A discrete function that takes the number of sides of a regular polygon and tells you the measure of one of its inner angles. A regular triangle has three sides and its inner angle is 60 degrees. A regular quadrilateral has four sides and its inner angle is 90 degrees. A regular pentagon has five sides and its inner angle is 108 degrees.

That's a recipe for a regular pentagon right there. Draw a 108 degree angle between two segments with the same length.

Then draw another 108 degree angle on the last segment.

And another, and another, until the segments reconnect and you have a regular polygon with five sides.

We can write a table:

We can graph those values:

We can also write an equation:

That equation *perfectly* describes the discrete values in that graph. But the equation is stupid. It doesn't know it's *only* supposed to describe those discrete values. We can put in *other* values and, like a sucker, it'll give us a number, even though it isn't supposed to and even though that number won't make any sense.

Like n = 3.5. A regular polygon with 3.5 sides? No such thing. But if we throw n = 3.5 into that function, it gives us the number 77.1 degrees.

Maybe that's just gibberish, the result of pushing this function machine beyond its warranty. But maybe it isn't.

What if we tried to draw a regular 3.5-gon in the same way we did the regular 5-gon up there?

When you make the shape, ask yourself the following question: But where is the 3.5 in that shape? Maybe you see how the number 3.5 turned into the number 77.1 and how the number 77.1 turned into that _____ shape. But where is the 3.5 in that shape?

**Edited by BMAD, 28 March 2013 - 05:17 PM.**