If you take a horzontal 2d slice of a human being near the center of the  
torso, you will get 3 objects: 2 almost circular objects for the arms and  
one ovalish shape for the torso. So if a 2d world could coincide with a 3d  
world, a 2d being on that plane would see 3 objects and could not imagine  
that it could be one object.

If we step it all up one dimension, you could have a 3-dimensional slice  
of 4-dimensional object that would seem to be more than one 3d object to  
us 3-dimensional beings.

Is this theoretically correct? Or am I missing something?

-- 
-Nekar Xenos-

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Am 22.07.2013 22:48, schrieb Nekar Xenos:
> If you take a horzontal 2d slice of a human being near the center of the
> torso, you will get 3 objects: 2 almost circular objects for the arms
> and one ovalish shape for the torso. So if a 2d world could coincide
> with a 3d world, a 2d being on that plane would see 3 objects and could
> not imagine that it could be one object.
>
> If we step it all up one dimension, you could have a 3-dimensional slice
> of 4-dimensional object that would seem to be more than one 3d object to
> us 3-dimensional beings.
>
> Is this theoretically correct? Or am I missing something?

You've gotten it absolutely right.

Think of, for instance, three blobs that over time merge into one. That 
/is/ a single object in 4D spacetime, but at any one moment we can only 
see a 3D slice of it, with some of the slices looking as if it were 
three separate objects.

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> If you take a horzontal 2d slice of a human being near the center of the
> torso, you will get 3 objects: 2 almost circular objects for the arms
> and one ovalish shape for the torso. So if a 2d world could coincide
> with a 3d world, a 2d being on that plane would see 3 objects and could
> not imagine that it could be one object.
>
> If we step it all up one dimension, you could have a 3-dimensional slice
> of 4-dimensional object that would seem to be more than one 3d object to
> us 3-dimensional beings.
>
> Is this theoretically correct? Or am I missing something?

Sounds correct to me - if you haven't already read Flatland then I 
recommend it:

http://en.wikipedia.org/wiki/Flatland

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On 22/07/2013 09:48 PM, Nekar Xenos wrote:
> Is this theoretically correct? Or am I missing something?

Nope, you're correct.

There are two main ways to transform a 4D object into something that our 
3D senses can comprehend. One is to take a 3D slice out of the object 
(which can obviously be done at different angles, and through different 
points). The other is a "projection", rather like the way POV-Ray 
transforms 3D scenes into 2D images. Like POV-Ray, you can do this two 
ways - a perspective projection, and an orthographic projection.

Fun fact: If you slice off one corner of a cube, the new facet is 
triangular. Similarly, if you slice off the corner of a hypercube, you 
get a tetrahedron...

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Nekar Xenos <nek### [at] gmailcom> wrote:
> If you take a horzontal 2d slice of a human being near the center of the  
> torso, you will get 3 objects: 2 almost circular objects for the arms and  
> one ovalish shape for the torso. So if a 2d world could coincide with a 3d  
> world, a 2d being on that plane would see 3 objects and could not imagine  
> that it could be one object.

> If we step it all up one dimension, you could have a 3-dimensional slice  
> of 4-dimensional object that would seem to be more than one 3d object to  
> us 3-dimensional beings.

It depends on whether you are seeing a slice or a projection.

-- 
                                                          - Warp

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On Tue, 23 Jul 2013 09:38:16 +0200, scott <sco### [at] scottcom> wrote:

>> If you take a horzontal 2d slice of a human being near the center of the

> Sounds correct to me - if you haven't already read Flatland then I  
> recommend it:
>
> http://en.wikipedia.org/wiki/Flatland
>
Very interesting. I will have to read it =)

-- 
-Nekar Xenos-

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On Tue, 23 Jul 2013 15:29:01 +0200, Warp <war### [at] tagpovrayorg> wrote:

> Nekar Xenos <nek### [at] gmailcom> wrote:
>> If you take a horzontal 2d slice of a human being near the center of the
>> torso, you will get 3 objects: 2 almost circular objects for the arms  
>> and
>> one ovalish shape for the torso. So if a 2d world could coincide with a  
>> 3d
>> world, a 2d being on that plane would see 3 objects and could not  
>> imagine
>> that it could be one object.
>
>> If we step it all up one dimension, you could have a 3-dimensional slice
>> of 4-dimensional object that would seem to be more than one 3d object to
>> us 3-dimensional beings.
>
> It depends on whether you are seeing a slice or a projection.
>

Yes. In this case it is a slice.


-- 
-Nekar Xenos-

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On Mon, 22 Jul 2013 22:48:15 +0200, Nekar Xenos <nek### [at] gmailcom>  
wrote:

If we take a vertical 2D slice of a tin can we get a rectangle.
If we take a horizontal 2D slice of the same tin, we get a rectangle.

So if we have a special shape that I will call a 4D Cylinder when taking  
3d slices we can get a sphere or a block depending on which direction it  
is being sliced.

So a 4d entity would tell you that that specific sphere and block are the  
same thing and any normal human being would think this 4d guy is just  
plain crazy ;)

-- 
-Nekar Xenos-

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Am 23.07.2013 22:06, schrieb Nekar Xenos:
> On Mon, 22 Jul 2013 22:48:15 +0200, Nekar Xenos <nek### [at] gmailcom>
> wrote:
>
> If we take a vertical 2D slice of a tin can we get a rectangle.
> If we take a horizontal 2D slice of the same tin, we get a rectangle.
>
> So if we have a special shape that I will call a 4D Cylinder when taking
> 3d slices we can get a sphere or a block depending on which direction it
> is being sliced.

Somehow my intuition tells me that depending on how you construct your 
4D cylinder you would either get (a) a sphere or a cylinder (by 
extruding a 3D-sphere along the 4th dimension), or (b) a cylinder or a 
box (by extruding a cylinder along the 4th dimension), but not what you 
describe.

After all a sphere is curved in 2 dimensions, so in order to hide all 
curvature you'd need 2 extra dimensions, i.e. a 5D space.

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> If we take a vertical 2D slice of a tin can we get a rectangle.
> If we take a horizontal 2D slice of the same tin, we get a rectangle.
>
> So if we have a special shape that I will call a 4D Cylinder when taking
> 3d slices we can get a sphere or a block depending on which direction it
> is being sliced.

You can do it purely from a mathematical point of view if you feel like 
it - you can do isosurfaces right? :-)

3D:
A cylinder is x^2+y^2<1 and 0<z<1
If you fix z to 0.5 (your 2D plane) you get a circle.
If you fix x to 0 you get y^2<1 and 0<z<1 (ie a rectangle)

4D:
Try something like x^2+y^2+z^2<1 and 0<w<1
If you fix w=0.5 (your hyper-plane) you get a sphere
If you fix z=0 you get x^2+y^2<1 and 0<w<1 (ie a cylinder)

Or how about x^2+y^2<1 and 0<z<1 and 0<w<1
z=0.5 gives a cylidner
y=0 gives a 3D block

So as clipka says, I don't think there is any 4D shape you can define 
that you can "hyper-slice" to get either a block or a sphere. But that 
isn't exactly a proof - feel free to experiment, look forward to your 
images in p.b.i :-)

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