Cosmo Player Virtual Reality Mathematics Geometric Objects
Virtual Reality
.....Make the effort of a download that takes about twenty minutes, which you will do once and for all: it's worth it. The "virtual reality" images are simply amazing, and you will "pilot" them. However, the objects themselves download in ten or twenty seconds. Moreover, on this site, a whole section dedicated to this virtual reality imaging will develop rapidly, thanks to the collaboration of a CAD professional, Christophe Tardy. Therefore, sooner or later you will have to download Cosmo Player. Since you're going to do it...
.- If Cosmo Player (free utility, downloadable) is already installed on your machine, proceed: Read the "piloting instructions" (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.
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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 website no longer allows downloading this player (File Not Found during download), so it's impossible to view the virtual representations of your site. However, I found a site that still has this program, if readers have problems, you can communicate this link:
http://www.cs.iupui.edu/~aharris/vrml/ (scroll down this page, in the Cosmo Player section)
****Crosscappolyhedral version
****Boy's Surfacepolyhedral version
****Klein Surfacepolyhedral version
****Steiner's Roman Surfacepolyhedral version
****Toruspolyhedral version
****Torus with zero gorge
****Whitney Umbrellapolyhedral version
****Lactoedron
****Moebius Strip with 1/2 turn
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, some brief comments about the object.
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(surface with variable curvature)
Geometric objects, do you have a soul?
Collaboration with Christophe Tardy:
http://ctardy.free.fr/jadore/sciences/boy/index.htm
Very nice images of the Boy on his site:
(We didn't have time to refine the parametric equations: my representation still has these unwanted folds)
...Christophe, in his early thirties, living in Aix-en-Provence, was getting a bit bored creating, in CAD, synthetic images of chemical complexes. The sphere inversion, which I described as early as 75 with drawings that would massacre your neurons, has only been understood by a very small number of people (article M. Morin J.P. Petit in January 79 in Pour la Science). Until recently, few people had seen a Boy's surface with their own eyes, and even fewer had been able to hold one in their hands, until I invented its representation with its system of elliptic meridians, then its polyhedral form, described in the Topologicon. If you happen to go to the Palais de la Découverte in Paris, you will find, in the center of the "p" room, a model of a Boy's surface that is finishing to oxidize. I don't know if they have now put a small sign somewhere to say that I invented this representation, it would have made me happy. On my last visit, there was nothing. But, well... ...Magazines La Recherche and Pour la Science having refused for fifteen years any article of mine dealing with such subjects, considered as an illegal exercise of mathematics, I finally got discouraged. It can't be said either that the idea of producing new black and white comic books sold at 90 F each, when you can find colored comics for less than 50 F, was a powerful incentive. A price so stupidly discouraging was likely to put off even the most well-intentioned of my regular customers: high school students, teachers, and researchers, at least in this type of comic. ...The possibility of showing new creations on an Internet site has rekindled my imagination. But the fact of stumbling, at the corner of a street, on a "country" that can transform cardboard models into objects inhabiting the world of virtual reality was a very pleasant surprise. You will find below the result of this beginning of collaboration with Christophe. Many other objects will soon be added to this list. He made a fantastic color film, where you walk inside a Boy's surface, entering through one of its ears. He was the first to do this. It will be put on a CD. It is a bit "heavy" to be downloadable. But what a wonder....
****The monodre
****Initial step
****Central model of the cube inversion
****The same, showing how it can be built
****The Moebius strip with three half turnsthe same, in polyhedral
****Polyhedral version of the trefoil knot
****Another polyhedral version of the Boy
(transparent). Model by J.P. Petit, imaging: C. Tardy
of one of the cube inversions. Model by J.P. Petit, imaging C. Tardy
. Model by J.P. Petit, imaging C. Tardy
. J.P. Petit, imaging C. Tardy
. J.P. Petit and C. Tardy.
J.P. Petit and C. Tardy
(A. Phillips and C. Tardy).