Skip to main content
SpringerLink
Log in

Adjusting degree of visual complexity: an interactive approach for exploring four-dimensional polytopes

  • original article
  • Published:
The Visual Computer Aims and scope Submit manuscript

Abstract

Few mathematical visualization tools support integrated, flexible interaction with complex, 4D mathematical concepts. This paper presents a solution to exploring uniform 4D polytopes through a mathematical visualization tool by introducing an approach for adjusting the degree of visual complexity of these complicated geometric structures. This approach introduces a number of interactive techniques: contextualizing, filtering, focus+scoping, and stacking-unstacking. Although these techniques can be effectively used in isolation, their integrated application provides highly specified and sophisticated interaction with polytopes, helping users make sense of these challenging mathematical structures. Exploring complicated structures from other domains such as chemistry and biology may benefit from this approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arcavi A, Hadas N (2000) Computer aided learning: an example of an approach. Int J Comput Math Learn 5(1):25–45

    Article  Google Scholar 

  2. Ball RW, Coxeter HSM (1974) Mathematical recreations and essays, 12th edn. Univerisity of Toronto Press, Toronto

  3. Banks D (1992) Interactive manipulation and display of two-dimensional surfaces in four-dimensional space. In: Zeltzer D (ed) 1992 Symposium on Interactive 3D Graphics 25(2):197–207

  4. Bertin J (1981) Graphics and graphic information processing. Walter de Gruyter, Berlin

  5. Borwein J, Morales MH, Polthier K, Rodrigues JF (eds) (2002) Multimedia tools for communicating mathematics. Springer, Berlin Heidelberg New York

  6. Bruter C (ed) (2002) Mathematics and art: mathematical visualization in art and education. Springer, Berlin Heidelberg New York

    Google Scholar 

  7. Card SK, Mackinlay JD, Shneiderman B (eds) (1999) Readings in information visualization: using vision to think. Morgan Kaufmann, San Francisco

    Google Scholar 

  8. Carpendale MST, Cowperthwaite DJ, Fracchia FD (1997) Extending distortion viewing from 2D to 3D. IEEE Comput Graph Appl July/August:42–51

  9. Chen M, Mountford SJ, Sellen A (1988) A study in interactive 3-D rotation using 2-D control devices. In: Dill J (ed) Proceedings of ACM SIGGRAPH 1988. ACM SIGGRAPH, Atlanta, pp 121–129

  10. Coxeter HSM (1991) Regular complex polytopes, 2nd edn. Cambridge University Press, New York

  11. Cross RA, Hanson AJ (1994) Virtual reality performance for virtual geometry. In: Proceedings of the IEEE Conference on Visualization. IEEE Computer Society Press, California, pp 156–163

  12. Dix AJ, Ellis G (1998) Starting simple – adding value to static visualization through simple interaction. In: AVI’ 98: 4th International Working Conference on Advanced Visual Interfaces. ACM Press, New York, pp 124–134

  13. Gawrilow E, Joswig M (2001) Polymake: an approach to modular software design in computational geometry. In: Proceedings of the Seventeenth Annual Symposium on Computational Geometry. ACM Press, New York , pp 222–231

  14. Hege HC, Polthier K (eds) (1998) Visualization and mathematics. Springer, Berlin Heidelberg New York

  15. Hege HC, Polthier K (eds) (2003) Visualization and mathematics III. Springer, Berlin Heidelberg New York

  16. Hanson AJ, Heng PA (1991) Visualizing the fourth dimension using geometry and light. In: Proceedings of Visualization ’91. IEEE Computer Society Press, California, pp 321–328

  17. Hanson AJ, Heng, PA (1992) Four-dimensional views of 3D scalar fields. In: Proceedings of Visualization ’92. IEEE Computer Society Press, California, pp 84–91

  18. Hanson AJ, Cross, RA (1993) Interactive visualization methods for four dimensions. In: Proceedings of Visualization ’93. IEEE Computer Society Press, California, pp 196–203

  19. Hanson AJ, Munzner T, Francis G (1994) Interactive methods for visualizable geometry. IEEE Computer 27(4): 78–83

    Article  Google Scholar 

  20. Hanson AJ (1995) Rotations for n-dimensional graphics. In: Graphics Gems V. Academic, Cambridge, pp 55–64

  21. Hanson AJ, Ma H (1995) Space walking. In: Proceedings of Visualization ’95. IEEE Computer Society Press, California, pp 126–133

  22. Hepting DH, Cao W, Russell RD (1998) An exploratory approach to mathematical visualization. In: Western Computer Graphics Symposium (April) 1998, pp 23–26

