Regular Heptagon is Smallest with no Compass and Straightedge Construction

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Theorem

The regular heptagon is the smallest regular polygon (smallest in the sense of having fewest sides) that cannot be constructed using a compass and straightedge construction.


Proof 1

The fact that it is impossible to construct a regular heptagon using a compass and straightedge construction is demonstrated in Compass and Straightedge Construction for Regular Heptagon does not exist.

From Construction of Equilateral Triangle, an equilateral triangle can be constructed.

From Inscribing Square in Circle, for example, a square can be constructed.

From Inscribing Regular Pentagon in Circle, a regular pentagon can be constructed.

From Inscribing Regular Hexagon in Circle, a regular hexagon can be constructed.

Thus all regular polygons with $3$ to $6$ sides can be so constructed, but not one with $7$ sides.

$\blacksquare$


Proof 2


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A theorem states that: "The n-gon is constructible by compass and straightedge construction if and only if $n = 2^kn_0$, with $k \in \mathbb{Z}_{\geq0}$ and $n_0$ the product of any number of distinct Fermat primes."

Note that 3 and 5 are both Fermat primes, so their respective n-gon are both constructible; and $4$ is just a power of $2$, so the 4-gon also constructible.

Lastly, since 6 is the product of 2 and 3, the 6-gon is constructible as well. Hence, we remain with 7, which does not satisfy the theorem's criteria; and thus the 7-gon is the smallest regular polygon that cannot be constructed using a compass and straightedge construction.

$\blacksquare$


Sources

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