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.
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
Tools:
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:
Dalbec's Docs
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!
Demo: jazzDunce
Tools: A better rtica should explain this. Currently available dunce.c, dunce.html. But this mini is basically more Javascript coding. There is nothing more available in C/IrisGL or Py/OpenGL that isn't already in the extant RTICA.
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".
Resources:
Tools: All of the 5 Morin-Apery homtopies were collected in a C/IrisGL rtica five.c of which there never was made an C/OpenGL or Python translation. So far, the following chapters for Project Five have been "liberated", i.e. reconstructed into rough, HTML5/Javascript/WebGL RTICAs.
jazzCuboctaversion: Denner's polyhedral eversion of the sphere.
jazzOptiverse: Eversion of the Morin-Froissart surface.
jazzNeck: Eversion of a cylinder.
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.
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.
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.
Stabilityof Quasicrystal Frameworks Projects 2017
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.
Comment
Note Yuliya's good use of animatedGIFsResources:
Tools: Python/OpenGL together with Javascript/HTML5/CANVAS (Gregg "Greggman" Tavares)
Resources:
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:
Resources:
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