Program -- Emeritus
Traditional Courses of Mine:
B581, Graduate Computer Graphics.
Public B581 syllabus: Overview of B581.
This was 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 emphasized creating interactive interfaces to help understand the graphics objects and techniques being studied. Lighting and simple material modeling were covered as an introduction to the creation of realistic images.
B689, Mathematical Modeling Methods for Scientific Visualization.Public B689 syllabus: Overview of B689.
This course focused on Mathematica-based methods of producing rapid prototypes solving complex software modeling problems. This class introduced the Mathematica programming environment, and incorporated 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.
My most recent research focuses on several areas: Mathematical Physics, Applications of Quaternions, Human Interfaces for Effective Learning, and Scientific Visualization.
|This WebGL App (implemented by Leif Christiansen from a corresponding OpenGL desktop application) uses the left-mouse (or 1-finger) drag to apply a 3D rotation to a triad of axes, each corresponding to a column of a 3D rotation matrix. As the rotations accumulate, the left-hand bar shows the corresponding quaternion q0 component, and the thick tube emanating from the origin shows the (qx,qy,qz) 3-vector component. If q0≥0, the components show in yellow, and if q0<0, the components show in blue. The entire viewpoint can be rotated without changing the matrix or its quaternion components using mouse-right, and alt-left-mouse restricts the rotation to the z-axis for pedagogical study. The application is here: The QuatRot App.|
|This WebGL App (implemented by Leif Christiansen from a corresponding OpenGL desktop application) uses the left-mouse (or 1-finger) drag to apply a 3D rotation to any chosen projection of a 4D object. On the desktop, shift-mouse-left rotates in the 4D zw plane, and alt-mouse-left uses both up/down and left-right mouse drags to rotate in the combined xw and zw directions. On a touch-screen device, check the boxes for fixed z-axis and xw-plane and yw-plane rotation to change from 1-finger 3D rotation to the above 4D operations. Among the available shapes are a 2-torus manifold embedded in 4D, a 4D Veronese embedding of Steiner's Roman Surface (the 2D projective plane), and a variety of Fermat surfaces in CP2 labeled as Calabi-Yau manifolds, with N=5 being the 2D cross-section of the 6-manifold that is a candidate for the hidden dimensions of 10-dimensional string theory. The application is here: The 4D Object Exploration App|
|4Dice is my (free) iPhone App first posted in 2012. 4DRoom, first posted in 2017, is an extension of the 4Dice context to the interior of a 4-dimensional room, including a virtual-reality mode that rotates in the 4D wx plane as you rotate your body with your iPhone. (More details below.)|
|Images of the Calabi-Yau Quintic: Hidden dimensions of string theory|
| Shown below are my graphical representations of the
Calabi-Yau quintic representing the hidden dimensions of string
on my 1994 paper. These were made available on the
Wiki Commons domain in 2014, and clicking on the images takes you to
the source material. The first is a 2-dimensional cross-section of
the quintic in CP2, and the second is a complete
6-dimensional representation of the (local C4) quintic
embedded in CP4 using 4D discrete samples in a
C2 subspace to produce a hypercubic array of the
corresponding varying 2D cross-sections.|
