Geometry Foundations and axiom systems Concept
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 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 or of the field of the old coordinates. This turns questions about constructions into questions about degrees of field extensions. It shows that doubling the cube, trisecting the angle and squaring the circle are impossible with these tools.
Where this leads
14 pages build on this one.
- Trisecting the angle
- Doubling the cube
- Squaring the circle
- Impossibility of doubling the cube
- Impossibility of trisecting the angle
- Constructible number
- The constructible numbers form a field closed under square roots
- The degree of a constructible number is a power of two
- Impossibility of squaring the circle
- Gauss–Wantzel theorem
- Poncelet–Steiner theorem
- Problem of Apollonius
- Mohr–Mascheroni theorem
- Impossibility of the classical construction problems
Further reading
- Web https://en.wikipedia.org/wiki/Straightedge_and_compass_construction, Straightedge and compass tools
- Web https://en.wikipedia.org/wiki/Straightedge_and_compass_construction, Constructible points
About this page
| Tag | 0DAECMZVRT |
|---|---|
| URI | m2://0DAECMZVRT |
| Aliases | ruler-and-compass-construction, compass-and-straightedge-construction |
| Kind | concept |
| Writing | W0 stub |
| MSC | 51M15 |
| Same as |
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What links here
Used by
- Trisecting the angle
- Doubling the cube
- Squaring the circle
- Constructible number
- Poncelet–Steiner theorem
- Problem of Apollonius
- Mohr–Mascheroni theorem