Introduction  to the JazzSnail                                                                                         12/15/2017
By Yi "Lisa" Li
For MA390 Fall 2017

The jazzSnail (a pronouncable derivative of jsSnail)  is a real-time interactive computer animation(RTICA) and 3D visualization of the snail shaped surface formed moving the boundary
of a Moebius Band into a planar circle, a.k.a. Sudanese Moebius Band.  It was created over
the past three decades by many members of the illiMath Collective. The original illiSnail was designed in C/IrisGL by George Francis,  Glenn Chappel, Chris Hartmann & Alex Bourd between1992 to 1997. Its concept is based on ideas of Michael Pintard, Blaine Lawson, Larry Siebenmann, Dan Asimov, and Ulrich Brehm, Conversion to HTML/WebGL was accomplished by Zach Reizner, Mohammad Amanzadeh and Yi "Lisa" Li from 2013 to 2017. The starfield is
due to  Stuart Levy, NCSA from 1994 to 2017. It was part of the VROOM for the 
CAVE presented by the  Electronic Visualization Lab (EVL)  at SIGGRAPH'94. Here is a three minutes video of the Post-Euclidean Walkabout.

Functions of the program:

To slow the rotation of the snail move the cursor towards the middle of the picture window. The surface is gapped rather than transparent and its colors indicate the values of its two 
angular parameters. The edge of the snail is a band with whose color indicates the
various navigation modes. The background of the snail is the night sky. The following keys change states: (key)name in parentheses
        1.	Toggle fl(y)mode  to moves the rotation center  from the snail to the observer.
      2.	Toggle for cross-eyed (b)inocular viewing.
      3.	Press (z)ap which reverts to  initial position. Useful if the snail disappears on you.
      4.         Morp(h)  through a succession of surfaces from the snail, through Steiner's 
                  surfaces, the torus, and to Brehm's Knotbox aka half of Lawson's Kleinbottle.
      5.        Cycle through other sha(p)es. See below.
      6.	Fly (f)oreward or (a)ft. Sorry for this poor substitute for the 6D CAVE wand.

References:
George Francis: Prof. emer. of  Mathematics at University of Illinois, Urbana.
Alex Bourd: erstwhile RA, now senior engineer at Qualcomm.
Chris Hartmann: RA, now math professor at University of Alaska, Fairbanks.
Glenn Chappell: RA, now math professor at University of Alaska, Fairbanks.
Michael Pintard: was a French Engineer who created the first wire model of the snail.
Blaine Lawson: A Professor of Mathematics at S.U.N.Y , Stony Brook, N.Y.
Larry Siebenmann: Mathematician based at U. Paris Sud, Orsay, France.
Dan Asimov: Mathematician at U.C. Berkeley.
Ulrich Brehm: Professor of Mathematics at Technische Universität Dresden
illiMath Collective: The illiMath Collective is a group of students, scholars, professors,   
artists, scientists, and others based at the University of Illinois at 
Urbana-Champaign and the University of Illinois at Chicago who have 
created mathematical artworks since the late 1980s. 

A technical and mathematical description of this RTICA appears in "The Hypergraphics Honor Seminar at Illinois", Chapter 13 of David A. Thomas, Ed., Scientific Visualization in Mathematics and Science Teaching, 1995. ISBN-1-880094-09-6.

A reprint of this publication can be found at http://new.math.uiuc.edu/glsimEck/SnailFrancis.pdf

The present HTML/WebGL version of the JazzSnail uses David Eck's glsim.js library to illustrate the versatility of  his simulation of the IrisGL/OpenGL graphics library by James Clark.

Subversion snA.html of 16dec17 has the trail making feature originally designed for the
Duke University  DiVE Cube in 2006.  With the t(r)ail toggle (the border turns red) the flight path leaves a ribbonlike yellow trail. A stored trail (white) of a flight through Brehm's Knotbox (cycle through the sha(p)es) shows you how it got its name. Can you fly through the knotbox without penetrating the skin? If you do you will have flown along a trefoil knot.  In homotopy theory, this 2D cell complex is the regular spine of the complement of a trefoil know in the 3-sphere. Imagine an topological innertube, but tied into a trefoil knot. Now inflate it until the interior of the innertube occupies  every 3D volume of the 3-sphere. The 2D complement could be the knotbox.