Compass and straightedge constructions | Wikipedia audio article
Channel: wikipedia tts
Date: 13/01/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Straightedge_and_compass_construction


00:02:16 1 Straightedge and compass tools
00:04:17 2 History
00:06:43 3 The basic constructions
00:08:25 4 Much used straightedge and compass constructions
00:09:11 5 Constructible points and lengths
00:09:22 5.1 Formal proof
00:11:10 6 Constructible angles
00:14:07 7 Straightedge and compass constructions as complex arithmetic
00:17:46 8 Impossible constructions
00:18:21 8.1 Squaring the circle
00:19:38 8.2 Doubling the cube
00:20:21 8.3 Angle trisection
00:20:51 9 Constructing regular polygons
00:21:06 10 Constructing a triangle from three given characteristic points or lengths
00:23:10 11 Distance to an ellipse
00:24:33 12 Constructing with only ruler or only compass
00:24:58 13 Extended constructions
00:25:51 13.1 Solid constructions
00:27:17 13.2 Angle trisection
00:30:18 13.3 Origami
00:30:56 13.4 Markable rulers
00:31:33 14 Computation of binary digits
00:33:32 15 See also
00:34:01 16 References



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SUMMARY
=======
Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.
The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it with only one edge. The compass is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with collapsing compass; see compass equivalence theorem.) More formally, the only permissible constructions are those granted by Euclid's first three postulates.
It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone.
The ancient Greek mathematicians first conceived straightedge and compass constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not.  Some of the most famous straightedge and compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields.
In spite of existing proofs of impossibility, some persist in trying to solve these problems. Many of these problems are easily solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone.
In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots.