Outside to inside ratios in the dimensions

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Outside to inside ratios in the dimensions

Postby anderscolingustafson » Sun Nov 14, 2010 3:08 am

I have noticed that when two objects have the same diameter but one object has one more dimension than the other the one with the extra dimension will have a greater outside to inside ratio than the one with one less dimension. For example if you had a square and a cube with the square about four smaller squares across and the cube about four smaller cubes across the square would have 12 squares on its outside and four on its inside making its outside to inside ratio about three to one wile the cube would have 56 cubes on its outside and eight on its inside making its outside to inside ratio 7 to one.
Last edited by anderscolingustafson on Fri Nov 19, 2010 5:08 pm, edited 1 time in total.
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Re: Outside to inside ratios in the dimensions

Postby PWrong » Wed Nov 17, 2010 11:52 am

It seems true for squares and cubes of equal side length. Try a square with 'x' smaller squares across and a cube with 'x' smaller cubes across, and see if you can prove that it's true whenever x is positive.

By the way, are you sure the diameters are equal?
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Re: Outside to inside ratios in the dimensions

Postby anderscolingustafson » Fri Nov 19, 2010 5:07 pm

I found that wile for objects of the same diameter with different numbers of dimensions the object with the most dimensions always has a greater outside to inside ratio this is not always true when the object with more dimensions has a greater diameter. For instance if you have a square that is three smaller squares across and a cube that is four smaller cubes across the square has eight squares on its outside and one on its inside making its outside to inside ratio 8 to 1 wile the cube has 56 cubes on its outside and eight on its inside making its outside to inside ratio seven. So the square in this case has a greater outside to inside ratio to spite having fewer dimensions. If there are two objects were the one with more dimensions has a smaller diameter than the one with less dimensions than the one with more dimensions will almost always have a greater outside to inside ratio as outside to inside ratios are inversely related to diameter.

By the way, are you sure the diameters are equal?


What exactly do you mean by this question?
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Re: Outside to inside ratios in the dimensions

Postby PWrong » Sun Nov 21, 2010 7:35 am

So in general, suppose your square is x squares across and your cube is y cubes across. The square has 4(x-1) small squares on the outside, and (x-2)^2 on the inside right? If you find a similar formula for the cube, you can compare the two and see exactly when one ratio is bigger than the other.

What exactly do you mean by this question?


Well the diameter of a unit square is √2, and the diameter of a unit cube is √3. You probably meant the side length when you say "diameter".
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Re: Outside to inside ratios in the dimensions

Postby anderscolingustafson » Wed May 18, 2011 1:16 am

This would also mean that the more the dimensions the greater the diameter a cell would probably be in terms of molecules across wile the fewer the dimensions the smaller the diameter of a cell would probably be in terms of molecules across.

2d cells would probably need to have a much smaller diameter in terms of molecules across than 3d cells in order to have the right circumference to area ratio. If a 2d cell was as many atoms across as a 3d cell it would probably be too difficult for it to get ride of enough waste and to observe enough nutrients across it's cell membrane to survive.

4d cells on the other hand would probably be able to have a much larger diameter than 3d cells in terms of molecules across and still have a healthy surface volume to hyper volume ratio. This is also partly because a 4d cell membrane could probably also be more purpose than a 3d cell membrane.
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Re: Outside to inside ratios in the dimensions

Postby wendy » Thu May 19, 2011 8:37 am

A sphere, of diameter sqrt(N), has something like 98% of its volume between parallel planes of +1 to -1.
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