Constructing a polygon within a given circle

Factual Questions
1

What is the regular polygon with the least number of sides that cannot be constructed within a given circle, using only a straightedge and a compass?

2

A heptagon (seven sides).

3

Yes, of course. I knew you could get close, but not precisely.

4

The complete list would start with:
7 sides
9 sides
11 sides
13 sides
14 sides
15 sides
18 sides
19 sides
21 sides

5

See Constructable Polygon. The only regular polygons that can be constructed are those where the number of sides is a product of a number of distinct Fermat primes multiplied by a power of 2.

–Mark

6

15 sides is easily doable, since both 5 and 3 are constructible, and are relatively prime.

7

Of course you are right – my mistake.

8

Of course you can construct n-sided polygons with a ruled straightedge; I wonder why the Greek geometrists considered that cheating.

9

Is there anything the Greeks cared about that couldn’t be constructed with a compass and ruler? If there isn’t, that takes most of the game out of the game.

10

You can make a ruled straightedge easily, using the Greeks’ own techniques (Euclid’s Proposition VI.9). The problem isn’t in putting the marks on the ruler; it’s in interpolating between the marks.

11

If Wikipedia is to be believed, you can trisect an angle or double a cube using a marked straightedge. You cannot, however, square the circle (which is a problem the Greeks knew of and cared about.) Roughly speaking, doubling the cube becomes possible because the cube root of two is an algebraic number, while π is a transcendental (i.e., non-algebraic) number and so squaring the circle is still impossible.

There’s also a list of the regular n-gons that can and cannot be constructed using a marked straightedge here. In particular, it is still impossible to construct a regular hendecagon.

12

I don’t know a Fermat prime from a hole in the wall, but would that “power of 2” include 2[sup]0[/sup]? Because otherwise I don’t know how you come up with 3, a number whose regular polygon can be drawn with only a compass.

13

Yup, 1 is a power of 2.

14

Yes, as Chronos says, 1 is a power of 2.

Also, “a number of Fermat primes” includes the case of zero Fermat primes, which is why you can construct a square (4 sides = 2^2, with no primes involved).

A Fermat prime is a prime of the form (2^(2^n))+1. There are only 5 known Fermat primes: 3, 5, 17, 257, and 65537.

–Mark