revision 3 of 12sep10
Mini Project Resource Page
IntroductionThis page has the resources for the mini projects. We began the discussion in week1. Because "projects" appears in too many contexts ( IGL, gitmath, etc) we will use the term mini project (miniproj, minip, just mine) I hope you will choose one. But first just consider these as a possibility. This list is by no means exaustive.
In research (mathematical and otherwise) one generally toys with an idea first. In our case, construct a proof-of-concept rtica. This page will grow as you, the TAs, and I formulate other minis to consider.
Once you choose a mini from this list, or propose an entirely different one of your own, you are not (yet) committed. But we will settle on your mini the week after I get back (week4 of the semester).
The format will include a nickname (for quick reference) and a descriptive name (for the IGL BOX), links, and in which languages/libraries the original sources are written in. I will also a suggest a format for the mini for it to become a part of the mathviz19 class IGLprojects webpage.
Aside which may reappear elsewhere once gitmath is up and running. Your gitmath private subproject (think of a repo only you and the instructors see) will have a special directory, named public_html/, which is easily placed on the new.math webpage, once all are satisfied with it. This is to get you used to the work environment mathematicians like to work in: including a one step "publication" of their webpages.
From Euclid's proof of the Pythagorean Theorem to the Law of CosinesThis a continuation of the 3 assignments made last week in the first lecture.
Background: There are many websites illustrating some proof of this fundamental theorem, some have even several different examples. But I have never found one that showed how Euclid's own proof extends to more theorems which become apparent with the interactive features of a geometry construction kit, like Geogebra (or GEX and KSEG we talked about last week.)
Resources: Euclid's Elements end with Pythagoras
- Learn enough Geogebra to construct the squares on the sides of a triangle, and then more constructions.
- Geogebra native or screen-captured video (.mp4or .mov)
- Make an animatedGIF teaser for the webpage
- Webpage (by hand, not with factory, if possible.
Improved RTICA illustrating John Dalbec's contraction Zeeman's uncollapsible 2D cell complex, the Dunce Hat.Remark: The concept of collapsing a triangulated cell-complex, and a homotopy retract of a topological cell-complex comes, in the end, to the same thing in an RTICA since any triangulation can be made fine enough so that every simplex has subpixel size. Then the collapse looks exactly how a "smooth" homotopy looks on a pixilated screen! It's all in the eye of the beholder. Background: The following Wikipedia article has an animatedGIF of a different way to imagine the dunce hat. Maybe the parametrization of the surface is available and you could illustrate the contraction in competition to Dalbec's 17 year old animation.
Zeeman's classical demonstration involves homotopy equivalence. This is still the gold-standard method of proof in topology. But an explicit contraction is still more desirable, if it is well illustrated. [insert rtica history] Resources:
One day I think I understand this homotopy, the next I develop doubts again. I can't say I "own" Dalbec's "proof" completely. Homotopies (in particular) are that way. Illustrations of them, often called "picture proofs", are deceptive. They can fool you in accepting a proof when it isn't one. That's why iconoclastic mathematicians, like the Bourbaki group, forbid the use of any, not even figures!
Sphere EversionsThe Optiverse: A video on how to turn the sphere inside out. Background This video has a remarkably complicated RTICA, avn.c, which was used to make (most of) the video. This rtica has been maintaned by one of its three five authors, Stuart Levy, and still compiles (at least on a mac and a linux, and possibly on a PC) more than 2 decades after its first version was written.
The RTICA avn.c (and subsequent, simplified versions) is a viewer of sequence of surfaces in space, called a topiary. The topiary is a database originally created as an application of Ken Brakke's Evolver. But avn can, with suitable modifications if needed, display any topiary, however it was created, of other homotopies.
A topiary is a display list of surfaces considered to be the stages of a morphing shape in space. Every computer animation is in fact a sequence of frames displayed sufficiently rapidly to insinuate the impression of a fluid motion in the mind of the viewer. The word "frame" in the context of a sequence of 3D surfaces would be confusing, so we coined the term "tope" for a a temporal slice of a homotopy. A database of topes, then, is a "topiary".
Background: The late Bernard Morin invented a large body of topology that others illustrated with great success. Bernard lost his sight as a child! In 1992, his student Fraçois Ápery, visited the NCSA and together with Chris Hartman and Glen Chappell, programmed two sphere eversions which lived in the CAVE and Cube at the University of Illinois until these two popular, public virtual environments were closed for the last time in the early teens.
Notes I think Mimi Tsuruga actually translated five.c/IrisGL into Calculus and Matematica to make the topiary which the HTML translation mavn.html can be modified to re-play.
Then figure out an adaptive mesh for the classical homotopy itself. Or, stick to .py and do a C/IrisGL to Python/OpenGL but with recursive (?) adaptation.
Resurrecting the classic C/IrisGL RTICA
Background: See above
Resources: The original, no-longer compilable code, videos, articles, still pictures, etc . Also, there are the topiaries made by Mimi Tsuruga of Morin-Apery Gastrula eversion.
Tools: This would be the definitive, exemplary "elemenatary" re-implementation of a classical C/IrisGL RTICA into HTML/WebGL.
Note I still do not know whether DeWitt's idea was correct. Did he desribe a valid eversion? At the conference at the Batelle institute in 1968 where Morin first heard Froissart idea, the physicist DeWitt sketched a few planar diagrams of a regular homotopy of 2D slices of a 3D sphere eversion. This led to Chris Hartmann constructing the Philever tool for stitching plane sections together into a surface. And therefore catenating such contructions together into another sphere everstion. Or not, if the homtopy developed "Morin Parasites", i.e. forbidden singularities.
Don't know if this is worth revisiting. There have been some eversions since Optiverse, but did not become very popular.
Lecture: Use whiteboard to explain how homotopies of closed curves in the plane "extrude" to surfaces in space. For example, the Whitney Bottle in one dimension higher.
Background: See earlier IGL Project
Comment: I still don't understand the DeBrujn-Robbin's scheme for creating 3D Penrose quasicrystals to compose a rigorous proof that the progam is correct.
Tools: This would be in Greg "Greggman" Tavares direction. It's pure CANVAS code and mostly line-graphics. This could be enhanced with WebGL. The geometry should be converted from Tavares's library to MacKenzieJones and WebGL enhancements added successively. This would contribute towards liberating the Restorations away from glsim.js.
Yulia Semibratova's 3D-Koch-Fractals is a .py project where recursion is essential. The recursion "converges" to an unexpected shape, why? Not clear, yet. But it should be an .html project too. Higher Dimensional Koch Kurves
CommentNote Yuliya's good use of animatedGIFs
Chris Hartman's Sewing Machine to stitch planar homotopies of curves into surfaces.Background:
Wendy Hubbard's Philever Background: There remains several sphere eversions that have not yet been made into RTICAs. Two (technically related) of these are the Tony Phillips eversion (Scientific American, 1966) which is the first published one, and Bryce DeWitt (maybe non-) eversion, which was proposed (but never "proved" in the sense of checke to be true) in 1967 at the same Batelle conference which spawned the Froissart-Morin eversion, which eventually led to the Optiverse. Both eversions are based on horizontal slices (plane curves) of a surface undergoing a regular homotopy.e> Tools:
Chris Hartman's Planar Cellular Automata FactoryBackground:
End of Week2