|The Quaternion-Based Translational and Rotational Coordinate Matching Problems, (Under review: Andrew J. Hanson). We review the problem of transforming matching collections of data points into optimal correspondence. The classic RMSD (root-mean-square deviation) method (sometimes referred to as the "Orthogonal Procrustes Problem,") calculates a 3D rotation that minimizes the RMSD of a set of test data points relative to a reference set of corresponding points. Similar literature in aeronautics, photogrammetry, and proteomics employs numerical methods to find the maximal eigenvalue of a particular 4x4 quaternion-based matrix, thus specifying the quaternion eigenvector corresponding to the optimal 3D rotation. Various approaches have been proposed to include orientation frames into this problem, which we also review. Extensions of the quaternion method are given for 4D spatial and rotational data sets as well as related applications.|
|[DQT3] Quantum Interval-Valued Probability: Contextuality and the Born Rule, by Yu-Tsung Tai, Andrew J. Hanson, Gerardo Ortiz, and Amr Sabry, (Physical Review A, 97, (5), 1 May 2018). DOI: 10.1103/PhysRevA.97.052121. We present a mathematical framework based on quantum interval-valued probability measures to study the effect of experimental imperfections and finite precision mesasurements on defining aspects of quantum mechanics such as contextuality and the Born rule. This work continues our systematic investigation into finite precision, limited resources, and errorful processes in quantum mechanics. An early preprint is posted on the arXiv: https://arxiv.org/abs/1712.09006.|
|Isometric Embedding of the A1 Gravitational Instanton, by Andrew J. Hanson and Ji-Ping Sha, pp. 95-111, appearing in "Memorial Volume for Kerson Huang, " Ed. K.K. Phua, H.B.Low, & C. Xiong, World Scientific Pub. Co., Singapore (2017). ISBN-13: 978-9813207424, ISBN-10: 9813207426, available here. A local copy can be found here.|
|Memories of Kerson Huang, by Andrew J. Hanson, pp. 13-16, appearing in "Memorial Volume for Kerson Huang, " Ed. K.K. Phua, H.B.Low, & C. Xiong, World Scientific Pub. Co., Singapore (2017). ISBN-13: 978-9813207424, ISBN-10: 9813207426, available here. A local copy can be found here.|
|Charting the Interstellar Magnetic Field behind the Interstellar Boundary Explorer (IBEX) Ribbon of Energetic Neutral Atoms, by P.C. Frisch, A. Berdyugin, V. Piirola, A.M. Magalhaes, D.B. Seriacopi, S.J. Wiktorowicz, B-G Andersson, H.O. Funsten, D.J. McComas, N.A. Schwadron, J.D. Slavin, A.J. Hanson, and C.-W. Fu, appearing in Astrophysical Journal, November, 2015. A local copy can be found here.|
|[DQT2] Discrete Quantum Theories, by Andrew J. Hanson, Gerardo Ortiz, Amr Sabry, and Yu-Tsung Tai, appearing in J. Phys. A: Math. Theor. 47 (2014) 115305 (20pp) (March, 2014). This work continues a systematic investigation of the formulation of discrete quantum computing using finite fields, and introduces that concept of Cardinal Probability as a way of dealing with probabilistic concepts in the absence of ordered numbers for finite fields. The DOI link is doi:10.1088/1751-8113/47/11/115305. A local copy can be found here.|
|[DQT1] Geometry of Discrete Quantum Computing, by Andrew J. Hanson, Gerardo Ortiz, Amr Sabry, and Yu-Tsung Tai, appearing in J. Phys. A: Math. Theor. 46, no. 18, pp. 185301 (22 pages), (2013). The DOI link is doi:10.1088/1751-8113/46/18/185301, This work presents the mathematical elements of the consequences of formulating quantum computation and qubits in terms of discrete, computable numbers, using complexifiable fields with finite characteristic. A local copy can be found here.|
|Scheduling Scaffolding: The Extent and Arrangement of Assistance During Training Impacts Test Performance, by Jonathan G. Tullis, Robert L. Goldstone, and Andrew J. Hanson, appearing in the Journal of Motor Behavior, pp 01--11, (2015) The DOI is DOI:10.1080/00222895.2015.1008686.|
