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.
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$\begingroup$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.
$\endgroup$ 5 $\begingroup$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.
$\endgroup$ 6 $\begingroup$You're looking for constructible angles; see this link:
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).
$\endgroup$ $\begingroup$Divide it in half, then in half again, and one more time. You now have eight slices.
One slice is the geometer's fee; the remaining seven slices belong to the customer.
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