Question

the sum of angles of a polygen is 1440degree what is the sum of angles of a polygen with one side lens? what about the polygen with one side more?​

Answer 1

Answer:

usual, all the other answers limit themselves to convex polygons, and they all find the obvious solution, n=10.

However, allowing non-convex polygons, such as star polygons, can lead to additional solutions.

For a convex polygon (winding number = 1), the number of sides, n, is related to the sum (in degrees) of interior angles, S, by the equation

n=360+S180

So for a regular pentagon (Schläfli symbol {5}), S=540, and each interior angle (since it is regular) is 108.

But consider the regular five-pointed star polygon (Schläfli symbol {5/2}).

It winds around the center twice, and so it has a winding number of 2. If we introduce the winding number, w, into our equation, we get

n=360w+S180

So, for {5/2}, we get S=180, and each interior angle is 36.

Now back to solutions with S = 1440…

Plugging 1440 in for S, our equation becomes

n=2w+8

So, the possible numbers of sides are 10, 12, 14, 16, 18, …

In other words, any even number greater than 8.

But what do these solutions look like?

The usual interpretation of {12/2}, since 2 divides 12, would be a pair

of distinct concentric hexagons in dual position to each other (figure on left below), similar to the way in which the Star of David, aka {6/2} can be seen as a pair of distinct triangles.

However, if we do not require regularity, then there is a solution in which we still have a single path connecting all twelve edges (figure on right below).

For n=14, w=3, so we have {14/3} which can be regular

So again, in the Euclidean plane, n can be any even number greater than 8.

But what about in spherical or hyperbolic space?

In spherical space, the interior angle of a regular convex n-gon, can approach 180. So in spherical space, a regular 9-gon can have a sum of interior angles equal to 1440. The regular 10-gon would have to be shrunk to a point, and the regular 8-gon would have to lie on a great cicrle (both cases can be considered degenerate).

So in spherical space the only solution for a non-degenerate regular convex polygon is n=9.

In hyperbolic space, interior angles of n-gons are less than their counterparts in Euclidean space, so n=10 would be degenerate (a point), and non-degenerate regular convex polygons would have solutions for any n>10.

I’ll leave it as an exercise for the reader to explore irregular and/or star polygons in spherical or hyperbolic space.