Andrew J. Hanson's Home Page

Computer Science Program -- Emeritus
School of Informatics and Computing
Indiana University, Bloomington
   (IU -- OneStart System)

See also my page in the Indiana University Cognitive Science Program.

Here is my capsule biography, as well as a more detailed curriculum vita.

Andrew J. Hanson
Professor Emeritus of Computer Science
School of Informatics and Computing
Indiana University
Bloomington, IN 47405
Office [+1] (812) 855-5855
LH215 Administrator [+1] (812) 855-6486
LH215 FAX [+1] (812) 855-4829
hansona (at) indiana [dot] edu
Generic URL:

My Erdös Number is 4, computed from Paul Erdös(0):Irving Kaplansky(1):Peter Freund(2):Tohru Eguchi(3):Andrew Hanson(4).

My Academic Genealogy traces back through Carl Friedrich Gauss, starting from my PhD thesis ("A Dual Resonance Model for Meson-Nucleon Scattering") under Professor Kerson Huang at MIT (August, 1971).

My Google Scholar Profile gives a nice picture of the various fields I have worked in. My DBLP Profile is another useful (but incomplete) list of my Computer Science publications compiled by the DBLP project.

Traditional Courses of Mine:


B581, Graduate Computer Graphics, taught about 30 times.

    Public B581 syllabus: Overview of B581.

This is an OpenGL-based course introducing the mathematical foundations and practical programming methods of modern interactive computer graphics. The homework involves coding in C using OpenGL and GLUT, and mastering the theoretical principles upon which OpenGL-like graphics is based. The course emphasizes creating interactive interfaces to help understand the graphics objects and techniques being studied. Lighting and simple material modeling are covered as an introduction to the creation of realistic images.


B689, Mathematical Modeling Methods, taught half a dozen times  

Public B689 syllabus: Overview of B689.

This course focused on Mathematica-based methods of producing rapid prototypes solving complex software modeling problems. This class will start with an introduction to the Mathematica programming environment, and will incorporate Mathematica prototyping methods implicitly into a broad survey of mathematical modeling methods, techniques, and folklore used widely throughout computer science, computer graphics, scientific visualization, mathematics, and physics.


Interactive Mathematics


Click here for a larger version
4Dice is (free) iPhone App. See our IEEE COMPUTER paper on the design of 4Dice, IEEE Computer, Volume 45, Number 9, pp.80-88 (September, 2012). Alternative site for PDF: see also here.

4Dice is an interactive application using our 4D Rolling Ball algorithm combined with the graphically correct 4D backface-culled representation of the hypercube introduced in the 4Dice video animation (see below). This is a collaborative project with Xiaoqi Yan and Prof. Philip Chi-Wing Fu at NTU, Singapore.

The 4Dice one-minute YouTube video provides a quick introduction to the issues of properly visualizing a back-face culled hypercube; the 4Dice iPhone App provides interactive exploration of all the concepts introduced in the animation.


Make a Hypercube in Lights



Click here for a larger version
Click here for printable PDF page

Click here for a larger version

The images you see in the 4Dice interactive application actually form a wire-frame torus when you turn off back-face culling. The deep reason for this is that the hypercube is a simple tessellation of a 3-sphere (which has Euler characteristic zero) surrounding a 4-ball, and a nice parameterization of the 3-sphere involves a nested family of tori; the wireframe hypercube is effectively a rectangular tessellation of the "center" member of this family of tori. It is known that such a set of edges, with four edges meeting at each vertex, admits an Eulerian path. The figure on the left shows one of many such paths that can be constructed, with a ratio of 1:3 for the inner cube edge relative to the outer cube edge, and diagonal edges with relative length sqrt(3). With a little work, one can use this diagram to construct a hypercube out of a single string of holiday lights, as shown on the right. (With a tip of the hat to Priscilla and Russell.)


Selected Publications of Interest

Visualizing Quaternions (Morgan-Kaufmann/Elsevier, 2006, ISBN 978-0-12-088400-1) is a comprehensive approach to the significance and applications of quaternions, and focuses on the exploitation of Quaternion Fields, a tool developed primarily by the author.

The official website for the book is maintained by the publisher, and provides background material, downloadable material from tables, and demonstration software.

I maintain a local companion website here, which may be more up to date.

Updates and Errata are maintained on the update and errata page. An example is the closed form double-reflection quaternion form   q = ( A · B, A × B ) inadvertantly omitted from the Clifford Algebra treatment in Chapter 31. There are several array indices that are transposed in the C Program in Table E.3, page 446.

