiu, Adrian A.. 2017. Multidimensional Descriptive Geometry. In: 16th edition National Technical-Scientific Conference, Modern Technologies for the 3rd Millennium, Romania: 257–266.Google Scholar
  15. Stachel, Hellmuth. 1990 The Right-Angle-Theorem in Four Dimensions. Journal for Theoretical Graphics and Computing 3, 1: 4–13.Google Scholar
  16. Volkert, Klaus. 2017. On Models for Visualizing Four-Dimensional Figures. The Mathematical Intelligencer 39, 2: 27–35. DOI 10.1007/s00283-016-9699-1Google Scholar
  17. Weiss, Gunther. 1997. (n,2)-Axonometries and the Contour of Hyperspaces. Journal for Geometry and Graphics 1, 2: 157–167.Google Scholar
  18. Zachariáš, Svätopluk, Velichová, Daniela. 2000. Projection from 4D to 3D. Journal for Geometry and Graphics 4, 1: 55–69.Google Scholar
  19. Zhou, Jianhua. 1991. Visualization of Four Dimensional Space and Its Applications (Ph.D. Thesis). Computer Science Technical Reports, Paper 922.Google Scholar

Copyright information

© Kim Williams Books, Turin 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

Personalised recommendations