![Lill_method_quartic_example](https://upload.wikimedia.org/wikipedia/commons/a/a4/Lill_method_quartic_example.svg)
In
mathematics, Lill's method is a visual method of finding the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of a univariate
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
of any
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
. It was developed by Austrian engineer
Eduard Lill in 1867. A later paper by Lill dealt with the problem of
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
roots.
[ ]
Lill's method involves drawing a path of straight line segments making
right angles
In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
, with lengths equal to the coefficients of the polynomial. The roots of the polynomial can then be found as the
slopes
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used ...
of other right-angle paths, also connecting the start to the terminus, but with vertices on the lines of the first path.
Description of the method
![LillsMethod](https://upload.wikimedia.org/wikipedia/commons/9/97/LillsMethod.svg)
To employ the method a diagram is drawn starting at the origin. A line segment is drawn rightwards by the magnitude of the first coefficient (the coefficient of the highest-power term) (so that with a negative coefficient the segment will end left of the origin). From the end of the first segment another segment is drawn upwards by the magnitude of the second coefficient, then left by the magnitude of the third, and down by the magnitude of the fourth, and so on. The sequence of directions (not turns) is always rightward, upward, leftward, downward, then repeating itself. Thus each turn is counterclockwise. The process continues for every coefficient of the polynomial including zeroes, with negative coefficients "walking backwards". The final point reached, at the end of the segment corresponding to the equation's constant term, is the terminus.
A line is then launched from the origin at some angle , reflected off of each line segment at a right angle (not necessarily the "natural" angle of reflection), and
refracted
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
at a right angle through the line through each segment (including a line for the zero coefficients) when the angled path does not hit the line segment on that line.
The vertical and horizontal lines are reflected off or refracted through in the following sequence: the line containing the segment corresponding to the coefficient of
then of
etc. Choosing so that the path lands on the terminus, the negative of the tangent of is a root of this polynomial. For every real zero of the polynomial there will be one unique initial angle and path that will land on the terminus. A quadratic with two real roots, for example, will have exactly two angles that satisfy the above conditions.
For complex roots, one also needs to find a series of
similar triangles, but with the vertices of the root path displaced from the polynomial path by a distance equal to the imaginary part of the root. In this case the root path will not be rectangular.
Explanation
The construction in effect evaluates the polynomial according to
Horner's method
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horn ...
. For the polynomial
the values of
,
,
are successively generated as distances between the vertices of the polynomial and root paths. For a root of the polynomial the final value is zero, so the last vertex coincides with the polynomial path terminus.
Additional properties
A solution line giving a root is similar to the Lill's construction for the polynomial with that root removed, because the visual construction is analogous to the
synthetic division
In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division.
It is mostly taught for division by linear monic polynomials (known as the Ruffini ...
of the polynomial by a linear (root)
monic (
Ruffini's rule
In mathematics, Ruffini's rule is a method for computation of the Euclidean division of a polynomial by a Binomial (polynomial), binomial of the form ''x – r''. It was described by Paolo Ruffini (mathematician), Paolo Ruffini in 1804. The rule i ...
).
From the symmetry of the diagram, it can easily be seen that the roots of the
reversed polynomial are the
reciprocals of the original roots.
The construction can also be done using clockwise turns instead of counterclockwise turns. When a path is interpreted using the other convention, it corresponds to the mirrored polynomial (every odd coefficient sign changed) and the roots are negated.
When the right-angle path is traversed in the other direction but the same direction convention, it corresponds to the reversed mirrored polynomial and the roots are the negative reciprocals of the original roots.
Finding quadratic roots using Thales's theorem
![Lill_method_quadratic_example](https://upload.wikimedia.org/wikipedia/commons/0/02/Lill_method_quadratic_example.svg)
Lill's method can be used with
Thales's theorem
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
to find the real roots of a quadratic polynomial.
In this example with 3''x''
2+5''x''−2, the polynomial's line segments are first drawn in black, as above. A circle is drawn with the straight line segment joining the start and end points forming a diameter.
According to Thales's theorem, the triangle containing these points and any other point on the circle is a
right triangle
A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
. Intersects of this circle with the middle segment of Lill's method, extended if needed, thus define the two angled paths in Lill's method, coloured blue and red.
The negative of the gradients of their first segments, ''m'', yield the real roots 1/3 and −2.
Finding roots using paper folding
![Lill_method_folding_example](https://upload.wikimedia.org/wikipedia/commons/a/a7/Lill_method_folding_example.svg)
In 1936
Margherita Piazzola Beloch
Margherita Beloch Piazzolla (12 July 1879, in Frascati – 28 September 1976, in Rome) was an Italian mathematician who worked in algebraic geometry, algebraic topology and photogrammetry.
Biography
Beloch was the daughter of the German hist ...
showed how Lill's method could be adapted to solve cubic equations using
paper folding
) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a f ...
. If simultaneous folds are allowed then any ''n''th degree equation with a real root can be solved using ''n''–2 simultaneous folds.
In this example with 3x
3+2x
2−7x+2, the polynomial's line segments are first drawn on a sheet of paper (black). Lines passing through reflections of the start and end points in the second and third segments, respectively (faint circle and square), and parallel to the them (grey lines) are drawn.
For each root, the paper is folded until the start point (black circle) and end point (black square) are reflected onto these lines. The axis of reflection (dash-dot line) defines the angled path corresponding to the root (blue, purple and red). The negative of the gradients of their first segments, ''m'', yield the real roots 1/3, 1 and −2.
See also
*
Carlyle circle
In mathematics, a Carlyle circle (named for Thomas Carlyle) 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 ...
, which is based on a slightly modified version of Lill's method for a normed quadratic.
References
External links
Animation for Lill's MethodMathologer video: "Solving equations by shooting turtles with lasers"
Geometry
Paper folding
Polynomials
{{Mathematics of paper folding