The Visual Computer

, Volume 20, Issue 8–9, pp 565–585 | Cite as

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

original article
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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.

Keywords

Mathematical visualization High-dimensional geometry Interactive techniques Computer-aided visual reasoning Dynamic exploration 
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References

  1. 1.
    Arcavi A, Hadas N (2000) Computer aided learning: an example of an approach. Int J Comput Math Learn 5(1):25–45CrossRefGoogle Scholar
  2. 2.
    Ball RW, Coxeter HSM (1974) Mathematical recreations and essays, 12th edn. Univerisity of Toronto Press, TorontoGoogle Scholar
  3. 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):197–207Google Scholar
  4. 4.
    Bertin J (1981) Graphics and graphic information processing. Walter de Gruyter, BerlinGoogle Scholar
  5. 5.
    Borwein J, Morales MH, Polthier K, Rodrigues JF (eds) (2002) Multimedia tools for communicating mathematics. Springer, Berlin Heidelberg New YorkGoogle Scholar
  6. 6.
    Bruter C (ed) (2002) Mathematics and art: mathematical visualization in art and education. Springer, Berlin Heidelberg New YorkGoogle Scholar
  7. 7.
    Card SK, Mackinlay JD, Shneiderman B (eds) (1999) Readings in information visualization: using vision to think. Morgan Kaufmann, San FranciscoGoogle Scholar
  8. 8.
    Carpendale MST, Cowperthwaite DJ, Fracchia FD (1997) Extending distortion viewing from 2D to 3D. IEEE Comput Graph Appl July/August:42–51Google Scholar
  9. 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 121–129Google Scholar
  10. 10.
    Coxeter HSM (1991) Regular complex polytopes, 2nd edn. Cambridge University Press, New YorkGoogle Scholar
  11. 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 156–163Google Scholar
  12. 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 124–134Google Scholar
  13. 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 222–231Google Scholar
  14. 14.
    Hege HC, Polthier K (eds) (1998) Visualization and mathematics. Springer, Berlin Heidelberg New YorkGoogle Scholar
  15. 15.
    Hege HC, Polthier K (eds) (2003) Visualization and mathematics III. Springer, Berlin Heidelberg New YorkGoogle Scholar
  16. 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 321–328Google Scholar
  17. 17.
    Hanson AJ, Heng, PA (1992) Four-dimensional views of 3D scalar fields. In: Proceedings of Visualization ’92. IEEE Computer Society Press, California, pp 84–91Google Scholar
  18. 18.
    Hanson AJ, Cross, RA (1993) Interactive visualization methods for four dimensions. In: Proceedings of Visualization ’93. IEEE Computer Society Press, California, pp 196–203Google Scholar
  19. 19.
    Hanson AJ, Munzner T, Francis G (1994) Interactive methods for visualizable geometry. IEEE Computer 27(4): 78–83CrossRefGoogle Scholar
  20. 20.
    Hanson AJ (1995) Rotations for n-dimensional graphics. In: Graphics Gems V. Academic, Cambridge, pp 55–64Google Scholar
  21. 21.
    Hanson AJ, Ma H (1995) Space walking. In: Proceedings of Visualization ’95. IEEE Computer Society Press, California, pp 126–133Google Scholar
  22. 22.
    Hepting DH, Cao W, Russell RD (1998) An exploratory approach to mathematical visualization. In: Western Computer Graphics Symposium (April) 1998, pp 23–26Google Scholar
  23. 23.
    Jackiw N (1995) The Geometer’s Sketchpad, v. 3.0. Key Curriculum, CaliforniaGoogle Scholar
  24. 24.
    Keller PR, Keller MM (1993) Visual Cues: Practical Data Visualization. IEEE Computer Society Press, CaliforniaGoogle Scholar
  25. 25.
    McMullen P, Schulte E (2002) Abstract regular polytopes. Cambridge University Press, New YorkGoogle Scholar
  26. 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 483–488Google Scholar
  27. 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 YorkGoogle Scholar
  28. 28.
    Muzner T (1996) Mathematical visualization: Standing at the crossroads. In: Proceedings of Visualization ’96. IEEE Computer Society and ACM Press, California, pp 451–453Google Scholar
  29. 29.
    Palais RS (1999) The visualization of mathematics: towards a mathematical exploratorium. Notices Amer Math Soc 46(6):647–658MathSciNetGoogle Scholar
  30. 30.
    Phillips M, Levy S, Munzner T (1993) Geomview: an interactive geometry viewer. Notices Amer Math Soc 40:985–988Google Scholar
  31. 31.
    Polthier K (2002) Visualizing mathematics – online. In: Bruter C (ed) Mathematics and Art. Springer, Berlin Heidelberg New York, pp 29–42Google Scholar
  32. 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 23–27Google Scholar
  33. 33.
    Scharein RG (1998) Interactive topological drawing. Dissertation, Department of Computer Science, The University of British Columbia, VancouverGoogle Scholar
  34. 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):34–59Google Scholar
  35. 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. ElsevierGoogle Scholar
  36. 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):142–159CrossRefGoogle Scholar
  37. 37.
    Spence R (2001) Information visualization. Pearson Education, HarlowGoogle Scholar
  38. 38.
    Strothotte T (1998) Computational visualization: graphics, abstraction, and interactivity. Springer, Berlin Heidelberg New YorkGoogle Scholar
  39. 39.
    Strothotte C, Strothotte T (1997) Seeing between the pixels: pictures in interactive systems. Springer, Berlin Heidelberg New YorkGoogle Scholar
  40. 40.
    Stylianou DA (2002) On the interaction of visualization and analysis: the negotiation of a visual representation in expert problem solving. J Math Behav 21:303–317CrossRefGoogle Scholar
  41. 41.
    Tufte ER (1997) Visual explanations: images and quantities, evidences and narratives. Graphics, CheshireGoogle Scholar
  42. 42.
    Tweedie L, Spence R, Dawkes H, Su H (1996) Externalizing abstract mathematical models. In: Proceedings of CHI’96. ACM Press, pp 406–412Google Scholar
  43. 43.
    Webb R (2000) Stella: polyhedron navigator. Symmetry: culture and science. pp 231–268, International Symmetry Foundation, BudapestGoogle Scholar
  44. 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 14–19Google Scholar
  45. 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, SwedenGoogle Scholar
  46. 46.
    Ziegler GM (1995) Lectures on polytopes. Springer, Berlin Heidelberg New YorkGoogle Scholar

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Department of Computer Science, Cognitive Engineering LaboratoryThe University of Western OntarioCanada
  2. 2.Department of Computer Science and Faculty of Information and Media Studies, Cognitive Engineering LaboratoryThe University of Western OntarioCanada

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