|Putting Science First: Distinguishing Visualizations from Pretty Pictures. Andrew J. Hanson, "Putting Science First: Distinguishing Visualizations from Pretty Pictures," in Visualization Viewpoints column, Theresa-Marie Rhyne, editor. IEEE Computer Graphics and Applications, Vol 34, No. 4, pages 63--69, (July/August 2014). A local copy is here.|
|Interactive Exploration of 4D Geometry with Volumetric Halos Weiming Wang, Xiaoqi Yan, Chi-Wing Fu, Andrew J. Hanson, and Pheng-Ann Heng. ``Interactive Exploration of 4D Geometry with Volumetric Halos.'' In Proceedings of Pacific Graphics 2013 (Singapore, October 7--9, 2013). The DOI link is DOI:10.2312/PE.PG.PG2013short.001-006. A local copy is here.|
|Multitouching the Fourth Dimension. By Xiaoqi Yan, Chi-Wing Fu, and Andrew J. Hanson, IEEE Computer, Volume 45, Number 9, pp.80-88 (September, 2012). The IEEE site for the article is here, and a local copy can be found here.|
|Localization of polymerase IV in Escherichia coli By Sarita Mallik, Ellen M. Popodi, Andrew J. Hanson, and Patricia L. Foster. "Interactions and localization of Escherichia coli error-prone DNA polymerase IV after DNA damage," J. Bacteriol. (June 2015). Accepted manuscript posted online 22 June 2015. DOI link is here doi:10.1128/JB.00101-15, Abstract is here, and the pre-publication manuscript is here.|
|Mutational Topology of the Bacterial Genome. By Patricia L. Foster, Andrew J. Hanson, Heewook Lee, Ellen Popodi, and Haixu Tang. "On the Mutational Topology of the Bacterial Genome," G3: Genes, Genomes, Genetics, Volume 3, no. 3, pp. 399--407 (March 2013). Pub Med link: is here, DOI link is here, and the Journal URL link is here.|
|"Quaternion maps of global protein structure," By A.J. Hanson and S. Thakur, appearing in Journal of Molecular Graphics and Modelling, Volume 38, September 2012, pp. 256--278. The paper PDF site is here, and a local copy can be found here.|
|Talk. Illustrating String Theory Using Fermat Surfaces, Andrew J. Hanson ``Illustrating String Theory Using Fermat Surfaces,'' 19 September, 2019. ICERM workshop " Illustrating Geometry and Topology," Brown University, 16-20 September, 2019. Exploring the path leading from attempts to visualize Fermat's "Last Theorem," via superquadrics, to illustrations of the Calabi-Yau quintic representing the hidden dimensions of String Theory.|
|Talk. Matching Paired Sets of Space Space and Orientation Data, Andrew J. Hanson ``Matching Paired Sets of Space and Orientation Data,'' 17 October, 2018. Wolfram Technology Conference 2018 (16--19 October 2018, Champaign, Illinois).|
|Talk. Matching Stuff with Quaternions and Hough, Andrew J. Hanson ``Matching Stuff with Quaternions and Hough,'' 22 June, 2018. ASIC 2018 (17--22 June 2018, Loano 3 Village, Italy).|
|Talk. The 4D Room, Andrew J. Hanson ``The 4D Room,'' 20 July, 2017. ASIC 2017 (15--20 July 2017, Interlaken, Switzerland).|
|Talk. Discrete Quantum Computing, Andrew J. Hanson (with Gerardo Ortiz, Amr Sabry, and Yu-Tsung Tai), Seminar for Quantum Computing Group, Computer Science, Oxford University, Oxford, UK (25 July, 2014).|
|Talk. The Bugcatcher. Andrew J. Hanson (with Jonathan Tullis and Rob Goldstone), ``The Bugcatcher,'' 25 June, 2014. ASIC 2014 (23--27 June 2014, Moab, Utah).|
|Talk. Multitouching the Fourth Dimension. Andrew J. Hanson, ``Multitouching the Fourth Dimension,'' ASIC 2013 (24--30 July 2013, Cortina d'Ampezzo, Italy).|
|Course. Quaternion Applications. Andrew J. Hanson: Presented at Siggraph Asia (Singapore, 29 November, 2012). New application topics included optimal, smoothly controllable tubing and tube texturing, quaternion protein maps, and how dual quaternions solve the century-old conundrum of how a quaternion acts on a vector.|
Click here for a larger version
4Dice is my (free) iPhone App first posted in 2012.
