Regular shapes in various numbers of dimensions

New readers:

• At this point you may like to click here. A new window will open where you can find my definition of the 4th dimension. This links to a simple explanation of a 4D cube ... how it relates to a point, a line, a square and an ordinary cube ... and how to draw it ... and how it rolls.

Here is a series of numbers, which are in this order for a reason.   (Some of them, I suppose, are arguable, in that you could claim they should be different, but these ones will do for me.)   Here they are. 

1;  1;  infinity;  5;  6;  3;  3;  3;  3;  and guess the next number.

Any ideas?   What they are supposed to be, and indeed what they are at least in my mind, is the numbers of convex regular shapes, bounded by straight lines, that are possible in various numbers of dimensions.   So the numbers of dimensions corresponding to the above are: 

0;  1;  2;  3;  4;  5;  6;  7;  8;  9;  ?

In zero dimensions the only shape possible is a point.   Hence the answer in zero dimensions is 1.   Now, this is what I mean by arguable.   Is a point, a simple dot on a piece of paper, truly a shape?   And  even if it is a shape, is it convex and is it bounded by straight lines?   Arguably not.   In which case the first number in my series should be 0, zero.

What about one dimension?   Here the only shape possible is a straight line.   Is that a convex shape bounded by straight lines?   I think arguably it is.   So for my part I'd leave the second number in the series as 1.   But you may feel differently.

In two dimensions we are relatively untrammelled.   We can draw an equilateral triangle, a square, a pentagon, a hexagon and so on.   Whatever number of edges we want, however big, it is theoretically possible to draw a regular polygon (that's what they're called) which is convex and has that number of edges, or straight line sides.   So the third number in the series is infinity.

Now, there is a special feature of infinity.   If we were in the business of drawing these dreaded regular polygons, we could draw them, or at any rate imagine them, as big or small as we like.   We could have a regular thousand-sided figure, or a million- or trillion-sided figure if we prefer, with each of its regular straight line sides a mile long, a millimetre long, or whatever we choose.   But what if we draw a circle?

It is perfectly permissible to think of a circle, say a circle one inch in diameter, as consisting of an infinite number of infinitely short straight lines all joined together in such a way as to make a circle. So, on that way of looking at it, a circle is in fact an example of a regular convex polygon with an infinite number of sides.   This may be relevant when we look at higher dimensions.

It is surprising and extraordinary in a way, is it not,  after the infinite pastures of two dimensions, to find that the regular convex shapes in three dimensions are, as Lewis Carroll put it, "provokingly few in number".   They consist of the tetrahedron, which has four triangular faces, the cube, with its six square faces, the octahedron, with its eight triangular faces, the dodecahedron, with its twelve pentagonal faces, and finally the icosahedron which has twenty triangular faces.    Surely someone could create some extra regular shapes if they put their mind to it and had a sharp knife and a raw potato?