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What is the Kolakoski Series?

OlimpiAkademi
8 min readNov 30, 2022

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References: Numberphile

Imagine a sequence of numbers consisting only of 1s and 2s: For example 1 1 1 2 1 2 2 1 1 2 2… it is a series that goes… Or 2 2 1 2 2 2 2 1 1 1 1 2 1 like… Now, create such a series, that is, write these 1s and 2s in such an order that when you write the number of consecutive 1s and the number of 2s as a separate series, we get exactly the same series as the first series. Is it complicated?

Let’s take the first series, which we wrote randomly: 1 1 1 2 1 2 2 1 1 2 2…

Let’s go from left to right in this series and write down as a separate series how many times 1s and 2s overlap:

  • There are 3 from 1, which means that the first number of our new series is 3.
  • Immediately after that comes 1 out of 2, which means that the second number of our new series is 2.
  • Then comes 1 out of 1, which means that our third number is 1.
  • Then comes 2 out of 2, which means that our fourth number is 2.
  • Then from 1 comes 2, which means that our fifth number is 2.
  • Finally, there are 2 out of 2, which means that our sixth number is 2.
    So our series is: 3 2 1 2 2 2… he goes.

As you can see, this series (3 2 1 2 2 2… the first series), which is our first series 1 1 1 2 1 2 2 1 1 2 2… it is not the same as the outgoing series. In the series we are looking for, these two series should be exactly the same. Do you think this can’t be done?

Consider the following series, known as the Kolakoski Sequence: 1 2 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 2 1 2 1 1 2 1 2 2 1 1 2 1 1 2 1 2 2 1 2 2 1 1…

The Kolakoski Sequence is a sequence of lengths consisting of an infinite number of 1s and 2s equal to itself. Let’s explain: If we call the first sequence K1 and the second sequence K2, then;

The first term of K1 is 1, and there is 1 from the number 1, then we write 1 to the sequence K2.

The second term of K1 is 2, and the continuation is 3 the term is also 2 (there have been 2 from 2) of the K2 sequence 2 we write 2 in the term; the reason is that there are 2 of the number 2 (which can also be the number 1) in K1.

4 And 5 of K1 since the term 1 is 2 of the number 3 of the sequence K2 we write in the term 2.

6 Of K1 since the term 2 is 1 of the number K2 of the sequence 4 we write in the term 1.

Accordingly, it can be said that each term in the Kolakoski sequence forms a sequence of one or two future terms. In other words, the second sequence is exactly the same as the first, because the first 1 of the sequence produces a sequence “1”, that is, itself; the first 2 produces a sequence “22” containing itself; the second 2 produces a sequence “11”, and so on… If we put it on the chart:

References: OEIS Foundation Inc.

The Kolakoski Sequence was first mentioned in a paper written in 1939 by Rufus Oldenburger (1908–1969), a mathematician and mechanical engineer who had studied with Albert Einstein at Princeton University. Rufus Oldenburger, who joined the faculty of Purdue University in 1956, took part in the space race, having the perfect background and position to influence the course of events. He established the University’s Automated Control Center and used Purdue’s facilities and his engineering skills to conduct pioneering research in the field. Derek Oldenburger, the son of Rufus Oldenburger, says the following about his father:

My dad worked quite a bit with NASA. They had a problem with vibration on the Saturn rocket and he solved it for them. He also worked on the guidance system for the Mariner spacecraft.

But it wasn’t Oldenburger who gave the series its name. The person who gave the series its name was William Kolakoski (1944–1997). Kolakoski was not originally a mathematician; he was a painter. He introduced the series to his friends who were students at the Carnegie Institute of Technology and presented the series to the American Mathematical Monthly (AMM). Kolakoski published this series under the title Advanced Problem 5304, in the following form.

In the Kolakoski sequence, each number in the sequence is the length of the next work to be created, and the element to be created Decays from 1 to 2:

1, 2 (sequence length l = 2; sum of terms s = 3)
1, 2, 2 (l=3, s=5)
1, 2, 2, 1, 1 ( l=5, s=7)
1, 2, 2, 1, 1, 2, 1 ( l=7, s=10)
1, 2, 2, 1, 1, 2, 1, 2, 2, 1 ( l=10, s=15)
1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2 ( l=15, s=23)

As can be seen, the length of the sequence at each stage is equal to the sum of the terms of the previous stage. As another illustration, you can review the image below:

References:SemanticScholar, Joel Nilsson

In order to clarify the subject a little more, we can look at the algorithm made by Jean Constant, a lecturer who conducts research in the fields of Mathematics, Science and Art. Here you see the application of the notation of the classical Kolakoski Sequence in the first, the length of which is up to 100.

