---
tag: 0DAECMZVRT
kind: concept
name: Straightedge and compass construction
aliases: [ruler-and-compass-construction, compass-and-straightedge-construction]
msc: [51M15]
uses: [0FMN2MN7TM]
sources:
  - url: https://en.wikipedia.org/wiki/Straightedge_and_compass_construction
    locator: Straightedge and compass tools
  - url: https://en.wikipedia.org/wiki/Straightedge_and_compass_construction
    locator: Constructible points
same_as:
  wikidata: Q115368
  wikipedia: Straightedge and compass construction
---

## Summary

A straightedge and compass construction draws points, lines and circles in the plane from given points with two ideal tools.
The straightedge draws the line through two constructed points, and it has no marks.
The compass draws the circle with a constructed centre through a constructed point.
New points are the intersections of the lines and circles drawn so far.
The [[constructible numbers|0FMN2MN7TM]] describe the points that these steps can reach.

## Remarks

Each step solves linear or quadratic equations.
So the coordinates of the new points lie in an extension of degree $1$ or $2$ of the field of the old coordinates.
This turns questions about constructions into questions about [[degrees of field extensions|0GS74FGFEK]].
It shows that [[doubling the cube|04SD3F4FCR]], [[trisecting the angle|00J3DCCEK5]] and [[squaring the circle|0BSXRER81B]] are impossible with these tools.

## Explanations

### L1 human

The ancient Greeks asked which figures can be drawn with an unmarked ruler and a compass alone.
Field theory gives the answer: a length can be drawn exactly when it comes from rational numbers by field operations and square roots.