Searching for Extra Dimensions
by Don V Black, PhD

My research into the visualization of multiple spatial dimensions has lead to a PhD in Computer Graphics and Visualization and the following dissertation:
Computational Techniques to Enable Visualizing Shapes of Objects of Extra Spatial Dimensions

  The following videos are cited in the above dissertation. Click the YouTube™ logo to play those videos on YouTube at a lower resolution. Otherwise there is a mirror copy at UCI that may be downloaded for your personal use. The videos© and dissertation© are copyright 2010, Don V Black.
  Figure 2.9 Two 4D Objects Crossing at 0.866c.
  Figure 6.2 Two-torus in 4D and sliced orthogonal to W by a three-flat.
  Figure 6.3 Two-torus in 4D and projected obliquely onto a three-flat.
  Figure 6.8 A 4D Sphere sliced along the W-axis by a 3D hyperplane.
  Figure 6.9 A 4D Torus sliced along the W-axis by a 3D hyperplane.
  Figure 6.10 A 5D Torus Shown in 15 3D Viewports.
 Figure 7.2 Spacetrace showing Three Viewports.
 Figure 8.1 Slice of a 4D Torus in a non-Euclidean 4-space.
 Figure 8.2 Slice of a 4D Torus in Euclidean 4-space..
  Figure 0.0 This is a concatentation of the above videos into one stream.
Recent Papers
Visualizing flat spacetime: Viewing optical versus relativistic effects., Black, Don V.; Gopi, M.; Wessel, Frank; Pajarola, Renato; Kuester, Falko American Journal of Physics, Volume 75, Issue 6, pp. 540-545 (2007).

  A visual representation of Minkowski spacetime appropriate for a student with a background in geometry and algebra is presented. Minkowski spacetime can be modeled with a Euclidean four-space to yield accurate visualizations as predicted by special relativity. The contributions of relativistic aberration, as compared to classical prerelativistic aberration, to the geometry are discussed in the context of its visual representation
Viewing Classical and Relativistic Spacetime - Click here for accompanying Animated Videos
Master of Science in Information and Computer Sciences
Visualization of Classical and Relativistic Spacetime Geometry
   Herein we devise, explore, and test an elegant theoretical spacetime model to enable realistic visualization of special relativistic effects. Images and animated videos are provided.
BackLight Software Package Animations - Click here for accompanying Animated Videos
Viewing Classical and Relativistic Spacetime - Click here for accompanying Animated Videos
Two early examples of interactive 4D models, follow. The first is a straight-forward Euclidean 4D space depicted by a display similar to that of a 3D CAD pacakge. The second is a 4D Minkowski spacetime displayed the same way, showing relativistic distortions and infinities at lightspeed.
Euclidean 4D Viewer (sin & cos) Minkowski 4D Viewer (sinh & cosh)


My Erdös Number - 5: The Paul Erdös number represents the separation of a scientist from the famous mathematician Paul Erdös in terms of co-authoring mathematical research articles, and is an easy way to lose a lot of time browsing bibliographies and the Internet (there are tools). I determined my Erdös number to be 5 as follows:

Don V Black -> Renato Pajarola -> Peter Widmayer -> Emo Welzl -> Boris Aronov / Paul Erdos


You are welcome to explore my various websites:

Go here for Physics & Education:
or here for Business:
or here to learn about Special Relativity & Visualization:
or here to explore another Dimension:
or here to learn about Flying:
I calculated my Erdös number to be 5 as follows:

Boris Aronov, Paul Erdös, Wayne Goddard, Daniel J. Kleitman, Michael Klugerman, János Pach, and Leonard J. Schulman. Crossing families. In Proceedings Symposium on Computational Geometry, pages 351–356. ACM, 1991.

  1. Artur Andrzejak, Boris Aronov, Sariel Har-Peled, Raimund Seidel, and Emo Welzl. Results on k-sets and j-facets via continuous motion. In Proceedings Symposium on Computational Geometry, pages 192–199. ACM, 1998.
  2. Tetsuo Asano, Desh Ranjan, Thomas Roos, Emo Welzl, and Peter Widmayer. Space-flling curves and their use in the design of geometric data structures. Theoretical Computer Science, 181(1):3–15, 1997.
  3. Renato Pajarola and Peter Widmayer. An image compression method for spatial search. IEEE Transactions on Image Processing, 9(3):357–365, March 2000.
  4. Don V. Black, M. Gopi, Falko Kuester, Frank Wessel, and Rentao Pajarola.Visualizing Flat Spacetime: Viewing Optical versus Special Relativistic Effects, American Journal of Physics, 75(6), pp. 540--545, June 2007.

09/22/07 by Digital Don V Black