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The Thue–Morse Constant

October 25, 2013

The most useful, and probably the most beloved, mathematical constant is pi (π). Pi is even celebrated as a holiday, Pi Day, March 14, since 3-1-4 are the three most significant decimal digits of pi.[1] As I wrote in a previous article (The Circle Ratio, April 12, 2013), pi is sometimes called Archimedes' constant, since the Greek mathematician, Archimedes, produced a good estimate of its value by inscribing and circumscribing regular polygons in and around a circle.

Archimedes' bath

Archimedes is famous also for his Eureka! moment. While in a bath, he realized that the volume of water displaced by a submerged object was equal to the volume of that object.

(Colored woodcut on paper, 1547, by Johann Petrejus, from the Deutsche Fotothek collection, via Wikimedia Commons.)

Wikipedia has a list of common mathematical constants. Without counting reciprocals, there are still quite a few constants with a value less than one. Perhaps some number theorist will one day prove that the number of mathematical constants less than one is equal to the number greater than one. If we allow reciprocals, that would be a certainty.

Here are five of the more easily described mathematical constants with a value less than one:

• 0.26149 72128 47642... Meissel–Mertens constant (M1). This is the limit of the difference between the harmonic series summed over the primes, only, and the natural logarithm of the natural logarithm. That last part always reminds me of Boss Hogg's percentage cut to Sheriff Rosco P. Coltrane, "ten percent of ten percent," in the Dukes of Hazzard television series.

• 0.56714 32904 09783..., the Omega constant (Ω). The Omega constant is defined by the equation, ΩeΩ = 1.

• 0.57721 56649 01532..., the Euler–Mascheroni constant (γ), is the limit of the difference between the harmonic series and the natural logarithm, so it's related to the Meissel–Mertens constant. It's larger, since the harmonic series sums the reciprocals of all positive integers, and not just the prime numbers.

• 0.66016 18158 46869..., the Twin prime constant (C2), is defined as the product of the following for p being a prime number greater than two:

• 0.87058 83800 (approximately), Brun's constant (B4), is the sum of the reciprocals of the twin primes.

As you've guessed by now, prime numbers are a recurrent theme in number theory. When you search arXiv for mathematics papers whose titles contain "prime" or "primes," you get this message: "Your query resulted in too many hits, only 1000 hits are being displayed."

The above constants are legitimate mathematical constants, since they represent solutions to formal conjectures. It's easy to manufacture a mathematical constant less than one by placing a decimal point in front of one of the many integer sequences in the On-Line Encyclopedia of Integer Sequences (OEIS).

The Champernowne constant in base 10,
0.12345 67891 01112 13141 51617...,
is built in such a way. If you haven't yet guessed it, this constant is formed by placing a decimal point in front of the positive integers. The individual decimal digits forming the Champernowne constant are sequence A033307 in the OEIS. Likewise, there's a Champernowne constant in base-2,
0.11011 10010 11101...
The Thue-Morse sequence (OEIS A010060) is especially understandable to computer scientists, since it involves the two's complement of numbers. Two ways in which this constant can be defined are as follow:
1) Start with a string that's just the single character 0, and repeatedly apply the replacement rules 0 -> 01 and 1 -> 10 to the characters of the string.

2) Start with a string that's just the single character 0, and repeatedly append the binary complement of the string to itself; that is, repeatedly concatenate copies of the string in which all ones have become zeros, and all zeros have become ones.
Not surprisingly, there's a lot of symmetry in such a number when it's viewed in chunks that are powers of two in length, as shown in the following figure. This figure was generated from a file created by my own program for generating the Thue-Morse sequence. The source code for the program can be found here.

Graphical representation of the Thue-Morse sequence

A graphical representation of the Thue-Morse sequence in which ones are represented as blue boxes in a 32x32 square.

The numbers were calculated by the author's program and mapped using a word processing program.

(Illustration by the author using LibreOffice)

The Thue–Morse sequence can be manufactured into a Thue-Morse Constant with value
0.01101 00110 01011... (base-2)
0.41245 40336 40107... (base-10)
The Thue-Morse sequence might not be mathematically important, but it illustrates how a very simple rule can generate some interesting results. Another example is the logistic map.


  1. The US House of Representatives, in its 2009-2010 session (the 111th Congress), passed a resolution supporting March 14 as Pi Day on March 12, 2009. The text of the resolution contains the following:
    "...Whereas Pi can be approximated as 3.14, and thus March 14, 2009, is an appropriate day for ''National Pi Day'': Now, therefore, be it Resolved, That the House of Representatives-(1) supports the designation of a ''Pi Day'' and its celebration around the world..."
    The full text of the resolution (H.Res. 224) can be found here.

  2. Eric W. Weisstein, "Thue-Morse Constant," From MathWorld--A Wolfram Web Resource

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