4DRoom, first posted in 2017, is an extension of
the 4Dice context to the interior of a 4-dimensional
room, including a virtual-reality mode that rotates in the
4D wx plane as you rotate your body with your iPhone.
( 4D Client, first posted in 2016, is an unsupported iPhone-based utility system that implements the 4D Multitouch controls of the 4Dice design to control a desktop 4D application over the internet. Example code for an elementary interface is available in "iPodLink.cpp" linked in the 4Dice web repository. This application is used for our internal research purposes; we do not have the resources to provide support for outside users, though anyone is welcome to use it.)
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
path. The figure on the left shows one of many such paths
that can be constructed, with a ratio of
Selected Publications of Interest
(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: All known errata have been corrected in the Elsevier eBook and print-on-demand hardcopies as of mid-2018, including the last known correction to several problems in Chapter 29 on 4D rotations using quaternions (see just those corrections in the revised Ch. 29 pdf file.) Note: We are still working with Elsevier to repair the copyright page of both the eBook and the print-on-demand harcopies to include some notation to distinguish the corrected copies from the original, uncorrected, copies of the book, but they have so far been apparently powerless to fulfill that request.
If you have an older copy of the book, please check the list of known corrections that is maintained on the update and errata page (supplemented by the fixed Chapter 29 pdf file). In the older errata, for example, note that there are several array indices that are transposed in the C Program in Table E.3, page 446.
Addition: An example of something I should have
included in the book is the closed form double-reflection quaternion
Winner with co-author Sidharth Thakur of the
2012 JMGM Graphics Prize.
See also the
MGMS/Elsevier Graphics Award.
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
The Solar Journey Movie is 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. A DVD version containing the Solar Journey animation and supplementary science materials exists but is no longer marketed.
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.
Some Selected Research Topics
My research has focused on several areas of science, including: Mathematical Visualization, Virtual Reality, and Astronomy.
|A Tessellation for Fermat Surfaces in CP3,
10.1016/j.jsc.2008.09.002, appears in the Journal of Symbolic Computation
(Sept 2008). This work presents an explicit algorithm for tessellating
the algebraic surfaces (real 4-manifolds) F(n) embedded in
by the "Fermat" equation
|Dual Five-Point Function Geometry.
This work investigates the geometry of two-complex-variable
contour integration using the classic 1960's Dual 5-Point Function of
the early string model as the critical example. The investigation
actually began with some questions introduced in a very early
paper, Dual N-Point Functions in
PGL(N-2,C)-Invariant Formalism (A. J. Hanson, Physical Review, 1972).
A number of new insights are given in our paper "A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6" by Andrew J. Hanson and Ji-Ping Sha, DOI link 10.1088/0305-4470/39/10/01, which is published in J. Phys. A: Mathematics and General., vol. 39, pages 2509-2537 (2006). There is also a version on the arXiv, math-ph/0510064. A local copy can be found here.
Other topics. Various long-term projects deal with techniques for modeling, depicting, and interacting with geometric structures of extreme complexity. Subject domains of interest range from mathematical objects in four dimensions to exploiting quaternions to represent orientation fields of geometric objects. Recent work concerns rephrasing some of the classical differential geometry of curves and surfaces directly in terms of quaternion fields; an application is the determination of optimal framings of curves and surfaces by minimizing appropriate energies of the quaternion frame fields ("quaternion Gauss maps") in the 3-sphere.
An example of our work on star rendering embedded in the Solar Journey package may be found in this QuickTime movie depicting simulated stars, which compares favorably to real images such as Akira Fujii's Orion.
