Concept of Construction with ruler and compass, Models of Computation

Construction with ruler and compass:

Plato (427-347 B.C.) considered ruler and compass the merely appropriate tools for geometric construction.

A) Primitive objects: straight line (finite or unbounded), point and circle.

B) Primitive operations:

•    Given 2 points P, Q construct a line passing via P and Q
•    Given 2 points P, Q construct a circle with center P which passes via Q
•    Given 2 lines L, L’and two circles C, C’ or a line L and a circle C,
•    Make a new point P at any intersection of such a pair.

C) Compound object: Any structure built from the primitive objects by following a series of primitive operations.

Illustrations:

A) Given a line L and a point P on L, then construct a line L’ passing via P and orthogonal to the L.

B) Construct a pair of Lines L, L’ which defines angles of 90º, 60º, 45º and 30º respectively.

C) The segment is a triple (L, P, Q) with points P and Q on L. The segment of length 1 might be given or it might be randomly constructed to define the unit of measurement.

The angles and lengths which can be constructed:

A) Beginning with given angles, construct additional angles through bisection, subtraction and addition.

B) Beginning with a segment of unit length, construct the additional lengths through bisection, addition, subtraction, division, multiplication and square root.

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Therefore, construction by ruler and compass is reduced the question whether certain desired quantity can be stated in terms of rational operations and square roots.

Theorem: The proposed construction is probable by ruler and compass if the numbers that define analytically the preferred geometric elements can be derived from such defining the given elements by the finite number of rational operations and extractions of the real square roots.

Possible construction: Equilateral polygon of 17 sides, ‘17-gon’ (that is, Carl Friedrich Gauss 1777-1855)

Impossible constructions: Equilateral heptagon, ‘7-gon’ (Gauss). Doubling the unit cube (simplify x3 = 2).

Theorem: There is no algorithm to trisect the arbitrary angle with the ruler and compass. And yet, the Archimedes (~287 B.C. - ~ 212 B.C) found a process to trisect the arbitrary angle:

Given any angle AOB = 3x in a unit circle. The ruler CB has a mark D at distance CD = radius = r = 1. Slide C all along the x-axis till D lies on circle. Then, angle ACB = x.

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Msg: Minor modifications in a model of computation might have drastic consequences- precise definition required!

Example: The quadratic equation x2 + bx + c = 0 consists of roots x1, x2 = (-b ± sqrt(b2 - 4c))/2. Now prove that in the ruler and compass construction illustrated below, the segments x1 and x2 are solutions of this equation.

135_quadratic equation.jpg

Solution: Replace xc = - b/2 and yc = (1 + c)/2 in the circle equation (x - xc)2 + (y - yc)2, set y = 0  to get the quadratic equation  x2 + bx + c = 0 whose roots are the preferred values  x1 and x2 .

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