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The question is about spatial dimensions.

It seems to me that every dimension adds something fundamentally new to space:

If you have one dimension, you have different positions and the ability to move an object (like a point). By adding the second dimension, the objects have the possibility to go around each other - they mustn't go through each other to change positions. The third dimension allows to tie a knot. Is there anything new like this if you add the fourth, fifth, ... dimension?

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Four dimensions makes you unable to tie a knot... –  Eric Stucky Apr 27 '13 at 22:16

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In four dimensions, you can not knot a line.

On the other hand, it is possible to build a city with roads, railways and footpaths, on the level with none of these crossing each other.

Seasons are still possible in four dimensions, but the day is 12 hours everywhere, and the world would be perfectly round. What does happen though, is that one gets 'season zones' like 'time zones'. One can think of 3d world as a line where the S hemisphere is one season, and the N hemisphere is six months later. In four dimensions, there is always somewhere that is in early spring, or late autumn, or mid-winter, or whatever.

You can weave 2d sheets in 4d, in general, to cut and weave one needs N-2 dimensional sheets and blades.

In four dimensions, it is possible for topologically different figures to have the same topological surface.

In five dimensions, it is possible to knot, but not weave, 2d sheets.

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With four dimensions, you can have an orthogonal transformations with determinant 1 which can not be expressed as a rotation around a given axis by a given angle. It is up to you whether or not you want to call such a thing a rotation nevertheless.

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