TITLE: Klein's Bottle
NAME: Thomas Willhalm
COUNTRY: Germany
EMAIL: Thomas.Willhalm@uni-konstanz.de
WEBPAGE: http://www.informatik.uni-konstanz.de/~willhalm/
TOPIC: Physics & Math
COPYRIGHT: I SUBMIT TO THE STANDARD RAYTRACING COMPETITION COPYRIGHT.
JPGFILE: twklein.jpg
RENDERER USED: 
    (modified) Povray 3.0

TOOLS USED: 
    paper, pencil, fractint, calculator, g++, maple

RENDER TIME: 
    2 hours 17 minutes 49.0 seconds

HARDWARE USED: 
    Sun Ultra Enterprise 4000/5000

IMAGE DESCRIPTION: 


My idea was to show an "everyday life" scene full of mathematical objects.

The main object is "Klein's Bottle" on the table. Up to isomorphism you get
this by identifying  0,1 x 0,1 /R where R is the equivalence relation
generated by

  { ((x,0),(x,1)) | x in  0,1  } U { ((0,x),(1,1-x)) | x in  0,1  }

Klein's Bottle has no border and can't be orientated. Since you can't
embed it in the 3-dimensional space the handle intersects with the body
of the bottle in my picture.


DESCRIPTION OF HOW THIS IMAGE WAS CREATED: 


The bottle itself is a CSG structure of several SORs, tori, planes
and spheres. It was designed on paper and modified using a text-editor
and Povray.

The flower is a 3-dimensional cut of a julia set. The parameters are:
        <0.2,0.4,-0.3,0.3>
        max_iteration 12
        slice <0,0,1,1> 0
The stock is composed of three cylinders.

The left image is part of the common mandelbrot set. The image was
created with fractint and used as height field as well as for
coloring of the image. The image itself can be retrieved from
http://www.cnam.fr/fractals/thomas-f.html

The right image shows the first digits of the number pi. It uses the fact 
that a rectangle with ratio 1:sqrt(2) can be devided by two and keeps its
ratio (well known from the A3, A4, A5, etc. sheets). The font used for
the numbers is "Schwaben Alt" designed by Peter Wiegel.

The frames of both images are made with CSG.

The wallpaper is "mesh1" provided by the "isosurface patch" of povray
with a plane behind it.
(http://www.public.usit.net/rsuzuki/e/povray/iso/index.html)

The table was created by a modified checker pattern. First, I wanted
to show you a nice tiling of the poincare disc. But since time hurries
I ended up with this "partial result". When I've finished this code
I will put it on my web page.

All textures are part of the povray distribution.

The illumination was done with four spotlights and two point light sources.

I didn't provide the source code - especially the modified povray code -
because the files need some "clean up" before I can distribute them.
But feel free to email me for asking about details of my scene.