Carlyle circle

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

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 (2iπ/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.

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. This is the Carlyle circle for x² + x − 1 = 0. 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. These are the points p₁ and p₂ mentioned above.
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. The figure to the right 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.

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; for example, it requires the construction of the Carlyle circle for the solution of the quadratic equation x² + x − 2¹⁴ = 0.

History

According to Howard Eves (1911–2004), the mathematician John Leslie (1766–1832) described the geometric construction of roots of a quadratic equation with a circle in his book Elements of Geometry and noted that this idea was provided by his former student Thomas Carlyle (1795–1881). However while the description in Leslie's book contains an analogous circle construction, it was presented solely in elementary geometric terms without the notion of a Cartesian coordinate system or a quadratic function and its roots:

To divide a straight line, whether internally or externally, so that the rectangle under its segments shall be equivalent to a given rectangle.

— John Leslie, Elements of Geometry, prop. XVII, p. 176

In 1867 the Austrian engineer Eduard Lill published a graphical method to determine the roots of a polynomial (Lill's method). If it is applied on a quadratic function, then it yields the trapezoid figure from Carlyle's solution to Leslie's problem (see graphic) with one of its sides being the diameter of the Carlyle circle. In an article from 1925 G. A. Miller pointed out that a slight modification of Lill's method applied to a normed quadratic function yields a circle that allows the geometric construction of the roots of that function and gave the explicit modern definition of what was later to be called Carlyle circle.

Eves used the circle in the modern sense in one of the exercises of his book Introduction to the History of Mathematics (1953) and pointed out the connection to Leslie and Carlyle. Later publications started to adopt the names Carlyle circle, Carlyle method or Carlyle algorithm, though in German speaking countries the term Lill circle (Lill-Kreis) is used as well. DeTemple used in 1989 and 1991 Carlyle circles to devise compass-and-straightedge constructions for regular polygons, in particular the pentagon, the heptadecagon, the 257-gon and the 65537-gon. Ladislav Beran described in 1999 how the Carlyle circle can be used to construct the complex roots of a normed quadratic function.