Prouhet-Thue-Morse constant
Encyclopedia
In mathematics
, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue
, and Marston Morse
, is the number—denoted by
—whose binary expansion .01101001100101101001011001101001... is given by the Thue–Morse sequence
. That is,
where
is the ith element of the Prouhet–Thue–Morse sequence.
The generating series for the
is given by
and can be expressed as
This is the product of Frobenius polynomials, and thus generalizes to arbitrary fields
.
The Prouhet–Thue–Morse constant was shown to be transcendental
by Kurt Mahler
in 1929.
at the end of the sequence of western bulbs of the Mandelbrot set
. This property is tied to the nature of period doubling in the Mandelbrot set.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue
Axel Thue
Axel Thue was a Norwegian mathematician, known for highly original work in diophantine approximation, and combinatorics....
, and Marston Morse
Marston Morse
Harold Calvin Marston Morse was an American mathematician best known for his work on the calculus of variations in the large, a subject where he introduced the technique of differential topology now known as Morse theory...
, is the number—denoted by
—whose binary expansion .01101001100101101001011001101001... is given by the Thue–Morse sequenceThue–Morse sequence
In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is a binary sequence that begins:Any other ordered pair of symbols may be used instead of 0 and 1; the logical structure of the Thue–Morse sequence does not depend on the symbols that are used to represent it.- Direct...
. That is,
where
is the ith element of the Prouhet–Thue–Morse sequence.The generating series for the
is given byand can be expressed as
This is the product of Frobenius polynomials, and thus generalizes to arbitrary fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
.
The Prouhet–Thue–Morse constant was shown to be transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
by Kurt Mahler
Kurt Mahler
Kurt Mahler was a mathematician and Fellow of the Royal Society.He was a student at the universities in Frankfurt and Göttingen, graduating with a Ph.D...
in 1929.
Applications
The Prouhet–Thue–Morse constant occurs as the angle of the Douady–Hubbard rayExternal ray
An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.This curve is only sometimes a half-line but is called ray because it is image of ray....
at the end of the sequence of western bulbs of the Mandelbrot set
Mandelbrot set
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
. This property is tied to the nature of period doubling in the Mandelbrot set.
External links
- Entry in the On-Line Encyclopedia of Integer SequencesOn-Line Encyclopedia of Integer SequencesThe On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...
- The ubiquitous Prouhet-Thue-Morse sequence, John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history
- PlanetMath entry