Virtual Reality
.....Make the effort of a download of about twenty minutes, which you will only do once and for all: the result is worth it. The "virtual reality" images are simply extraordinary, and it is you who will "pilot" them. Yet the objects themselves download in ten or twenty seconds. Moreover, on this site, a whole section focused on this virtual reality imagery will develop at great speed, thanks to the collaboration of a CAD professional, Christophe Tardy. So, sooner or later you will have to download Cosmo Player. While you're at it...
.- If Cosmo Player (free utility, downloadable) is already installed on your machine, move on to the next part: Read the "instructions for piloting" (unless you already know them), then choose an object and click on it.
...- If this shareware utility is not already installed on your machine, read the instructions for its free download.
Then click on this icon::
http://www.cai.com/cosmo (archive of 01/08/2003)
August 19, 2004: Received this message from a reader:
Hello,
On your Virtual Reality page (http://www.jp-petit.com/science/maths_f/virt_real.htm), the official Cosmo Player site no longer allows downloading this Reader (File Not Found during download), so it is impossible to see the virtual representations on your site. Nevertheless, I found, with some difficulty, a site that still had this program; if readers have problems, you can share this link:
http://www.cs.iupui.edu/~aharris/vrml/ (Scroll to the bottom of this page, in the Cosmo player section)
...Click on the geometric object of your choice. On the right, its "polyhedral representation", using flat facets.
...Objects created by Frédéric Wescamp-Lorenz
On each page, a few brief comments on the object.
1- Crosscap ( and polyhedral version)
2- Boy's Surface ( and polyhedral version)
3- Klein Surface ( and polyhedral version)
4- Steiner's Roman Surface ( and polyhedral version)
5- Torus ( and polyhedral version)
6- Torus with zero throat
7- Whitney Umbrella ( and polyhedral version)
8- Lactohedron (surface with variable curvature)
9- Möbius strip with a 1/2 twist
*Geometric objects, do you then have a soul? *
Collaboration with Christophe Tardy:
Very pretty images of the Boy on his site:
http://ctardy.free.fr/jadore/sciences/boy/index.htm (archive of 30/12/2008)
(We did not have time to retouch the parametric equations: my representation still shows these unwanted creases)
...Christophe, in his early thirties, living in Aix-en-Provence, was a little bored creating, in CAD, computer-generated images of chemical complexes. The turning inside out of the sphere, which I described as early as 75 using drawings to fry your neurons, has still only been understood by a very small number of people (article M.Morin J.P.Petit published in January 79 in Pour la Science). Until recently, rare were the people who had seen a Boy's surface with their own eyes, and even fewer those who had been able to hold one in their hands, until I invented its representation with its system of elliptical meridians, then its polyhedral form, described in the Topologicon. If by chance you go to the Palais de la Découverte in Paris, you will find, in the center of "room p", a model of a Boy's surface that is finishing oxidizing. I don't know if a little card has now been put somewhere to say that it was I who invented this representation, that would have pleased me. On my last visit, there was nothing. But, oh well.... ... The journals La Recherche and Pour la Science having for fifteen years refused any article by me dealing with subjects of this kind, considered an illegal practice of mathematics, I had ended up getting discouraged. Nor can it be said that the idea of producing new comic strips sold in black and white at 90 F apiece, when color comics can be found for less than 50 F, was a powerful stimulus. Such a stupidly dissuasive price was bound to put off my usual best-intentioned customers: high school students, teachers and researchers, at least for this type of comic. ... The possibility of showing new creations on an Internet site rekindled my imagination. But coming across, at a street corner, a "country" that knows how to turn cardboard models into objects inhabiting the world of virtual reality in two shakes of a lamb's tail was a very pleasant surprise. You will find in what follows the result of this beginning of collaboration with Christophe. Many other objects will quickly be added to this list. He made a fantastic film, in color, where you stroll inside a Boy's surface, entering through one of its ears. He was thus the first to take this approach. We will put it on a cd. It is a little "heavy" to be put online for download. But what a marvel....
1-The monhedron (transparent). Model by J.P.Petit, imagery: C.Tardy
2- Initial step of one of the cube eversions. Model by J.P.Petit, imagery C.Tardy
3- Central model of the cube eversion. Model by J.P.Petit, imagery C.Tardy
4- The same, showing how it can be built. J.P.Petit and C.Tardy.
5-The Moebius strip with three half-twists (the same, in polyhedral form).
6- Polyhedral version of the trefoil knot J.P.Petit and C.Tardy
7- Another polyhedral version of the Boy(A.Phillips and C.Tardy).
