Lill's method

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In mathematics, Lill's method is a visual method of finding the real roots of polynomials of any degree.[1] It was developed by Austrian engineer Eduard Lill in 1867.[2] A later paper by Lill dealt with the problem of imaginary roots.[3]

Lill's method involves expressing the coefficients of a polynomial, in right angle paths from the origin, right or left depending on the sign of the coefficient, to a terminus, then finding a path from the start to the terminus changing direction these lines.

Description of the method[edit]

Solution of the cubic 4x3+2x2−2x−1 using Lill's method. Solutions are −1/2, −1/√2, 1/√2.

To employ the method a diagram is drawn starting at the origin. A line is drawn rightwards by the space of the first coefficient (so that with a negative coefficient the line will end left of the origin). From the end of the first line another line is drawn upwards the space of the second coefficent, then left the space of the third, and down the space of the fourth. The direction turns counterclockwise 90° for each positive coefficient and negative coefficients are drawn in the opposite direction. The process continues for every coefficient of the polynomial including zeroes. This final point reached is the terminus.

A line is then launched from the origin at some angle θ, reflected off of the line segments at right angle paths, and refracted through the line through each segment (including a line for the zero coefficients) when the path does not hit the line segment on that line.[4] Choosing θ so that the path lands on the terminus, the negative of the tangent of θ is a root of this polynomial. For every real zero of the polynomial there will be one unique path and angle that will land on the terminus. A quadratic with two real roots, for example, will have exactly two angles that satisfy the above conditions.

The construction in effect evaluates the polynomial according to Horner's method. For the polynomial a_n x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+ \cdots the values of a_n x, (a_n x+a_{n-1})x, ((a_n x+a_{n-1})x+a_{n-2})x,\ \dots are successively generated. A solution line giving a root is similar to the Lill's construction for the polynomial with that root removed.

In 1936 Margharita P. Beloch showed how Lill's method could be adapted to solve cubic equations using paper folding.[5] If simultaneous folds are allowed then any Nth degree equation with a real root can be solved using N-2 simultaneous folds.[6]


  1. ^ Dan Kalman (2009). Uncommon Mathematical Excursions: Polynomia and Related Realms. AMS. pp. 13–22. ISBN 978-0-88385-341-2. 
  2. ^ M. E. Lill (1867). "Résolution graphique des équations numériques de tous degrés à une seule inconnue, et description d'un instrument inventé dans ce but". Nouvelles Annales de Mathématiques. 2 6: 359–362. 
  3. ^ M. E. Lill (1868). "Résolution graphique des équations algébriques qui ont des racines imaginaires". Nouvelles Annales de Mathématiques. 2 7: 363–367. 
  4. ^ Phillips Verner Bradford, Sc.D.. Visualizing solutions to n-th degree algebraic equations using right-angle geometric paths.
  5. ^ Thomas C. Hull (April 2011). "Solving Cubics With Creases: The Work of Beloch and Lill". American Mathematical Monthly: 307–315. doi:10.4169/amer.math.monthly.118.04.307. 
  6. ^ Roger C. Alperin; Robert J. Lang (2009). "One-, Two-, and Multi-Fold Origami Axioms". 4OSME (A K Peters). 

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