My mathematical activities arise out of an enthusiasm for projective geometry and its connection to the human being and the world.

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    "conform!" update

    May 13, 2018: The "conform!" DVD is finally available in the Springer catalog here. te move forward. The covers and labels were approved in March and today I received copies of the finished product (NTSC version) in the mail.

    PGA code running in browser (Java script)

    August, 2017: Steven De Keninck, a programmer in Belgium, has made a big contribution to the development of projective geometric algebra (PGA). He has programmed it in JavaScript so it now runs "in the browser". GitHub project ganja is here. Not only that, he has prepared a number of valuable examples that can be run directly with a source code window for direct editing. Visit s site. Very cool, Steven!

    Path curves and pivot transformations

    March 30, 2017: I've extended my jReality path curve app to include the pivot transformation. All this is explained in Lawrence Edwards' book The Vortex of Life. The image shows a stereo pair of images; the green path curve system acts on the white wireframe funnel pathcurve surface to produce the yellow surface.

    Ribbon-edge sculpture photos

    March 23, 2017: I recently discovered that I had not ever photographed the best 3D print arising from my article on ribbon edges . So I've taken some photos to rectify that.

    "Geometric algebras for Euclidean Geometry" paper publication

    March 8, 2017: Just a bit over a year after the on-line version was published, my article "Geometric Algebras for Euclidean Geometry" has been published in the print version of the journal Advances in Applied Clifford Algebras (Vol 27, Nr. 1, pp. 185-208). The official online version remains here, the arXiv version is here.

    "Doing euclidean plane geometry with projective geometric algebra" published

    October 20, 2016: The journal Advances in Applied Clifford Algebra has published this article, based on a workshop presentation at ICCA10 in Tartu, Estonia (in August 2014). You can read it online here or here (author's copy on The article is aimed at a broad audience and is intended to give a gentle introduction to using projective geometric algebra for real-world applications.

    More Turing patterns with wallpaper symmetries

    March, 2016: I've gone back to work on the generating Turing reaction-diffusion patterns with wallpaper group symmetry. A photo album of these results can be found here. This collage appeared in the Art Show at Bridges 2016 conference in Finland.

    Graphics-in-the-browser project with Mark Phillips

    January, 2016: I spent a week in NYC with old friend Mark Phillips that mixed business with pleasure. The business was a collaborative programming project: Mark taught me about how to use ClojureScript to do graphics in the browser, and I taught Mark about 2D geometric algebra to program graphics. A running demo of our work can be found here.

    Mathematical visualization student projects

    May 20, 2014: The student projects for the Mathematical Visualization course from WS 2013/4 are now online. They are Java webstart applications and can be accessed on the course webpage.

    Conformal Movie update

    March 12, 2014: We have moved from planar domains w/ boundary to 3D surfaces w/o boundary. The picture shows a conformal map from an inflated 'B' to a hyperbolic octagon. The texture map represents the voronoi partition with respect to a distribution of 500 points on this octagon wrt the hyperbolic metric. The official website of the movie is here.

    Ten-cell movie update

    October 28, 2013: I've continued to work on the 10-cell project (see below 4 or 5 items). I've made a short movie "Ten-cell Views" (1280x720 format footage with soundtrack w/o narration). [P. S. that was accepted in the Short Movie Show of the Bridges 2016 Conference.] There are also a new picture galleries (focusing on two configurations of associated lines) here.

    Rendering Tessellations with Curvilinear Perspective

    August 14, 2013: I recently returned from Enschede, Holland where I attended Bridges 2013. I presented a paper on rendering anything but particularly 3D tesselations as spherical panoramas. Here's a short movie showing the transition from standard rendering to this method based on stereographic projection of the viewable sphere.


    April 12, 2013: That's not a command, that's the name of a movie we are producing here in the discrete geometry group. We are gearing up again to work on it after a semester busy with teaching. Here's the first fruit of the renewed activity: title sequence. More footage is on the way! We are looking for artistic collaborators (video editing with Premiere Pro) please let me know if you know someone who might be interested in contributing.

    Tetrahedra helices

    December 7, 2012: There's an arrangement of regular tetrahedra in euclidean space called a Boerdijk-Coxeter helix. Two of these towers almost fit together. In spherical space the gap can be made to disappear. Gluing spherical tetrahedra together as in the B-C helix leads to a closed helix of 30 tetrahedra with dihedral angles of 72 degrees. Twentry of these helices fit together without overlap to form the 600 cell. Here's a short movie of the result (mobile format) of one of these spherical towers.

    Visualization of discrete groups

    November 30, 2012: This semester I'm helping out with a mathematical visualization course here at the TU-Berlin. This includes a blog for the students. The course focuses on the visualization of discrete groups; I'm having a blast recycling software I've written over the years for this purpose. Today's assignment involved designing a fundamental domain for one of the 3D point groups . Here's a short movie of the result.

    Ten-cell research

    June 1, 2012: With Professor Tom Banchoff, old acquaintance and currently guest professor at Berlin Mathematical School, I've been working on understanding the 10-cell, a tessellation of the 3-sphere by 10 identical truncated tetrahedra. A movie (well, rough footage) can be found here, a webstart here, and more pictures (focusing on two configurations of associated lines) here.

    Stereographic Projection

    March 3, 2012: As part of an animation project, I've recently been working on improving a stereographic projection application (java webstart). Venturing out into writing GLSL shading language shaders for jReality, I've implemented exact ambient occlusion for the occlusion of the plane by the sphere! Otherwise I'm just fascinated how the picture of the earth one obtains shakes up my fixed ideas of size and proximity.