  23. Jackiw N (1995) The Geometer’s Sketchpad, v. 3.0. Key Curriculum, California

  24. Keller PR, Keller MM (1993) Visual Cues: Practical Data Visualization. IEEE Computer Society Press, California

    Google Scholar 

  25. McMullen P, Schulte E (2002) Abstract regular polytopes. Cambridge University Press, New York

  26. Morey J, Sedig K, Mercer R (2001) Interactive metamorphic visuals: Exploring polyhedral relationships. In: Proceedings of IEEE Information Visualization ’01. IEEE Computer Society Press, California, pp 483–488

  27. Morey J, Sedig K (2003) Archimedean kaleidoscope: a cognitive tool to support thinking and reasoning about geometric solids. In: Sarfraz M (ed) Geometric modeling: techniques, applications, systems and tools. Kluwer, New York

  28. Muzner T (1996) Mathematical visualization: Standing at the crossroads. In: Proceedings of Visualization ’96. IEEE Computer Society and ACM Press, California, pp 451–453

  29. Palais RS (1999) The visualization of mathematics: towards a mathematical exploratorium. Notices Amer Math Soc 46(6):647–658

    MathSciNet  Google Scholar 

  30. Phillips M, Levy S, Munzner T (1993) Geomview: an interactive geometry viewer. Notices Amer Math Soc 40:985–988

    Google Scholar 

  31. Polthier K (2002) Visualizing mathematics – online. In: Bruter C (ed) Mathematics and Art. Springer, Berlin Heidelberg New York, pp 29–42

  32. Presmeg NC (1998) On visualization and generalization in mathematics. In: Proceedings of Twentieth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, North Carolina, pp 23–27

  33. Scharein RG (1998) Interactive topological drawing. Dissertation, Department of Computer Science, The University of British Columbia, Vancouver

  34. Sedig K, Klawe M, Westrom M (2001) Role of interface manipulation style and scaffolding on cognition and concept learning in learnware. ACM Trans Comput-Human Interact 1(8):34–59

    Google Scholar 

  35. Sedig K, Morey J (2004) A descriptive framework for designing interaction for visual abstractions. In: Malcolm G (ed) Multidisciplinary approaches to visual representations and interpretations. Elsevier

  36. Sedig K, Rowhani S, Morey J, Liang HN (2003) Application of information visualization techniques to the design of a mathematical mindtool: a usability study. Inform Visual 2(3):142–159

    Article  Google Scholar 

  37. Spence R (2001) Information visualization. Pearson Education, Harlow

  38. Strothotte T (1998) Computational visualization: graphics, abstraction, and interactivity. Springer, Berlin Heidelberg New York

    Google Scholar 

  39. Strothotte C, Strothotte T (1997) Seeing between the pixels: pictures in interactive systems. Springer, Berlin Heidelberg New York

    Google Scholar 

  40. Stylianou DA (2002) On the interaction of visualization and analysis: the negotiation of a visual representation in expert problem solving. J Math Behav 21:303–317

    Article  Google Scholar 

  41. Tufte ER (1997) Visual explanations: images and quantities, evidences and narratives. Graphics, Cheshire

    Google Scholar 

  42. Tweedie L, Spence R, Dawkes H, Su H (1996) Externalizing abstract mathematical models. In: Proceedings of CHI’96. ACM Press, pp 406–412

  43. Webb R (2000) Stella: polyhedron navigator. Symmetry: culture and science. pp 231–268, International Symmetry Foundation, Budapest

  44. West TG (1995) Forward into the past: a revival of old visual talents with computer visualization. In: ACM SIGGRAPH Computer Graphics 1995, New York, pp 14–19

  45. Wiss U, Carr D (1998) A cognitive classification framework for 3-dimensional information visualization. In: Research report LTU-TR–1998/4–SE. Luleå University of Technology, Luleå, Sweden

  46. Ziegler GM (1995) Lectures on polytopes. Springer, Berlin Heidelberg New York

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Cognitive Engineering Laboratory, The University of Western Ontario, Canada

    Jim Morey

  2. Department of Computer Science and Faculty of Information and Media Studies, Cognitive Engineering Laboratory, The University of Western Ontario, Canada

    Kamran Sedig

Authors
  1. Jim Morey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kamran Sedig
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jim Morey.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Morey, J., Sedig, K. Adjusting degree of visual complexity: an interactive approach for exploring four-dimensional polytopes. Vis Comput 20, 565–585 (2004). https://doi.org/10.1007/s00371-004-0259-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00371-004-0259-x

Keywords

Search

Navigation