Compass-and-straightedge constructions

What just happened?

The compass and straightedge (unmarked) are the classical tools of Greek geometry — but not everything can be constructed with them. Some famous constructions are feasible, while others — despite attempts spanning millennia — turned out to be impossible, which was only proved in the 19th century.

Which lengths are constructible?

Mathematically: the lengths constructible with compass and straightedge are exactly those obtainable from a unit segment by addition, subtraction, multiplication, division and square roots:

So the feasible ones include: dividing a segment into any number of equal parts (Thales’ theorem), constructing $\sqrt{n}$ (as a geometric mean), bisecting an angle, and regular polygons with a suitable number of sides.

Gauss’s breakthrough (1796):

The young Carl Gauss proved that the regular 17-gon is constructible — unknown since antiquity. In general a regular $n$-gon is constructible when $n$ is a product of a power of two and distinct Fermat primes. Gauss thus gave the full condition for the constructibility of regular polygons.

What was proved to be IMPOSSIBLE:

Some constructions require numbers that cannot be obtained with square roots alone — e.g. $\sqrt[3]{2}$ requires a cube root. That is why the following are impossible:

• Trisecting (dividing into 3 equal parts) an arbitrary angle — Pierre Wantzel, 1837
• Doubling the cube (constructing the side of a cube of double volume, i.e. $\sqrt[3]{2}$) — Pierre Wantzel, 1837
• Squaring the circle — Ferdinand Lindemann, 1882

The invariant:

Only numbers obtainable from one by arithmetic operations ($+,-,\times,\div$) and square roots ($\sqrt{\ }$) are constructible with compass and straightedge — not, for example, a cube root.

Why this matters for the exam:

This story shows the power of algebra: purely geometric questions (“can it be drawn?”) are settled by number theory and the theory of algebraic fields, turning “we cannot” into “it has been proved impossible”. Remember the criterion — only arithmetic operations and the square root are available.

Remember: with compass and straightedge you can construct what arises from 1 by $+,-,\times,\div$ and $\sqrt{\ }$. Dividing a segment, $\sqrt{n}$ and the 17-gon — yes; trisecting an angle and doubling the cube ($\sqrt[3]{2}$) — proved impossible.