....also known as Klein's Surface, this is a test of the almost finished new
version of the bastard tiling shapes from hell include file.

Thanks in advance for your comments.

B. Gimeno
pd. The old BTSFH.inc site is NOT updated yet, sorry.

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B. Gimeno wrote:
> ....also known as Klein's Surface, this is a test of the almost finished new
> version of the bastard tiling shapes from hell include file.

  Personally I have always had a problem accepting klein's bottle as a
true *borderless* surface because the surface intersects with itself,
and the intersection could be interpreted as a border.

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Warp wrote:
> B. Gimeno wrote:
>> ....also known as Klein's Surface, this is a test of the almost
>> finished new version of the bastard tiling shapes from hell include
>> file.
> 
>   Personally I have always had a problem accepting klein's bottle as a
> true *borderless* surface because the surface intersects with itself,
> and the intersection could be interpreted as a border.

It only intersects itself in 3D because it can not be embedded in 3D.
But it can be embedded in 4 dimensions where it shows up as a true
borderless surface.

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"Warp" <war### [at] tagpovrayorg> wrote in message 
news:488a1829@news.povray.org...
> B. Gimeno wrote:
>> ....also known as Klein's Surface, this is a test of the almost finished 
>> new
>> version of the bastard tiling shapes from hell include file.
>
>  Personally I have always had a problem accepting klein's bottle as a
> true *borderless* surface because the surface intersects with itself,
> and the intersection could be interpreted as a border.

 Can't it be raised up though? So nothing intersects? Like a coil, I guess?

   ~Steve~

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"Urs Holzer" <urs### [at] andonyarcom> wrote in message
news:488a193c@news.povray.org...
> Warp wrote:
> > B. Gimeno wrote:

> >> ....also known as Klein's Surface, this is a test of the almost
> >> finished new version of the bastard tiling shapes from hell include
> >> file.

> >   Personally I have always had a problem accepting klein's bottle as a
> > true *borderless* surface because the surface intersects with itself,
> > and the intersection could be interpreted as a border.

Yes, it's a little disingenuous to call it a surface in 3D.

> It only intersects itself in 3D because it can not be embedded in 3D.
> But it can be embedded in 4 dimensions where it shows up as a true
> borderless surface.

Of course in 4D (which is inherently more interesting than 3D), it becomes
just a regular surface/volume, not much more interesting than the Moebius
band or even a torus in 3D.

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somebody wrote:
> Of course in 4D (which is inherently more interesting than 3D), it
> becomes just a regular surface/volume, not much more interesting than
> the Moebius band or even a torus in 3D.

The interesting property is that klein's bottle can not be "oriented".
It has only one "side" like the moebius band. So it is a little bit
more strange than a torus. Though I do not know whether it is more
interesting from a mathematical point of view. (Whatever that
means ...)

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St. wrote:

>  Can't it be raised up though? So nothing intersects? Like a coil, I guess?

Only if you got an extra dimension to raise it up in ;)

Nice image, BTW!

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