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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] anonymous org> 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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> 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.
FWIW I think an equation for such an object might be:
x^(1/w) + y^(1/w) + z^(1/w) < 1 and 0 < w <= 0.5
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On Thu, 25 Jul 2013 09:11:03 +0200, scott <sco### [at] scottcom> wrote:
>> 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.
>
> FWIW I think an equation for such an object might be:
>
> x^(1/w) + y^(1/w) + z^(1/w) < 1 and 0 < w <= 0.5
>
Ingenius! =)
I will give that a try. I want to make an image that will depict how a 4
dimensional object can look totally different depending on where it is
sliced.
--
-Nekar Xenos-
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Now for the part that I don't understand at all:
I have heard it mentioned in scientific news that scientists have found
the fourth dimension and measured it.
How can you measure a dimension? I don't understand that. If the 4th
dimension has a "thickness", what then is the thickness of the 3rd
dimension. It makes absolutely no sense to me at all.
--
-Nekar Xenos-
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Am 25.07.2013 19:24, schrieb Nekar Xenos:
> Now for the part that I don't understand at all:
>
> I have heard it mentioned in scientific news that scientists have found
> the fourth dimension and measured it.
> How can you measure a dimension? I don't understand that. If the 4th
> dimension has a "thickness", what then is the thickness of the 3rd
> dimension. It makes absolutely no sense to me at all.
Those news actually make no sense for yet another reason: Scientists
around the world should know better than to call a newly discovered
dimension the "4th dimension", as the term is already firmly associated
with time.
But yes, there is some sense to measuring a dimension: Imagine the
universe was made up of only one spacetime dimension, and one additional
dimension curled up in a small loop; the universe would then have the
shape of a cylinder surface stretching into infinity(*) along the
spacetime dimension. But the other dimension would be finite, and could
probably be measured.
(*Alternatively, spacetime might also be finite, but on a much larger
scale, in which case we'd get a torus surface.)
String theory postulates that there are indeed - AFAIR - about half a
dozen extra dimensions, and it is suggested that they may indeed all be
curled up in this manner, with sizes roughly on the scale of sub-atomic
particles.
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>> FWIW I think an equation for such an object might be:
>>
>> x^(1/w) + y^(1/w) + z^(1/w) < 1 and 0 < w <= 0.5
>>
> Ingenius! =)
Thinking about it, there should probably be a modulus function around
x,y,z (to get the absolute value), otherwise x=-5,y=5,z=0,w=0.2 is part
of the solid (which it shouldn't be).
> I will give that a try. I want to make an image that will depict how a 4
> dimensional object can look totally different depending on where it is
> sliced.
Slicing it along one of the coordinate axes will be easy, the hard (and
more interesting) bit will be taking angled slices :-)
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