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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On 23/07/2013 09:06 PM, Nekar Xenos 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.

Try this:

A line is a 1D object. A line has two ends. Ends are points, which are 
0D objects.

A square is a 2D object. A square has four edges. Edges are 1D objects. 
By extruding a line into a square, you're turning one line into two, and 
you're turning the two end-points into additional lines as well.

A cube is a 3D object. A cube has six surfaces. Surfaces are 2D objects. 
Again, when you extrude a square into a cube, that one square becomes a 
pair of squares, but in addition all four edges become new squares. So 
you have 2 + 4 = 6 new surfaces.

Now, come with me on this journey. A 4-cube is a 4D object. If it allows 
the pattern above, that means it should have 3D *volumes* as its "faces" 
(weird as that sounds). Presumably each one is cube-shaped. It also 
means that by extruding a cube into a 4-cube, the original cube becomes 
two cubes, and each surface on the original cube becomes an additional, 
new cube. That gives us 2 + 6 = 8 cuboid volumes.

A simpler way to work this out: A cube is given by
   -1 <= x <= +1
   -1 <= y <= +1
   -1 <= z <= +1
This defines a 3D region of space. By setting (say) x = -1, we now have 
only two degrees of freedom left (y and z). If we choose to fix one 
variable, there are three variables to choose, and two values to fix it 
to (-1 or +1), giving a total of six possibilities - which is why a cube 
has six sides.

We can extend this idea; what if we fix *two* variables? Then only one 
degree of freedom remains. Taking this approach, we see that we can pick 
two variables out of three - which is equivalent to NOT picking one out 
of the three. We can then set each variable to one of two values, for a 
total of four combinations. 4 * 3 = 12, and hence a cube has 12 
boundaries of the 1D type - i.e., 12 edges.

Fixing all three variables, we have 2^3 = 8 possible combinations, and 
sure enough a cube has eight 1D corners.

Taking all this, we see that a hypercube ought to have

   (4 choose 1) * 2^1 = 4 *  2 =  8 volumes
   (4 choose 2) * 2^2 = 6 *  4 = 24 surfaces
   (4 choose 3) * 2^3 = 4 *  8 = 32 edges
   (4 choose 4) * 2^4 = 1 * 16 = 16 corners

Another way of saying all this is that each time you extrude an N-cube 
into an (N+1)-cube, the number of K-dimensional things is doubled, and 
the number of (K-1)-dimensional things is then added to that.

          |   Points |         Lines |      Squares |       Cubes |
   -------+----------+---------------+--------------+-------------+
   Point  |        1 |               |              |             |
   Line   | 1*2 =  2 |             1 |              |             |
   Square | 2*2 =  4 |  1*2 + 2 =  4 |            1 |             |
   Cube   | 4*2 =  8 |  4*2 + 4 = 12 | 1*2 + 4 =  6 |           1 |
   4cube  | 8*2 = 16 | 12*2 + 8 = 32 | 8*2 + 6 = 24 | 2*1 + 6 = 8 |

Reassuringly, this agrees with the previous calculation.

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Am 24.07.2013 09:33, schrieb scott:

> 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 :-)

Well, I'm actually pretty sure there are such shapes - but they would be 
too irregular to deserve being called a "4D cylinder".

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On Tue, 23 Jul 2013 23:14:16 +0200, clipka <ano### [at] anonymousorg> wrote:

> 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.
>

You're right about the cylinder. I got a bit mixed up between 2 ideas.

I want to find a 4d object that has a cube slice in one place and a sphere  
slice in another place.

So I'll start with 2d slices from 3d objects. 3 dimensional space is made  
up of 2-d planes stacked on top of each other. The top plane of the object  
is a square. The next layer is slightly curved. Each layer is more curved  
than the previous. The bottom layer is a perfect circle. The one side is  
round and the other side still has corners. An unfinished table leg on a  
lathe :)

So shifting the dimensions up we get a cube on the first 3d section of  
this "4d table leg" with each cube being curved more than the previous and  
the last 3d section is a perfect sphere.

On the first object we can see that the first and last 2d sections look  
totally different. But because we can see the whole object, it doesn't  
seem strange to us. But because we cannot see into the 4d realm the 4d  
table leg can look like a cube if we see the first slice or a sphere if we  
see the last slice.

Is this correct now?

-- 
-Nekar Xenos-

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Le 24/07/2013 20:54, Nekar Xenos nous fit lire :
> 
> I want to find a 4d object that has a cube slice in one place and a
> sphere slice in another place.

sound like a superellipsoid going from <0,0> to <1,1>
Many paths connect these two slice, including the 3 obvious intermediate
slices: <0.5,0.5>, <1,0> and <0,1>

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Am 24.07.2013 20:54, schrieb Nekar Xenos:

> I want to find a 4d object that has a cube slice in one place and a
> sphere slice in another place.
>
> So I'll start with 2d slices from 3d objects. 3 dimensional space is
> made up of 2-d planes stacked on top of each other. The top plane of the
> object is a square. The next layer is slightly curved. Each layer is
> more curved than the previous. The bottom layer is a perfect circle. The
> one side is round and the other side still has corners. An unfinished
> table leg on a lathe :)
>
> So shifting the dimensions up we get a cube on the first 3d section of
> this "4d table leg" with each cube being curved more than the previous
> and the last 3d section is a perfect sphere.
>
> On the first object we can see that the first and last 2d sections look
> totally different. But because we can see the whole object, it doesn't
> seem strange to us. But because we cannot see into the 4d realm the 4d
> table leg can look like a cube if we see the first slice or a sphere if
> we see the last slice.
>
> Is this correct now?

Spot-on.

You might imagine a sphere morphing into a cube over time, with time 
being the 4th dimension in that picture.

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