This page contains links and bibliography to 4-dimensional visualization resources on the web and in various scientific proceedings. Any resource that can be found on the web is linked to the source.

There are four main categories here:
1. Literature: Serious Articles etc.
2. Visualization: Various tools and methods.
3. Scientific Computing: Visualization of simulations requiring 4D displays.
4. (Pre-)College: High School and Undergraduate math level.

Literature in proceedings and the web.

1. David Banks, "Interactive Manipulation and Display of Two-Dimensional Surfaces in Four-Dimensional Space" in 1992 Symposium on Interactive 3D Graphics, pp. 197
Comment: A paper primarily on intuitive manipulation of surfaces in four dimensions. Usage of multiple devices for input and intersecting with planes to allow seeing the internal structure of the surface.

2. AJ. Hanson and RA. Cross, "Interactive Visualization Methods for Four Dimensions" in Proceedings of Visualization '93, San Jose, CA, October 25-29, 1993, pp. 196
Comment: Not yet available.

3. SR. Hollasch, " Four-Space Visualization of 4D Objects " Master's Thesis, Arizona State University, 1991
Comment: A good introduction in extending 3D graphics techniques and mathematical background into 4D.

4. AJ. Hanson and PA. Heng, "Illuminating the Fourth Dimension ", IEEE Computer Graphics and Appl., Vol. 12, No. 4, pp. 54
Comment: Not yet available.

5. G. Ferrar, Arbitrary-Dimensional Solid Object Display Algorithm.
Comment: Not yet available.

Visualizing 4D objects and surfaces.

1. T. Banchoff, Brown University, Flatland-Hypergraphics.
Comment: A leader in 4D visualization.

2. Center for Innovative Computer Applications, Indiana University, Four-Dimensional Dice Simulation
Comment: A very nice movie of a rotation of a 4-dimensional dice, which appeared in SIGGRAPH

3. Center for Innovative Computer Applications, Indiana University, Visualizing Fermat's Last Theorem
Comment: Fermat's Last Theorem, before the proof was known.

4. R. Koch, UOregon, Java program that can display and rotate all possible three and four dimensional regular solids
Comment: Cubes, tetrahedra, octahedra and so on in 3 and 4 dimensions.

5. P. Fleckenstein, Rochester Institute of Technology N-dimensional Ray Tracing
Comment: Intersecting a 4-D object with a space, and making a movie out of it.

6. N. Jackiw, The Geometer's Sketchpad
Comment: A very nice Java Applet focusing on how to start from 1D cube and work your way up to a 4D cube.

7. H. van der Wal, HyperCube Game.
Comment: A neat game in three and four dimensions, of trying to avoid hitting a hypercube with a ball. Allows for stereo vision as well.

8. J Bailey, Four Dimensional Rubik's cube.
Comment: One way of thinking about a 4D rubik cube.

9. Another such list.
Comment: Geometry-concerned list of 4D stuff.

Scientific Computing.

1. Lattice QCD at the Ohio State university: Eigenmode of the Dirac operator.
Comment: Fourth dimension treated as time, and turned into a movie, a common practice.

2. TR. Nelson and DH. Pretorius, Interactive Acquisition, Analysis and Visualization of Sonographic Volume Data, Intl. Journal of Imaging Systems and Technologies, Vol. 8, 1997, pp. 26
Comment: Another common practice, data acquired in a 3D cubical lattice, with an associated value at each point, requires four-dimensional visualization. Also known as volume rendering.

3. Yale Medical School, Shape-Based Analysis of Nonrigid Motion list of articles.
Comment: Some examples include: Myocardial Motion and Function Assessment; Shape-Based 4D Left Ventricular Myocardial Function Analysis; A Sequential Filter for Temporal Analysis of Cardial Motion and much more.

4.A. Anderson, Watermodeling: A 4-Dimensional GIS/CADD-Based Decision Support System for Managing Environmental Remediation Projects
Comment: Another area in which 4D vis. is of importance is GIS systems, dealing with more than just land, eg. atmosphere, water bodies etc.

5. Medical Imaging, MRI, PET, CT, SECT and Ultrasound Volume Segmentation.

High School and Undergraduate Interest.

1. The Geometer's Sketchpad.
Comment: A very nice Java Applet focusing on how to start from 1D cube and work your way up to a 4D cube.

2. A Hypergame!.
Comment: A neat game in three and four dimensions, of trying to avoid hitting a hypercube with a ball. Allows for stereo vision as well.

3. Question corner at the University of Toronto Mathematics Network: Euclidean Geometry in Higher Dimensions.
Comment: A simple guide into higher dimensions.

4. A. Sheppard, Hampshire College Summer Studies in Mathematics, outline of a course for gifted high school students: Images From The Fourth Dimension.
Comment: Not yet available.

Dimitrios Mitsouras
Last modified: Mon Nov 23 05:14:53 EST 1998