    Ames room construction

    January, 2012: During January visited the Karlsruhe Waldorf School and taught a main lesson in Projection Geometry. During this time, I collaborated with my friend Thomas Neukirchner, a teacher there, to build an Ames room on the stage of the school. Over a third of the students in the school visited this optical illusion during its 3-day life. View this photo gallery to share the happening. Run this Java webstart to see the computer simulation used to calculate the Ames room we built.

    Charles Gunn, Ph. D.

    September 21, 2011: I successfully defended my Ph. D. thesis, Geometry, Kinematics, and Rigid Body Motion in Cayley-Klein Geometries before my examining committee at the Technical University Berlin. It can be downloaded here. This degree follows hot on the heels of my M. Sc. (Mathematics., UNC-CH, 1983).

    Guide to GA in Practice published

    September, 2011: This new arrival from Springer, a collection of articles based on the conference "Applications of Geometric Algebra in Computer Science and Engineering", contains a chapter by yours truly entitled "On the Homogeneous Model of Euclidean Geometry." This is basically an extract of my thesis (see previous post), restricted to the case of euclidean metric. You can download (a free copy of) this article here.

    Kemper Polyhedra: Geometry in a Bundle

    February 25, 2011: My latest endeavors in 3D printing focus on a way of polarizing traditional regular polyhedra. One projects the polyhedron onto the ideal plane, then forms the polar image in a bundle. The images shown here all involve polar forms of the dodecahedron/icosahedron family. More pictures can be found here.

    Ribbon Edges: A New Impulse for Geometric Sculpture

    March 24, 2010: A paper with the above title has been accepted by the Bridges 2010 conference. More pictures can be found here.

    Visualization of Discrete Groups

    June 24, 2009: I've initiated a web-page where I'm collecting all the resources I have created to visualize discrete groups. In plain text, that means computer programs which make 2D and 3D pictures of repeating patterns of various sorts. A gallery of images from these programs can be found here.

    Non-euclidean GL Shading language shader

    Feb.10, 2009: I continue to develop an OpenGL shading language shader for realistic rendering of non-euclidean spaces. A similar picture of spherical space can be found here .

    6-Point Perspective

    March 23, 2009: Inspired by a meeting with the American artist Dick Termes, I have developed a module for jReality to create a new kind of perspective image which, following Termes, I call six-point perspective. Compare image on left with traditional flat image here. It's particularly helpful in visualizing space-filling tessellations. More pictures here. You can play with a demo webstart here.

    Rigid body mechanics in noneuclidean spaces

    Jan. 14, 2009: The image on the left shows the path of a rigid body (represented by the colored box) moving in hyperbolic space under the influence of a fixed momentum state (carried by the polar line of the vertical purple line). The green line represents the current velocity state. [Sept. 2011: See my thesis here].

    Mathematical visualization and physics

    Nov. 19, 2008: This semester the mathematical visualization course here is focused on physics. The picture on the left shows a screen shot of an exercise on rigid body motion. Run it as a Java webstart.

    3d Prints

    Here are some examples of 3d prints I've made at the 3D Lab here at the Technical University, using software I've developed to convert 2D surface descriptions into printable 3D volumes.

    jReality tutorials

    July 14, 2008: I've begun to work on expanding and updating the tutorial help for our jReality 3D scene graph project. If you're a user and have suggestions, please send them to me at the above email address!

    UniCat Bear

    May 31, 2008: For the inauguration of the UniCat excellence cluster I helped prepare an animation which was shown in 3D stereo as a prequel to the event. The animation features the UniCat Berlin buddy bear with three internal organs. A real version can soon be seen on the Str. des 17 Juni! For details see June 2008 issue of TU Intern magazine.

    Projective Geometry: Paradoxes, Polarities, Pictures

    Details of a talk given at TU-Berlin April 26, 2008.

    Schatz linkage

    February 13, 2008: In the 1920's Bricard and Schatz (independently) discovered a linkage consisting of 6 members which had an unexpected degree of freedom. The movement of the linkage also produces the ruled surface known as the oloid. All this can be explored in this interactive jReality webstart application.

    Woodcut shader

    February 13, 2008: Inspired by a talk in our weekly seminar, I played around with some non-photorealistic rendering using the RenderMan backend in jReality. Here's a hybrid image where the sphere and torus are rendered with a shader that simulates how a woodcut is made.

    Hyperbolic tessellation

    November 30, 2007: This image was prepared for an exhibit of mathematical images in Munich. The image was generated by jReality and rendered by RenderMan. Click here to see the full 8000 by 8000 image.

    Christmas Tree and Snow

    November 30, 2007: The ornaments on this Christmas tree include the Platonic and Archimedean solids and their duals, plus some Kepler star solids. If that's not enough, you can turn on a snow machine, too!

    Stereographic Projection Demo

    November 9, 2007: This webstart showcases stereographic projection from the sphere to the plane. Rotate the earth and watch the stereographic map follow along. Optional: replace the earth with Mars! This demo was created for my Geometry I class.

    Spherical Triangle Demo

    October 20, 2007: This webstart allows the user to drag the vertices of a spherical triangle, and see how the polar triangle changes in response. This demo was created for my Geometry I class.

    Animated Proof of Pythagorean Theorem

    November 1, 2007: This webstart starts with a scan of an old copy of Euclid's Elements opened to the Pythagorean Thm, and animates the proof. You just have to press the '1' key to start and stop the animation.

    A logo for the International Mathematical Union

    On August 22, 2006 the IMU officially gets a logo and this video explains its mathematical genesis.