Exploring, Investigating and Discovering in Mathematics pp Cite as.
Problems of geometric constructions using ruler and compass, or only ruler, form a very special class of problems which, in order to be solved, require not only a very good knowledge of basic results in geometry but also special skills and cleverness. Regretably in the last decades geometry has lost the important position that it had in the curriculum of both secondary and high schools in Romania.
This is unfortunate because, in order to be well assimilated, geometry requires the student to extensively exercise problem solving skills and invest time in understanding the fundamental truths that lie subtly hidden behind a geometric figure. Unable to display preview. Download preview PDF. Skip to main content. Advertisement Hide.
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Heuristics and Didactics of Exact Sciences Donetsk , no. How do you prove something impossible? There are many different ways, but this particular problem we carefully demarcate the limit of the possible, and show that to solve these problems you must transgress that limit. Using a ruler and compass, you can impose coordinates on the plane.
Draw two points, and draw the line through them. Call that the x -axis, and define the length between the two points to be one. One construction that you can do is draw perpendiculars, so draw a perpendicular to your x -axis, and call it your y -axis. We now have a Cartesian coordinate system on the plane. In ruler and compass construction, one starts with a line segment of length one. If one can construct a given point on the complex plane, then one says that the point is constructible.
This shows that the constructible points form a field , a subfield of the complex numbers. Moreover, one can show that the given a constructible length one can construct its complex conjugate and square root.
The only way to construct points is as the intersection of two lines, of a line and a circle, or of two circles. Using the equations for lines and circles, one can show that the points at which they intersect lie in a quadratic extension of the smallest field F containing two points on the line, the center of the circle, and the radius of the circle.
Since the field of constructible points is closed under square roots , it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients. By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions. In particular, any constructible point or length is an algebraic number.
The most famous of these problems, " squaring the circle ", involves constructing a square with the same area as a given circle using only ruler and compass. Only algebraic ratios can be constructed with ruler and compass alone.
The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason. Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and has been solved many times in antiquity.
Doubling the cube : using only ruler and compass, construct the side of a cube that has twice the volume of a cube with a given side. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. Angle trisection : using only ruler and compass, construct an angle that is one-third of a given arbitrary angle.
The minimal polynomial for x is a factor of this, but if it were not irreducible, then it would have a rational root which, by the rational root theorem , must be 1 or -1, which are clearly not roots. Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the more powerful but physically easy operations of paper folding, or origami.
Huzita's axioms types of folding operations can construct cubic extensions cube roots of given lengths, whereas ruler-and-compass can construct only quadratic extensions square roots. See Mathematics of origami. Some regular polygons e.