Is it possible to divide a circle into $7$ equal "pizza slices" (using geometrical methods)?

Or is it possible to divide a circle into n equal "pizza slices" (I don't know how to call these parts, but I think you'll know what I mean), where n hasn't got a common divider with $360$? Or are the $360$ degrees just "arbitrarily" chosen, in such way that it's also possible to make a system with $7$ "degrees" in a circle?

The main question is actually if it's possible with a ruler and a pair of compasses to divide a circle into any number of slices, and if there's a condition for a number (e.g. a common divider with $360$ as I suggested) so it's possible to slice the circle in such a number of pieces.

Solution 1:

The following result answers your question. We can divide a circle into $N$ parts by straightedge and compass if and only if $$N=2^kp_1p_2\dots p_t,$$ where the $p_i$ are distinct Fermat primes. (We can have $k=0$, or $t=0$.)

A Fermat prime is a prime of the form $2^{2^j}+1$. There are only $5$ known Fermat primes: $3$, $5$, $17$, $257$, and $65537$.

Since $7$ is not a Fermat prime, we cannot by straightedge and compass do the division you seek.

Remark: The result was first published by Wantzel. Some people give Gauss credit for the result. Gauss certainly was the first to prove that the circle can be divided into $17$ equal parts by straightedge and compass. He almost certainly knew that any $360^\circ/N$ angle, where $N$ is of the shape described above, is constructible. There is no evidence that he knew that nothing else is.

Put $N=9$. Then $N$ is not of the shape described above, since $3$ occurs twice in the factorization. This shows that the $20^\circ$ angle is not constructible. Since $60^\circ$ is certainly constructible, that shows we cannot trisect the general angle by straightedge and compass.

Many books have proofs of the Wantzel result, for example Allan Clark's Elements of Abstract Algebra.

Solution 2:

The degree units have nothing to do with constructability using ruler and compasses.

The regular $n$- gon can be constructed if and only if $n$ is the product of a power of two and zero or more distinct odd primes of the form $p=2^k+1$. There are not so many such primes (called Fermat primes), namely $3, 5, 17, 257, 65537$ - and as far as is known today no more. Thus it is possible to construct the $6$-gon and the $8$-gon, but not the $7$-gon and not the $9$-gon. So I suggest you invite another (single!) friend to your pizza.

Exercise: If $2^k+1$ is prime then $k$ is a power of two.

Solution 3:

You're looking for constructible angles; see this link:http://en.wikipedia.org/wiki/Compass-and-straightedge_construction

The only numbers of pizza slices you can create are powers of two times distinct Fermat primes (so 8 times 3 times 17 would work).