Compass & Straightedge: What Two Tools Can and Cannot Build

Two tools, an unmarked ruler and a collapsing compass

Compass-and-straightedge construction is geometry stripped to its bare mechanism: a straightedge that draws the infinite line through two known points, and a compass that draws the circle centred at a known point through another known point. No ruler markings, no protractor, no measuring — every new point in the figure has to arise as an intersection of two lines, two circles, or a line and a circle. Euclid built the first three books of the Elements almost entirely from these two moves, and the discipline of using nothing else is not a historical accident: it is a precise model of what can be built from rational operations plus square roots, which is exactly why some constructions — trisecting an arbitrary angle, squaring the circle, doubling the cube — are provably impossible with this toolkit no matter how cleverly you try.

The perpendicular bisector, in five moves

Every serious construction reduces to a handful of primitive moves repeated and combined. The perpendicular bisector of a segment AB is the cleanest example: draw a circle centred at A through B, draw a circle centred at B through A (same radius, so they are congruent), and the two circles cross at two points. The line through those two crossing points is exactly the perpendicular bisector — it is equidistant from A and B by construction, because every point on either circle is a fixed distance from its centre.

1. circle(center=A, radius=AB)
2. circle(center=B, radius=AB)
3. P, Q = intersections of circle 1 and circle 2
4. line(P, Q)          <- perpendicular bisector of AB, midpoint = line ∩ AB

The angle bisector of angle ABC uses the same trick one level up: strike an arc centred at the vertex B that crosses both rays at equal distance, then bisect the segment joining those two crossing points exactly as above. Every classical construction — dropping a perpendicular from a point, copying an angle, erecting a tangent to a circle from an external point — decomposes into this same small vocabulary of equal-radius arcs and their intersections.

Building a regular hexagon for free

The regular hexagon is the construction that makes beginners fall in love with the compass, because it needs almost no cleverness: open the compass to any radius r, draw a circle, then, without changing the radius, step around the circumference marking off six arcs. They close up perfectly on the sixth step. The reason is a coincidence with real teeth — a chord equal to the radius subtends a 60° arc, since the triangle formed by the two radii and that chord is equilateral (two sides are radii, the third equals the radius by construction, so all three angles are 60°). Six copies of 60° tile the 360° circle exactly, which is why this construction, alone among the regular polygons, needs no auxiliary points at all.

Richmond's pentagon: the first genuinely hard case

The regular pentagon is where compass-and-straightedge stops being a party trick and starts being algebra. Its vertex angle needs cos 36° = (1+√5)/4, and √5 is a length you cannot pull out of a circle the way you pull a chord — you have to build it as the hypotenuse of a right triangle with legs 1 and 2, via the Pythagorean relation. Herbert Richmond's 1893 construction does exactly that: it bisects a radius, uses the bisected point as the centre of an auxiliary circle of radius equal to the distance to a point already on the main circle (which is length-√5⁄2 away by Pythagoras), and that one buried square root is what makes cos 36° constructible and cos(360°/7) — for the regular heptagon — not.

constructible with compass & straightedge  <=>  the required lengths lie in a tower of
field extensions each of degree 2 over the rationals (started at 1, closed under +,-,*,/,sqrt)

regular n-gon constructible  <=>  n = 2^k * p1 * p2 * ... (distinct Fermat primes 3,5,17,257,65537)
=> 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20 ... YES     7, 9, 11, 13, 14, 18 ... NO

Why some things can never be built

Every point you can reach with straightedge and compass, starting from two given points, has coordinates that live in a tower of fields each obtained from the previous one by adjoining a single square root — this is a theorem, not folklore, proved rigorously once Descartes turned geometry into algebra and Wantzel formalised the field-theory argument in 1837. Angle trisection needs the real root of a genuinely cubic equation, cos(θ/3) in terms of cos θ, and a generic cubic's root does not live in any tower of square-root extensions — it needs a cube root, which no straightedge-and-compass sequence can produce. Squaring the circle needs constructing a length of √π, and π is transcendental (Lindemann, 1882), so it is not even the root of any polynomial with rational coefficients, let alone one reachable by iterated square roots. These are not failures of ingenuity; they are the same statement Gauss made about the 17-gon in reverse — the tools define an exact algebraic sandbox, and either your target length lives inside it or it provably does not.

Frequently asked questions

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