1. MATH CLUB TALK
3/3/2019

2. Chapter 1
The Calkin-Wilf Tree

3. “Every child knows the rationals are countable…”
-(Possibly) Erdos

4. Calkin-Wilf Ordering (2000, Neil Calkin and Herbert S. Wilf)
But there is a listing that is even more elegant and systematic, and which contains no duplicates — found only quite recently by Neil Calkin and Herbert Wilf.
Here the denominator of the n-th rational number equals the numerator of the (n + 1)-st number.

5. Calkin-Wilf Tree

6. 4 Properties
Give inductive proofs using the height of a rational: h(a/b) = a+b

7. 2 Questions

8. Recursive rule

9. Stern’s diatomic array
“Pascal with memory”
-Andreas Kauf
Diatomic - consisting of 2 atoms

11. Stern’s Diatomic Series (1858, Moritz Abraham Stern)
This sequence has first been studied by a German mathematician, Moritz Abraham Stern, in a paper from 1858
Moritz Abraham Stern (29 June 1807 – 30 January 1894) was a German mathematician. Stern became Ordinarius (full professor) at Göttingen University in 1858, succeeding Carl Friedrich Gauss. Stern was the first Jewish full professor at a German university, who attained the position without changing his Jewish religion.[1]
As a professor, Stern taught Gauss's student Bernhard Riemann. Stern was very helpful to Ferdinand Eisenstein in formulating a proof of the quadratic reciprocity theorem. Stern was interested in primes that cannot be expressed as the sum of a prime and twice a square (now known as Stern primes).
He is known for formulating Stern's diatomic series[2]
1, 1, 2, 1, 3, 2, 3, 1, 4, … (sequence A in the OEIS)
that counts the number of ways to write a number as a sum of powers of two with no power used more than twice.
He is also known for the Stern–Brocot tree which he wrote about in 1858 and which Brocot independently discovered in 1861.
Stern prove many fascinating properties about the sequence, which I won't mention here

12. Hyper-binary partitions

13. Shift in perspective

16. Successor Formula (Moshe Newman, 2003)
Attributed to Moshe Newman, in a paper by Knuth

17. Calculation

18. Harmonices Mundi (1619, Johannes Kepler)

19. Chapter 2
The Stern-Brocot Tree, Tree of Matrices & The Topograph of Conway and Fung

20. Names Moritz Abraham Stern (1800s) John Farey Sr. ()
Achille Brocot (1800s)
He is known for his discovery (contemporaneously with, but independently of, German number theorist Moritz Stern) of the Stern–Brocot tree, a mathematical structure useful in approximating real numbers by rational numbers; this sort of approximation is an important part of the design of gear ratios for clocks.
The latter name is in honour of two independent descriptions of related ideas in the mid 1800’s by Stern [15] and Brocot [4]. Brocot was a french clockmaker who created an array of fractions for the purpose of designing clockwork gears . Stern studied an array of integers, which can be used to generate both the Stern-Brocot tree and the Calkin-Wilf tree (it has been quite reasonably suggested the latter tree be called the Eisenstein-Stern tree, but this name is not prevalent [2]). The name ‘Farey tree’ comes from its relationship to Farey sequences (which were themselves likely invented by Charles Haros [8]). For more on the muddy historical waters, and the trees themselves, see

21. Stern-Brocot Tree

22. Comparison
One tree can be obtained from the other by performing a bit-reversal permutation on the numbers at each level of the trees.

24. Tree of Matrices

25. Mobius Transformations, Projective Linear Group

26. Rephrase the theorem

27. Topograph of Conway and Fung
The proof proceeds by labelling the topograph two ways: first, to create the Farey tree, and second, to create the matrix tree. Comparing the two labellings generates the rule given in the Theorem.

28. Flow -> Stern-Brocot Tree
Each vertex has one incoming edge; with respect to this direction, there’s a left, right and forward region. If we label a vertex with the region bounded by the two outgoing edges (i.e. moving the region labels up to the ‘peaks’ of their respective regions), we obtain the Farey tree

29. Back to matrices -> Tree of matrices

30. Chapter 3
Farey Diagram

31. Farey Diagram

32. Farey Series
We can build the set of rational numbers by starting with the integers and then inserting in succession all the halves, thirds, fourths, fifths, sixths, and so on. Let us look at what happens if we restrict to rational numbers between 0 and 1. Starting with 0 and 1 we first insert 1/2, then 1/3 and 2/3, then 1/4 and 3/4, skipping 2/4 which we already have, then inserting 1/5, 2/5, 3/5, and 4/5, then 1/6 and 5/6, etc. This process can be pictured as in the following diagram:
Each time a new number is inserted, it forms the third vertex of a triangle whose other two vertices are its two nearest neighbors among the numbers already listed, and if these two neighbors are a/b and c/d then the new vertex is exactly the mediant a+c b+d .

33. For a vertical line segment in the diagram whose lower endpoint is at the point a b , 0 on the x -axis, the upper endpoint is at the point a b , 1 b

34. References Katherine Stange (An Arborist's guide to the rationals)
Katherine Stange (An Arborist's guide to the rationals)
Melvyn B Nathanson (A forest of linear fractional transformations)
Calkin-Wilf (Recounting the rationals)
Allen Hatcher (Topology of numbers)
(Proofs from THE BOOK)

35. Extra References
Lionel Ponton, Two trees enumerating the positive rationals ((ternary + quinary))
Donald Knuth, Recounting the Rationals, continued
Thomas Garrity, A Multidimensional Continued Fraction
Generalization of Stern’s Diatomic Sequence
Timothy B. Flowers and Shannon R Lockard, Identifying an m-ary Partition Identity Through an m-ary tree