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I've tried googling, but I can't find an answer: Is a Penrose Triangle a
possible object in 4 dimensions?
--
-Nekar Xenos-
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On 16/06/2013 07:40 PM, Nekar Xenos wrote:
> I've tried googling, but I can't find an answer: Is a Penrose Triangle a
> possible object in 4 dimensions?
The Penrose triangle is "impossible" in that it violates the rules of
perspective.
An object in 3D space can be projected into a 2D figure. The human brain
is well accustomed to this. The Penrose triangle suggests a 3D shape
projected into 2D, but then it violates the expected rules.
In fact, you *can* construct various 3D shapes which yield the Penrose
triangle if viewed from the right angle. The easiest way is to just have
warped beams.
Now, can we build this thing in 4D? Well, that would depend on what it
means to be a "Penrose triangle". Can we build something that, when
somehow reduced to 2D, gives the familiar figure? Well, yes we can, but
it depends on exactly how the dimension reduction is supposed to happen.
There's more than one way to turn 4D into 2D.
(The same happens with 3D, incidentally. You can take an orthographic
projection. You can do a perspective projection. And you can just cut a
2D slice out. All these options exist for reducing 4D to 3D, and then
again for 3D to 2D - or you can do it all in a single step...)
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On Sun, 16 Jun 2013 22:36:05 +0200, Orchid Win7 v1 <voi### [at] dev null> wrote:
> On 16/06/2013 07:40 PM, Nekar Xenos wrote:
>> I've tried googling, but I can't find an answer: Is a Penrose Triangle a
>> possible object in 4 dimensions?
>
> The Penrose triangle is "impossible" in that it violates the rules of
> perspective.
>
> An object in 3D space can be projected into a 2D figure. The human brain
> is well accustomed to this. The Penrose triangle suggests a 3D shape
> projected into 2D, but then it violates the expected rules.
>
> In fact, you *can* construct various 3D shapes which yield the Penrose
> triangle if viewed from the right angle. The easiest way is to just have
> warped beams.
>
> Now, can we build this thing in 4D? Well, that would depend on what it
> means to be a "Penrose triangle". Can we build something that, when
> somehow reduced to 2D, gives the familiar figure?
Yes, that is my question.
> Well, yes we can, but it depends on exactly how the dimension reduction
> is supposed to happen. There's more than one way to turn 4D into 2D.
>
> (The same happens with 3D, incidentally. You can take an orthographic
> projection. You can do a perspective projection. And you can just cut a
> 2D slice out. All these options exist for reducing 4D to 3D, and then
> again for 3D to 2D - or you can do it all in a single step...)
I would assume whatever would be considered the same as a perspective
projection by a 4d being. But then we don't even know of any 4d beings to
ask ;)
--
-Nekar Xenos-
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