Carlyle circle

In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis.[1] The idea of using such a circle to solve a quadratic equation is attributed to Thomas Carlyle (1795–1881).[2] Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.

Definition

Carlyle circle of the quadratic equation x2 − sx + p = 0.

Given the quadratic equation

x2 − sx + p = 0

the circle in the coordinate plane having the line segment joining the points A(0, 1) and B(s, p) as a diameter is called the Carlyle circle of the quadratic equation.

Defining property

The defining property of the Carlyle circle can be established thus: the equation of the circle having the line segment AB as diameter is

x(x − s) + (y − 1)(y − p) = 0.

The abscissas of the points where the circle intersects the x-axis are the roots of the equation (obtained by setting y = 0 in the equation of the circle)

x2 − sx + p = 0.

Construction of regular polygons

Construction of regular pentagon using Carlyle circles
Construction of a regular heptadecagon using Carlyle circles
Construction of a regular 257-gon using Carlyle circles

Regular pentagon

The problem of constructing a regular pentagon is equivalent to the problem of constructing the roots of the equation

z5 − 1 = 0.

One root of this equation is z0 = 1 which corresponds to the point P0(1, 0). Removing the factor corresponding to this root, the other roots turn out to be roots of the equation

z4 + z3 + z2 + z + 1 = 0.

These roots can be represented in the form ω, ω2, ω3, ω4 where ω = exp(2πi/5). Let these correspond to the points P1, P2, P3, P4. Letting

p1 = ω + ω4, p2 = ω2 + ω3

we have

p1 + p2 = −1, p1p2 = −1. (These can be quickly shown to be true by direct substitution into the quartic above and noting that ω6 = ω, and ω7 = ω2.)

So p1 and p2 are the roots of the quadratic equation

x2 + x − 1 = 0.

The Carlyle circle associated with this quadratic has a diameter with endpoints at (0, 1) and (-1, -1) and center at (-1/2, 0). Carlyle circles are used to construct p1 and p2. From the definitions of p1 and p2 it also follows that

p1 = 2 cos (2π/5), p2 = 2 cos (4π/5).

These are then used to construct the points P1, P2, P3, P4.

This detailed procedure involving Carlyle circles for the construction of regular pentagons is given below.[2]

  1. Draw a circle in which to inscribe the pentagon and mark the center point O.
  2. Draw a horizontal line through the center of the circle. Mark one intersection with the circle as point B.
  3. Construct a vertical line through the center. Mark one intersection with the circle as point A.
  4. Construct the point M as the midpoint of O and B.
  5. Draw a circle centered at M through the point A. Mark its intersection with the horizontal line (inside the original circle) as the point W and its intersection outside the circle as the point V.
  6. Draw a circle of radius OA and center W. It intersects the original circle at two of the vertices of the pentagon.
  7. Draw a circle of radius OA and center V. It intersects the original circle at two of the vertices of the pentagon.
  8. The fifth vertex is the intersection of the horizontal axis with the original circle.

Regular heptadecagon

There is a similar method involving Carlyle circles to construct regular heptadecagons.[2] The attached figure illustrates the procedure.

Regular 257-gon

To construct a regular 257-gon using Carlyle circles, as many as 24 Carlyle circles are to be constructed. One of these is the circle to solve the quadratic equation x2 + x − 64 = 0.[2]

Regular 65537-gon

There is a procedure involving Carlyle circles for the construction of a regular 65537-gon. However there are practical problems for the implementation of the procedure, as, for example, it requires the construction of the Carlyle circle for the solution of the quadratic equation x2 + x − 214 = 0.[2]

References

  1. ↑ Weisstein, Eric W. "Carlyle Circle". From MathWorld—A Wolfram Web Resource. Retrieved 21 May 2013.
  2. ↑ 2.0 2.1 2.2 2.3 2.4 DeTemple, Duane W. (Feb 1991). "Carlyle circles and Lemoine simplicity of polygon constructions". The American Mathematical Monthly 98 (2): 97–208. doi:10.2307/2323939. Retrieved 6 November 2011.