Compass & Straightedge Explorer

Compass and straightedge constructions are geometric drawings created using only two tools: an unmarked straightedge (for drawing lines through two points) and a compass (for drawing circles centered at one point and passing through another).

These constraints were formalized in Euclid's Elements around 300 BCE. The five postulates at the start of that work describe exactly what a straightedge and compass can do. By limiting ourselves to these two operations, every construction serves as a proof that a geometric relationship follows from the axioms of Euclidean geometry alone.

For example, bisecting a line segment $\overline{AB}$ with compass arcs proves that the midpoint and perpendicular bisector exist without relying on measurement or arithmetic.

Three famous problems from antiquity were eventually proved impossible with compass and straightedge alone:

• Trisecting an arbitrary angle. Splitting any given angle into three equal parts.
• Doubling a cube. Constructing a cube with exactly twice the volume of a given cube, which requires constructing a segment of length $\sqrt[3]{2}$.
• Squaring a circle. Constructing a square with the same area as a given circle, which requires constructing $\sqrt{\pi}$.

The proofs of impossibility, completed in the 19th century by Wantzel and Lindemann, rest on a key algebraic insight: the set of constructible numbers consists exactly of those reachable from the rationals by addition, subtraction, multiplication, division, and taking square roots. Since $\sqrt[3]{2}$ and $\pi$ are not constructible, neither are the constructions that require them.

Most compass and straightedge constructions combine a small set of recurring patterns. Keep these in mind as you build:

• Equal radii create equilateral triangles. Drawing two circles of the same radius from the endpoints of a segment gives you the vertices of an equilateral triangle, which is the basis for 60-degree angles.
• Arc intersections find equidistant points. Two arcs of equal radius from different centers meet at points equidistant from both centers. This underlies perpendicular bisectors, angle bisectors, and most other constructions.
• Transferring lengths preserves measurement. Setting the compass to a known segment and marking it elsewhere lets you copy distances without an actual ruler.
• Perpendicular bisectors find midpoints. The perpendicular bisector of a segment always passes through its midpoint, giving you an exact halfway point for free.
• Work outward from what you know. Start each construction from the given elements and add one circle or line at a time, using only previously constructed points.

Euclid's Elements (c. 300 BCE) organized centuries of Greek geometric knowledge into 13 books. The first four books deal almost entirely with compass and straightedge constructions, and every proposition is either a construction or a theorem proved with the help of one.

Constructions served a deeper purpose than drawing: they proved existence. To show that an equilateral triangle exists on a given segment, Euclid constructed one. This constructive approach influenced mathematical rigor for over two thousand years.

In the 19th century, mathematicians translated construction problems into algebra. The constructible numbers (those you can reach from the rationals using $+, -, \times, \div$ and $\sqrt{\phantom{x}}$) form a field, and a length is constructible if and only if its minimal polynomial has degree that is a power of 2. This algebraic framework settled the three classical impossibility problems and connected ancient geometry to modern abstract algebra and Galois theory.