Hi all,

Here's a fun little animation I recently made. It's small enough to be uploaded
as an animated GIF. Hopefully that will work.

The model is a mesh, generated entirely within POVRay via a few custom macros.

I will shortly have this printed in 3D via Shapeways. For this, I exported the
meshes from POVRay into ASCII .STL files (because it's an easy format). These
were converted into .3DS using MeshLab and subsequently imported in Sketchup,
where I merged the meshes into a single, closed volume without any interior
vertices (something POVRay does not mind, but Shapeway does). I am curious to
see how well the illusion will work in real life.

Cheers,
Erwin

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Attachments:
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Preview of image 'animatedtriangle-withpauses.gif'

Eriban wrote:

> Here's a fun little animation I recently made. It's small enough to be uploaded
> as an animated GIF. Hopefully that will work.

yes it works, great object! Is the shape your own design?

> I am curious to see how well the illusion will work in real life.

At least with one eye closed you should be good ;)

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Hi Christian,

Thanks for the response.

> yes it works, great object! Is the shape your own design?

No and yes. The basic idea of the shape is not new [1] and was apparently
already known in 1969 [2]. However, I did model it from scratch and I have not
seen this shape broken into cubes before. The latter does make it more difficult
to model but I am hoping it will also help to re-enforce the illusion.

> > I am curious to see how well the illusion will work in real life.
> At least with one eye closed you should be good ;)

That will help, but even then, the illusion won't be perfect. There will still
be differences in shading due to the curvature. Nevertheless, I am hoping that
these second order clues will be masked by the overall outline of the shape and
the bigger shade differences caused by the main face orientations.

Cheers,
Erwin

[1] http://www.anopticalillusion.com/2012/09/impossible-triangle-sculpture/
[2] http://im-possible.info/english/art/sculpture/vyacheslav-koleychuk.html

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On 15-6-2013 0:36, Eriban wrote:
> Hi Christian,
>
> Thanks for the response.
>
>> yes it works, great object! Is the shape your own design?
>
> No and yes. The basic idea of the shape is not new [1] and was apparently
> already known in 1969 [2]. However, I did model it from scratch and I have not
> seen this shape broken into cubes before. The latter does make it more difficult
> to model but I am hoping it will also help to re-enforce the illusion.

Smart! Years ago, I made a wood model of the triangle but it was an open 
structure of three straight branches. This is a much more interesting 
object.

Thomas

>
> [1] http://www.anopticalillusion.com/2012/09/impossible-triangle-sculpture/
> [2] http://im-possible.info/english/art/sculpture/vyacheslav-koleychuk.html
>
>
>

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On Thu, 13 Jun 2013 22:48:12 +0200, Eriban  
<pov### [at] spamgourmetcom> wrote:

> Hi all,
>
> Here's a fun little animation I recently made. It's small enough to be  
> uploaded
> as an animated GIF. Hopefully that will work.
>
> The model is a mesh, generated entirely within POVRay via a few custom  
> macros.
>
> I will shortly have this printed in 3D via Shapeways. For this, I  
> exported the
> meshes from POVRay into ASCII .STL files (because it's an easy format).  
> These
> were converted into .3DS using MeshLab and subsequently imported in  
> Sketchup,
> where I merged the meshes into a single, closed volume without any  
> interior
> vertices (something POVRay does not mind, but Shapeway does). I am  
> curious to
> see how well the illusion will work in real life.
>
> Cheers,
> Erwin

I wonder if the impossible triangle would be possible in 4d space. Looking  
at an animation of a rotating 4d cube makes it seem possible to me, but I  
don't have a clue if it is mathematically possible. .. and with straight  
sides of course.
-- 
-Nekar Xenos-

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Thomas de Groot <tho### [at] degrootorg> wrote:
> Smart! Years ago, I made a wood model of the triangle but it was an open
> structure of three straight branches. This is a much more interesting
> object.

Yes, in terms of interestingness, the connected-curvy model beats the
open-straight-edged one hands down. Also, the former model should work better as
a pendant. Less dangerous. Nevertheless, you can't beat the simplicity of the
latter. :-)

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"Nekar Xenos" <nek### [at] gmailcom> wrote:
> I wonder if the impossible triangle would be possible in 4d space. Looking
> at an animation of a rotating 4d cube makes it seem possible to me, but I
> don't have a clue if it is mathematically possible. .. and with straight
> sides of course.

To be honest, I have no clue. I find 3D often already hard enough to handle,
that I fear 4D is out of my league. So instead your question triggered me to go
down one dimension and so I started wondering if there are impossible 2D shapes
that can be drawn as illusions in 1D. Nothing fruitful came out of that exercise
though... ;-)

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Eriban wrote:

> "Nekar Xenos" <nek### [at] gmailcom> wrote:

>> I wonder if the impossible triangle would be possible in 4d space. 

> so I started wondering if there are impossible 2D shapes

both ideas don't seem to work because "impossible objects"
are not really geometric entities, they are mind tricks played
on a brain evolved to interpret 2d projections of 3d scenes.

Your animation depicts a perfectly valid object that you can
send to a 3d printer and use as a pendant. Any further confusion
caused is more a matter of neuroscience than mathematics.

So it doesn't work in 4d simply because there is no reference
observer to interpret the object in a wrong way. And it doesn't
work in 2d because a 1d projection of a 2d object is too boring
to have complex expectation assocatiated with it (for each set
of 1d lines you just assume you are seeing some unknown thin
shapes sideways and that is always a possible object).

Now, that being said you can certainly define some 4d cubelike
thing and a position in 4d space that causes 3d projections to align
in an analogous way to a specific 3d "impossible object". But I think
there is no longer any specific strangeness associated with that.

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On Mon, 17 Jun 2013 03:05:53 +0200, Christian Froeschlin <chr### [at] chrfrde>  
wrote:


> Now, that being said you can certainly define some 4d cubelike
> thing and a position in 4d space that causes 3d projections to align
> in an analogous way to a specific 3d "impossible object". But I think
> there is no longer any specific strangeness associated with that.

That is exactly what I was asking. Not for it to be an impossible object,  
but rather an assertion that the impossible object is not impossible, it  
is 4d. My question is where can I find a formula to prove/disprove this?

-- 
-Nekar Xenos-

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Nekar Xenos wrote:

> That is exactly what I was asking. Not for it to be an impossible 
> object, but rather an assertion that the impossible object is not 
> impossible, it is 4d. My question is where can I find a formula to 
> prove/disprove this?

That was not quite what I meant. I was pointing out that the
question is not well-defined because the "impossible" object is
not necessarily any sort of geometric object (not even a mathematical
abstraction in n-dimensional space). It is just a conflict in your
brain because it locally misinterprets the 3d structure of the image
and can't get to a consistent model.

Of course you can define a 4d object that looks like the impossible
object in 2d projection, but so you can in 3d (it's the possible object
Eriban is actually building). Who can say it is or is not the object
that you (fail to) imagine when you encounter the illusion.

Of course, there exist objects in 4d space that have 3 or less
dimensions and still cannot be embedded in 3d space. A famous example
is the Klein Bottle. But I think this is a different matter.

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