2012 Winner: JMGM Graphics Prize.
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Another example image from this paper.
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Winner with co-author Sidharth Thakur of the 2012 JMGM Graphics Prize. See also the MGMS/Elsevier Graphics Award.

Quaternion Approaches to Visualizing Protein Structure are worked out in detail in our JMGM article: A.J. Hanson and S. Thakur, "Quaternion maps of global protein structure," Journal of Molecular Graphics and Modelling, Volume 38, September 2012, pp. 256--278. (Alternate URL here.)

Quaternion maps of protein amino acid residues provide an alternative to Ramachandran plots for orientation analysis. Several alternative orientation frame systems can be chosen, with the residue-local Cα-centered frame being the default. Quaternion maps are noteworthy for their ability to compare the orientations of arbitrary sets of sequential or non-sequential residues located anywhere on the protein, and for the resulting opportunity to observe and analyze the statistical properties of global orientation clusters. Only the quaternion representation of orientation frames embodies a natural rigorous measure for comparing properties of sets of global orientation frames.


Quaternion Applications were covered in our updated quaternion tutorial lectures presented at Siggraph Asia 2012, 29 November in Singapore. Special application topics included optimal, smoothly controllable tubing and tube texturing, quaternion protein maps, and how dual quaternions solve the conundrum of how a quaternion acts on a vector. The latter is a long-standing controversy that pitted Hamilton against many contemporaries, and has been described in wondrous detail by Altmann in "Hamilton, Rodrigues, and the Quaternion Scandal.";

The solution is simply to replace Hamilton's impossible candidate for a "Vector," the binary rotation quaternion (0,  (x, y, z) ) by the dual quaternion vector, namely (0,  ε(x, y, z) ), with ε having the dimension of inverse length and satisfying ε2 = 0 .

Visualizing Relativity using complexified quaternions was part of the material covered by Andrew Hanson and Daniel Weiskopf in their Siggraph 2001 Course 15 Notes.

The Solar Journey DVD contains an educational computer animated film on the astronomy of the local neighborhood of the Earth and the Sun developed as part of our NASA-sponsored research work. The DVD version containing the Solar Journey animation and supplementary science materials is distributed by Finley-Holiday Films.


Some day I'll put together an annotated bibliography, but for now see the Google Scholar and DBLP project links at the top of the web page. Here is a summary of my historically most highly cited work, and together with some media links.

The Eguchi-Hanson metric (Physics Letters 74B, pp. 249--251 (1978)) is a vacuum solution of the Euclidean Einstein equations that is the first known instance of an important class of metrics now commonly referred to as ALE or Asymptotically Locally Euclidean metrics. A comprehensive review of Euclidean Einstein metrics and the context of the Eguchi-Hanson metric is given in our 1979 Annals of Physics review article. This work won the Second Prize in the 1979 Gravity Research Foundation Competition; see T. Eguchi and A.J.Hanson, "Gravitational Instantons," Journal of General Relativity and Gravitation, 11, pp. 315--320 (1979).

Our comprehensive introduction to the ways in which the languages of the theoretical physics and mathematics communities became inseparably connected after a long history of going their separate ways is available in the 1980 Physics Reports article "Gravitation, Gauge Theories and Differential Geometry" by Eguchi, Gilkey, and Hanson.

Constrained Hamiltonian Systems, is a short book by Hanson, Regge, and Teitelboim, originally published in 1976 by the Accademia Nazionale dei Lincei (Contributi del Centro Linceo Interdisc. di Scienze Matem. e loro Applic., No.22, Accademia Nazionale dei Lincei, Rome, 135 pages (1976)). Actual print copies of this work are rare and generally unavailable.

Media. The four-dimensional MeshView viewer is described in the Meshview tech note, and downloadable software is located HERE. Supported fully under X-windows/Motif only. Precompiled for Linux, Macintosh, SUN SOLARIS, and SGI IRIX. Recently available: reduced functionality Windows XP version. The shortcuts work in the Windows version, but you need to look at the Linux version to see what they are.


My most recent research focuses on several areas: Mathematical Visualization, Virtual Reality, and Astronomy.

