Exploring Low Dimensional Objects in High Dimensional Spaces

  • Dennis Roseman


We discuss general principles and a software implementation for visualizing low dimensional objects in high dimensional spaces. By high dimensional space we mean euclidian space of dimension greater than four. The low dimensional objects are modeled, mathematically, by simplicial complexes of dimension 4 or less. A particular software visualization project, named Hew, is discussed. An example of an embedded three dimensional projective space is featured in the figures.


Simplicial Complex High Dimensional Space Morse Theory Color Scheme Lens Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    T. Banchoff, Beyond the Third Dimension: Geometry, Computer Graphics and Higher Dimensions, Scientific American Library (1990).Google Scholar
  2. 2.
    D. C. Banks, Interactive Manipulation and Display of Surfaces in Four Dimensions, Symposium on Interactive 3D Graphics, Association for Computing Machinery (1992).Google Scholar
  3. 3.
    S. Bergman, The Kernel Function and Conformal Mapping, A.M.S. Math. Surv. 5 (1950).Google Scholar
  4. 4.
    C. Cruz-Neira, D. J. Sandin, T. A. Defanti Surround-Screen Projection-Based Virtual Reality: The Design and Implementation of the CAVE, Computer Graphics (Proceedings of SIGGRAPH –93), ACM SIGGRAPH (1993), 135 - 142.Google Scholar
  5. 5.
    G. K. Francis, R. Kusner, J. Sullivan, D. Roseman, K. Brakke, Laterna matheMagica, Entry No. 6 in Virtual Environments and Distributed Computing Global Information Infrastructure Testbeds. ACM/IEEE Supercomputing Conference, San Diego, CA, (1995).Google Scholar
  6. 6.
    R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics, Prentice Hall 120 - 167 (1961).Google Scholar
  7. 7.
    M. Hirsch, Differential Topology, Grad. Texts in Math. No. 33, Springer Verlag, 1976.Google Scholar
  8. 8.
    M. Hirsch, Embeddings and compressions of polyhedra and smooth manifolds, Topology 4 (1966), 361 - 369.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    J. F. P. Hudson, Piecewise Linear Topology, Benjamin, 1969.Google Scholar
  10. 10.
    S. J. JR. LOMONOCO, Five dimensional knot theory, Low Dimensional Topology, A.M.S. Contemp. Math 20 (1983).Google Scholar
  11. 11.
    J. H. Maddocks, R. F. Manning, R. C. Paffenroth, K. A. Rogers, J. A. Wasrner, Interactive Computation, Parameter Continuation and Visualization, preprint (1996).Google Scholar
  12. 12.
    Maddocks, J. H., Sachs, R. L., Constrained variational principles and stability in Hamiltonian systems, Hamiltonian Dynamical Systems 63 I.M.A. Math. and Its Appl. pp. 231-264, Springer Verlag, 1995.Google Scholar
  13. 13.
    J. Milnor, Morse Theory, Ann. of Math Study 51, Princeton Univ. Press, 1963.Google Scholar
  14. 14.
    J. R. Munkres, Elements of Algebraic Topology, Benjamin/Cummings, 1984.Google Scholar
  15. 15.
    M. Phillips, S. Levy, AND T. Munzner, Geomview - An Interactive Geometry Viewer, Notices A.M.S., October 1993.Google Scholar
  16. 16.
    V. A. Rohlin, The embedding of non-orientable three-manifolds into five-dimensional Euclidean space, Dokl. Akad. Nauk. SSSR 160 (1965), 549-551; English translation, Soviet Math. Dokl. 6 (1965), 153 - 156.MATHGoogle Scholar
  17. 17.
    D. Roseman, Spinning knots about submanifolds; spinning knots about projections of knots, Topology and its Applications 31 (1989), 225 - 241.MathSciNetCrossRefGoogle Scholar
  18. 18.
    D. Roseman, Motions of Flexible Objects, Modern Geometric Computing for Visualization (Eds. T.L. Kunii and Y. Shinagawa) Springer-Verlag, 1992, 91120.Google Scholar
  19. 19.
    D. Roseman, Wiener’s thought on the computer as an aid in visualizing higher-dimensional forms A.M.S. Proc. of Symp. in Appl. Math. 52 (1997), 441 – 471.Google Scholar
  20. 20.
    D. Roseman, What Should a Surface in 4-space Look Like?, Visualization and Mathematics, Springer Verlag, Edited by Hege and Polthier (1997).Google Scholar
  21. 21.
    C. T. C. Wall, All 3-manifolds imbed in 5-space, Bull. Am. Math. Soc. 71 (1965), 564 - 567.MATHCrossRefGoogle Scholar
  22. 22.
    N. Steenrod, Topology of Fibre Bundles, Princeton Univ. Press, 1951.Google Scholar

Listing of mathematical videos

  1. 23.
    STAFF OF GEOMETRY CENTER, Not Knot,A.K. Peters, Wellesley MA.Google Scholar
  2. 24.
    Staff OF Geometry Center, Shape of Space,Geometry Center, Minneapolis, MN.Google Scholar
  3. 25.
    D. Roseman, (with D. Mayer), Viewing Knotted Spheres in 4-space, video (8 mins) Produced at the Geometry Center, (June 1992).Google Scholar
  4. 26.
    D. Roseman, (With D. Mayer And O. Holt), Twisting and Turning in 4 Dimensions, video (19 mins) produced at the Geometry Center, distributed by Great Media, Nicassio CA, ( August 1993 ).Google Scholar
  5. 27.
    D. Roseman, Unraveling in 4 Dimensions, video (18 mins) produced at the Geometry Center, distributed by Great Media, Nicassio CA, ( July 1994 )Google Scholar

Listing of software

  1. 28.
    Staff OF Geometry Center, Geomview, software, Geometry Center, Minneapolis, Minn. Available via anonymous ftp from geom umn edu.Google Scholar
  2. 29.
    D. Roseman, J. Berdine, Hew, written University of Iowa (1996)Google Scholar
  3. 30.
    A.J. Hanson, ET AL, Mesh View 4D, a 4d surface viewer for meshes for SGI machines, available via ftp from the Geometry Center (1994).Google Scholar
  4. 31.
    G. Chapell, G. Francis, C. Hartman, slice, written at NCSA (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Dennis Roseman
    • 1
  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA

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