What is the formula for the largest number of parallel lines p in a regular shape with s sides where s is an odd number?

Answer 1

The formula is simple: p = 0.

There is no regular 3-d shape with an odd number of sides and so the question does not arise, so that only leave 2-d shapes.

A regular 2-d shape with an odd number of sides has no parallel sides.

Abbreviated proof:

s is odd, so let s = 2n +1 where n is any integer.

Each exterior angle of a regular polygon is 360/s degrees = 360/(2n+1) degrees. So, starting from any side, the successive sides are at angles of 360/(2n+1), 2*360/(2n+1), ... s*360/(2n+1) degrees from it.

A second side is therefore parallel to the first only if it is at an angle of 180 degrees. That is, if there is an integer k (0

This implies that 360*k = 180*(2n+1) = 360n + 180 degrees

Dividing both sides by 360 gives k = n + 1/2.

But that is impossible since both n and k are integers.

That contradiction proves that there cannot be a parallel line.