References:Jean Constant, Observable

Here, again, a representation of the length of the array up to 100 is given; however, a circular shape is obtained as an illustration.

References:Jean Constant, Observable

So far, the Kolakoski sequence, of course, looks a little different compared to other types of arrays, so the question arises in our mind: Should there be only 1 and 2?

The Kolakoski sequence is formally based on an alphabet of integers and can be constructed with any number group. For example, the classical Kolakoski sequence, which we gave above, has the alphabet {1,2}. But we can also generate Kolakoski sequences with other alphabets. We will give a few examples:

{1,3} alphabetically: 1,3,3,3,1,1,1,3,3,3,1,3,3,3,1,1,1,3,3,3,1,3,3,3,3,3,1,1,1,3,3,1,3,3,3,1,3,1,3,3,1,3,3,1,3,3,1,3,3,3,3,3,3,3,1,1,1,3,3,3,3,3,1,1,3,3,3,3,…
{2,3} alphabetically: 2,2,3,3,2,2,2,3,3,3,2,2,3,2,2,3,3,3,3,2,2,2,3,3,3,2,2,2,3,3,2,2,2,3,3,3,2,3,2,3,2,2,3,2,3,2,3,2,2,3,3,2,2,3,3,2,2,3,3,3,2,2,3,3,…
{1,2,3} alphabetically: 1,2,2,3,3,1,1,1,2,2,2,3,1,2,3,3,1,2,2,3,3,1,2,3,1,2,3,1,1,1,2,3,1,1,2,3,1,1,2,2,3,3,3,3,1,1,2,2,2,3,1,2,2,3,1,1,2,3,3,1,1,1,2,2,2,2,…

So the Kolakoski sequence is not just about 1s and 2s; different alphabets of different lengths have been obtained. The second question that comes to our mind at this point is: No matter how long the Kolakoski classical sequence is, how many of the 1s and 2s are there?

The question of whether the number of 1’s is “asymptotic” equal to the number of 2’s in the classical Kolakoski Sequence is unsolvable, but the following graph (which shows the fraction of 1’s as a function of the number of digits) seems to be consistent with the evenly distributed distribution of 1 and 2.

Density of the array, References:WolframMathWolrd

Kolakoski Turtle Curve

Let’s look once again at the sequence Kolakoski, which by definition is the sequence of 1s and 2s, in which the term nth is equal to the length of the nth row of consecutive equal numbers in the same order. When a series has only two inputs, it can be visualized with the help of a “turtle” that turns left when input 1 is or right when input 2. This visualization method seems particularly suitable for the Kolakoski sequence; because there is no equal sequence of 3 inputs, that is, the turtle will never move around a square of edge length equal to its step. Here is the sequence that goes like 1,2,2,1,1,2,1… left-right-right-left-left-left… The graph drawn by a turtle for the first 300 terms (or steps):

Kolakoski turtle curve (300th term), References:calculus7

No self-crossing yet… But at step 366 it finally happens.

Kolakoski turtle curve (term 366), References:calculus7

The self-intersections continue after that:

Kolakoski turtle curve (1000th term), References:calculus7

So why bother with such a thing? What is the significance of this directory? A glimpse into the life of William Kolakoski can answer that.

William Kolakoski, References:Wikimedia

Kolakoski was a schizophrenic and was preoccupied with free will and determinism all his life. Despite his high intelligence and ability to master many different skills with little effort, his illness was, in Mike Vargo’s words, “this thing that lives within and that always threatens to take total control.” He carried his mind into regions of chaos and delusion. Wanting to feel free, Kolakoski was aware that he could not control his own brain without the help of drugs and had to accept determinism. He thought that the Kolakoski series was a possible expression, an optimistic order in the universe. The array is purely deterministic, but behaves unpredictably and strangely. Kolakoski continued to work on the series for years and created a corpus on the subject. These works are held in the Carnegie Mellon University Libraries as the William Kolakoski Collection.

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References:

Numbers
Mathematics
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OlimpiAkademi
OlimpiAkademi

Written by OlimpiAkademi

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Mathematics Education, Mathematics, Geometry, Education, Science, History of mathematics.

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