Selected papers of ours in this area:
|Visualizing Multiwavelength Astrophysical Data, Hongwei Li, Chi-Wing Fu, and Andrew J. Hanson, TVCG, Nov/Dec 2008, 14, no. 6, pp. 1555-1562, Proceedings of IEEE Visualization 2008. Describes a unique interactive GPU-driven volume-rendering paradigm tailored to the study of all-sky multispectral astrophysical data. Paper web site.|
|Visualizing Large-Scale Uncertainty in Astrophysical Data, Hongwei Li, Chi-Wing Fu, Yinggang Li, and Andrew J. Hanson, TVCG, Nov/Dec 2007, 13, no. 6, pp. 1540-1647; Proceedings of IEEE Visualization 2007. Astrophysical data is characterized by a wide variety of uncertainties and error sources; this work provides a set of tools for examining and visualizing these features. Paper and web site.|
|A Transparently Scalable Visualization Architecture for Exploring the Universe, TVCG, Jan/Feb 2007, is a full description of work done mainly by Chi-Wing Fu in my laboratory. This framework supports transparent interactive navigation across enormous scale ranges such as those naturally occurring in astronomy. Paper and web site.|
|Scalable WIM: Effective Exploration in Large-scale Astrophysical Environments, TVCG, Sept/Oct 2006, 12, pp. 1005-1011; Proceedings of IEEE Visualization 2006. Describes a World-in-Miniature interface design for astrophysical exploration whose development was led by Yinggang Li in my laboratory. Paper and web site.|
Approach to the Black Hole at the Galactic Core. This animation represents an assembly of data collected by astronomers all over the world to examine the surroundings of the Black Hole suspected to be present at the center of our Milky Way galaxy. A variety of scientific instruments, using different methods and different wavelengths have been utilized to get many orders of magnitude of image precision. This permits us to make a continuous fly-in starting from the familiar constellations and going all the way in to the stars actually orbiting the Black Hole itself.
The Sun's interaction with its environment. This image of the heliosphere, representing the interaction of the Solar wind with the surrounding interstellar material, is taken from our short film "Solar Journey;" an extended version of the film will be produced for public distribution on videotape and DVD during the coming year. The shapes depicted here utilize a theoretical model by Timur Linde from the University of Chicago. The image has appeared as the Astronomy Picture of the Day, APOD 2002 June 24, and was used as an illustration in a recent astronomy news article in Science Magazine, page 2005 of Vol. 300, 27 June 2003. (The image credit is very obscure, in tiny vertically-aligned print along the spine of page 2005.)
Satellites in our Sky (GMT June 24th 2003 2:21pm). We have over a thousand satellites flying through the sky over our heads. This image is from a brief animation representing a user's interaction with our Earthday graphics program. The animation shows a large portion of these at a selected time, and then zooms in for a closeup of the International Space station (ISS). We can clearly see the ring structure of geo-stationary (deep-space) satellites rotating with the Earth, located 38,500km above the Earth's surface (about 6 times the radius of the Earth). The entire animation appeared as the Astronomy Picture of the Day on 14 July 2003. See APOD 2003 July 14.
Our initial work on handling very large scales of spacetime in interactive virtual reality environments is described in our paper, Very large scale visualization methods for astrophysical data, which appears in Proceedings of Joint Eurographics-IEEE TVCG Symposium on Visualization, May 29-31, 2000, Amsterdam, the Netherlands. This paper is part of the published proceedings, © Springer-Verlag.
Our most extensive recent work, which appeared in TVCG in January 2007, describes the maturation of the scaling framework developed in Philip Chi-Wing Fu's PhD thesis, and is entitled "A Transparently Scalable Visualization Architecture for Exploring the Universe." For details, see the summary web site.
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. This method is now obsolete, as the WebGL "4D Explorer" App runs on more platforms and has a complete 4D interactive interface.
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. Most features of the latter
are now available on the WebGL
"4D Explorer" App, implemented by Leif
Christiansen, which runs on more
platforms and has a complete 4D interactive interface.
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.
Cover picture: IEEE Computer 27 (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: