Make a 2D polygon - but all the side lengths (bar one) are whole numbers, and all angles are multiples of 22.5

Hello!

Here is a trig problem I simply do not have the knowledge to solve, please help! Here are the rules:

TASK: Create a complex or simple polygon with the smallest possible perimeter adhering the following limitations:

a) Exactly one side must have a unit length equal to any whole number +/- exactly 0.5 (eg: 0.5, 1.5, 2.5 etc). All other sides must have a unit length equal to a whole number.

b) All interior and exterior angles at vertices must be multiples of 22.5 degrees (22.5, 45, 67.5, 90 etc).

c) The polygon can have any number of total sides and vertices.

d) The polygon must be closed, and follow an unbroken chain of sides connected at vertices, but the sides can intersect and overlap each other to form a complex polygon.

The point of this question is to prove whether it is mathematically possible to create a shape using a set of line lengths of only whole numbers, snapping to 16 degrees of freedom in rotation, that terminates exactly 0.5 units from its origin.

 

A possible approach to the solution could be a repeating pattern that finds ways to reach factors of 0.5, for example a polygon that apparently violates rule A by having a side length of 0.25. By repeating all the lines of that polygon again it would terminate 0.5 units from origin. 

 

I leave this in your capable hands, wise mathematicians of the internet! Please let me know if I need to clarify any part of this.

 

Thank you for your time!

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