Does Shape Negate The Concept Of Infinity?

[JAX]

While doing some reading about the size and shape of the universe I found that quite a few supposed physicists postulate that the universe has a shape and infinite size. Now my very finite brain finds that counterintuitive. So my question to you much more learned than I, is,..... Can something have infinite size and a shape? If so,...how? It seems to me that something could be one or the other, not both. I would really like to learn more about his, thanks.

[Turtle]

While doing some reading about the size and shape of the universe I found that quite a few supposed physicists postulate that the universe has a shape and infinite size. Now my very finite brain finds that counterintuitive. So my question to you much more learned than I, is,..... Can something have infinite size and a shape? If so,...how? It seems to me that something could be one or the other, not both. I would really like to learn more about his, thanks.

 

Menger/Sierpinski Sponge

...The Menger sponge simultaneously exhibits an infinite surface area and encloses zero volume. ...

 

[JMJones0424]

Turtle (or any other more mathematically inclined than I am)-

I don't understand how the Menger sponge can possibly enclose zero volume.

 

If, as the wiki page says, "The number of cubes is 20ⁿ, with n being the number of iterations performed on the first cube"

 

Let x be the length of one of those cubes, then as long as x is not zero, isn't there always some space enclosed? I understand that the volume enclosed will approach zero as n increases assuming the overall dimensions of the sponge is fixed, but approaching zero is not the same thing as zero... right?

 

In other words: if starting volume, at n=0 iterations, is 1

then n=1 the volume would be 20/27

n=2 the volume would be (20/27)²

 

so volume equals (20/27)ⁿ, for what n is volume zero? The obvious answer would be infinity, but I don't understand how.

 

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JAX- I don't know about supposed physicists, but actual physicists when talking about the shape of a possibly infinite universe may be referring to spatial curvature. In short, in a flat, Euclidean space, the sum of all angles of every triangle equals 180 degrees. This seems trivially true, but we are not entirely sure as scale increases to cosmic scales whether or not a sufficiently large triangle would have 180 degrees. For instance, if you draw an equilateral triangle on your kitchen table, it is easy to show that the angles add up to 180, assuming your table is flat. However, if you were to start at the equator and draw a line due north 10,000 km to the north pole, then make a 90 degree angle and travel south another 10,000 km, and finally complete the triangle by going back to your starting point, you would find that you had a triangle with three 90 degree angles, giving a total of 270 degrees. This would be proof that the surface of the earth is not a flat euclidean plane (not that you need that as proof :))

 

The Universe appears to be homogeneous and isotropic, and there are only three possible geometries that are homogeneous and isotropic as shown in Part 3. A flat space has Euclidean geometry, where the sum of the angles in a triangle is 180^o. A curved space has non-Euclidean geometry. In a positively curved, or hyperspherical space, the sum of the angles in a triangle is bigger than 180^o, and this angle excess gives the area of the triangle divided by the square of the radius of the surface. In a negatively curved or hyperbolic space, the sum of the angles in a triangle is less than 180^o. When Gauss invented this non-Euclidean geometry he actually tried measuring a large triangle, but he got an angle sum of 180^o because the radius of the Universe is very large (if not infinite) so the angle excess or deficit has to be tiny for any triangle we can measure. If the radius is infinite, then the Universe is flat.

http://www.astro.ucla.edu/~wright/cosmology_faq.html#FLAT

 

 

 

Ned Wright's Cosmology Tutorial is an excellent starting point for further reading on modern cosmology.

[Rade]

If the universe has an infinite size, then as with the holes removed from the Menger sponge at each stage (M1,M2..Mn), this must mean that some part of the universe can always be taken outside what has already been taken, for this is the definition of what it means for a quantity to be infinite. And, as with what is taken from the Menger sponge at each stage to form a hole, what is taken outside of the universe is of finite shape, and it is then what is taken that gives finite shape to the universe, and not that the universe as a whole has finite shape. Also, in this view, what is taken from the infinite must go somewhere--it goes outside--and for the universe this would mean it goes to form other possible universes (perhaps via black holes ? The possible correspondence between a Menger hole and a Black hole is of interest). And it is the prediction of M-Theory, that there are an infinite number of other universes. But, is the Menger sponge a good model for this process ?

 

I wonder if the opposite situation of the Menger sponge has been considered mathematically ? That is, rather than divide the M1 stage into nine areas and remove the center 1/9th on and on for each face of the cube in such a way that the shape of the hole disappears, instead add the shape of what is taken to the center 1/9th at the M1 stage that has x,y,z dimensions of proper size for each face of the cube and continue the process on and on. Seems to me this also would result in something with infinite size, but now with shape increasing to infinite volume, whereas the Menger sponge has infinite size and shape decreasing to zero volume. Now, this reverse Menger situation, it seems to me, presents a more valid model for a concept of infinite universes using M-Theory since it leads to infinite volume possible for the many universes having infinite size, rather than zero volume. Does any of this make any sense ?

[Turtle]

Turtle (or any other more mathematically inclined than I am)-

I don't understand how the Menger sponge can possibly enclose zero volume. ...

 

it is decidedly counter-intuitive & i don't think i'm the one to try explaining it. i would recommend benoit mandelbrot's The Fractal Geometry of Nature for getting a handle on fractals & their peculiar habits & haunts.

 

mind you that i am not suggesting the universe is a menger sponge -or any particular fractal form for that matter -, rather i am simply giving an example i think succinctly answers jax's question "Can something have infinite size and a shape?". yes jax; somethings can.

[LaurieAG]

Each face of the Menger sponge is a Sierpinski carpet;

...

the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane.

 

The Wiki page says that this conceptual sponge is only complete after an infinite number of cycles.

 

Also, considering that each completed cycle increase each component cubes surfaces by +24 and changes its volume by -7/27, the sponge should not be mistaken for a representation of 3D physical volume as such.

[Turtle]

The Wiki page says that this conceptual sponge is only complete after an infinite number of cycles.

 

Also, considering that each completed cycle increase each component cubes surfaces by +24 and changes its volume by -7/27, the sponge should not be mistaken for a representation of 3D physical volume as such.

 

well, since the original question specifies infinity, should we not mistake it as a representation of inquiry as such?