Calabi-Yau Cross Sections:

I have also created a variety of graphics images derived from the Fermat Equation (see below) that are relevant to the Calabi-Yau spaces that may lie at the smallest scales of the unseen dimensions in String Theory; these have appeared in Brian Greene's books, The Elegant Universe and The Fabric of the Cosmos,, and in the book by Callender and Huggins, Physics Meets Philosophy at the Planck Scale. The writhing purple shapes in the October/ November 2003 NOVA production Elegant Universe, as well as the cover of the November 2003 Scientific American, were derived from software models I supplied to the NOVA graphics providers.

These images show equivalent renderings of a 2D cross-section of the 6D manifold embedded in CP4 described in string theory calculations by the homogeneous equation in five complex variables:
z05 + z15 + z25 + z35 + z45 = 0
The surface is computed by assuming that some pair of complex inhomogenous variables, say z3/z0 and z4/z0, are constant (thus defining a 2-manifold slice of the 6-manifold), normalizing the resulting inhomogeneous equations a second time, and plotting the solutions to
z15 + z25 = 1
The resulting surface is embedded in 4D and projected to 3D using Mathematica (left image) and our own interactive MeshView 4D viewer (right image). If you have CosmoPlayer, you can also interact with this VRML version of the quintic Calabi-Yau cross-section.

In the right-hand image, each point on the surface where five different-colored patches come together is a fixed point of a complex phase transformation; the colors are weighted by the amount of the phase displacement in z1 (red) and in z2 (green) from the fundamental domain, which is drawn in blue and is partially visible in the background. Thus the fact that there are five regions fanning out from each fixed point clearly emphasizes the quintic nature of this surface.

For further information, see: A.J. Hanson. A construction for computer visualization of certain complex curves. Notices of the Amer. Math. Soc., 41(9):1156-1163, November/December 1994.

An interactive version is available at the Wolfram Demonstrations Project Calabi-Yau Space page, based on the Hanson paper cited above, with assistance from Jeff Bryant.

Arbitrary Genus Surfaces:

This image shows my computer graphics construction of a four-hole torus described by an equation in complex two-space given by H. Blaine Lawson, "Complete Minimal Surfaces in S3," Ann. of Math. 92, pp.~335--374 (1970), with m = n = 2,

Im z1(m + 1) + |z2|(m-n) Im z2(n+1) = 0


|z1|2 + |z2|2 = 1

In general, the genus is m*n, and this surface is not actually minimal in S3 except for
m = n = 0 and m = n = 1.

Review article

Cover picture: IEEE Computer 27 (July 1994)

For more information about mathematical visualization in general, see the Web version of the review article Interactive Methods for Visualizable Geometry, by A.J. Hanson, T. Munzner, and G. Francis, published in IEEE Computer 27 , No. 7, pp. 73--83 (IEEE Computer Society Press, Los Alamitos, CA, July, 1994).

Mathematics and Physics Animations

We have produced a number of short video animations with mathematical and physical content. Some of my favorite projects are the following:

Cosmic Clock:

Observing the Universe using the finite speed of light to place measured objects in their correct temporal context.
The Cosmic Bloom excerpt from the movie as available here; it seems to play OK on PC's with QuickTime, but has troubles on some other platforms.
This 3:35 minute animation contains a visualization of the entire Universe from three different points of view: the time spectrum of observable photon radiation arriving at the earth, the constant-time shells of light sources represented in "comoving coordinates " (as though the Universe had always been the size it is today), and in "physical coordinates" (which incorporate the Hubble expansion since the "Great Flash", when the Universe was about 300,000 years old). This film was one of a select few chosen for showing at the Siggraph 2000 Electronic Theater in July 2000, and appears in Siggraph Video Review 134, Scene 5 (2000).

4Dice: Hypercube
4Dice: A Glimpse into the 4th Dimension (MPG silent version)
4Dice: Local version with narration.
4Dice: YouTube version with narration.
This short (1:00 minute) animation of a back-face-culled 4D die or hypercube, which has eight sphere-containing cubes as the analogs of faces, was shown at the Siggraph '95 Computer Animation Festival and appears in Siggraph Video Review 114, Scene 14 (1995).

Visualizing Fermat's Last Theorem:
Visualizing Fermat's Last Theorem Video
Fermat Video YouTube version:
Fermat's last theorem was proven at last in 1995 by Andrew Wiles, but the mystique lives on. This film was made in 1990, when it was still unclear that Fermat's conjecture would ever actually be proven --- in a way it was more romantic when we could believe that, wasn't it?
The film was shown in the Siggraph '90 Animation Screening Room, and appears in Siggraph Video Review 61, Scene 4 (1990).

Andrew J. Hanson: Last revised 9 January 2014.