Fibonacci Series for Genealogy – Estimating Generations of a Family Tree

Distant cousins want to know how far back in time they must search to find a common ancestor. Fibonnaci series, data analysis and probability can be used to estimate an answer.

The mathematician Fibonacci sought to answer a similar question dealing with the growth of rabbits. Honey bees live by his pattern too, it has been shown. Does the pattern apply to human genealogy?

Mathematician Sandra Lach Arlinghaus demonstrated1 the connection between urban population growth and Fibonacci series. Let us use this work as a starting point for our analysis and see what clues it can provide.

What is the Current Population?

First we need to rephrase the question, as it does not provide enough information. We need a starting point. We need to know:

How many descendants are there today for our common ancestor?

Rephrasing the question this way means that we start with a known set of current potential relatives and work back in time to a common ancestor.  Other approaches to find the most-recent common ancestor focus on descendants of that ancestor.  My approach is a simpler problem to solve.

I can use social media to estimate the number of potential common ancestors alive today. In my particular case, I am interested in the Yaroker family from Kanev. Searching the web for "Yaroker" or "Ярокер" identifies about 30 individuals who are visible on the internet. However, I know that there are many more that have a different family name or no web presence.

Let's assume that the visible number (e.g. those appearing in web searches) of Yarokers represent only 20% of today's total Yaroker population (following the 80/20 split that comes up so frequently in life). Therefore, I estimate:

There are about 150 Yaroker descendants alive today. — Order of Magnitude Estimate

This feels right, as I know there are more than 15. If there were 1500, they would be easier to find (which they are not).

Fibonacci Series in Population Growth

The next step in the problem is to understand clues that Fibonacci gives us about population growth. The chart below is one such clue. It is reproduced from the Golden Number website and is based on the work of Dr. Arlinghous.

Geographic Area
Phi Ratio
Percent Difference
New York, NY116,206,841
LA Long Beach CA28,351,26610,016,37920%
Chicago NW IN36,714,5786,190,4628%
Detroit, MI53,970,5843,825,9164%
Washington DC82,481,4592,364,5465%
Houston, TX131,677,8631,461,37013%
Cincinnati, OH211,110,514903,17619%
Dayton, OH34685,942558,19419%
Richmond, VA55416,563344,98317%
Las Vegas, NV89236,681213,21110%
New London, CT144139,121131,7725%
Great Falls, MT23370,90581,43915%

The chart compares actual population levels for major cities with Fibonacci population levels derived by dividing the largest city's population successively by φ. The correlation is not bad.

Interpreting the data in this chart, we see that there are units of population across major cities that match a Fibonacci pattern. Phrasing this in terms of physics, there are population "quanta" for different Fibonacci "energy" levels. We can make the hypothesis that:

Population numbers appear to settle into Fibonacci levels as well as transition between them, much like an electron jumps between energy bands in silicon. — Hypothesis

Observing that population "quanta" fit into Fibonacci "levels" allows us to make predictions, such as "Dayton, OH" once had a population similar to "Richmond, VA"; or conversely, "Richmond, VA" will transition to a population size of a "Dayton, OH".

Dr. Arlinghous's work is a clue, but it is not enough to answer our original question because this geographic analysis does not include time. We don't know how long it takes a city to transition between various Fibonacci population levels. We need to answer,

How long do transitions between Fibonacci population levels last?
Time of Transitions between Fibonacci Population Levels

World population growth can be used to find the transition time between Fibonacci population levels, assuming we agree that there are natural "quanta" of population levels. We also need to assume that the world's population correlates with my Yaroker family's population. One can argue that there are historical reasons why these two sets of data would differ considerably, but for a first-order approximation lets ignore historical influences.

The chart below2 shows world population levels for different time periods as well as the Fibonacci levels calculated in the same manner as described above. These Fibonacci population levels are then matched to the actual world population. This matching allows us to associate Fibonacci population levels with time period.

World Population (millions)
Phi Ratio for World Population
Phi Ratio for Yaroker Population
Average Family Size
Number of Yaroker Families
Probability of Finding Common Yaroker Ancestor
10,000 B.C.1
5,000 B.C.5
2,000 B.C.27
1,000 B.C.50
0 A.D.2002225121100%
500 A.D.300
1000 A.D.4003598121100%
1500 A.D.500
1650 A.D.6005821412250%
1750 A.D.750
1800 A.D.9009412212250%
1810 1,000
1850 1,171
1900 1,608 1,523 3510425%
1920 1,834
1930 2,008
1940 2,216
1950 2,406 2,464 575128%
1960 2,972
1970 3,700 3,986 934244%
1980 4,400
1990 5,100
1997 5,852
2000 6,080
2005 6,450 6,450 1503502%

Number of Generations to a Common Ancestor

The next-to-last column of the above chart gives the Yaroker family population by year, calculated using the same Fibonacci approach as mapping of Fibonacci levels described above. Using this chart, we can now ask the question,

What is the probability that the Yaroker distant cousins come from siblings of one family, given a known Yaroker population?

The probability is shown in the last column of the above table. In the worst-case scenario, where the distant cousins each descended from a different sibling, we would need to research back to the year 400 A.D. From this chart, we can now say,

There is a 25% chance that we will find a common ancestor in the year 1900; 50% chance in the year 1800; and, certainty in the year 400 A.D.


Fibonacci Approach to Common Ancestor Probability


  1. See source citation here: []
  2. World population data was taken from this site: []
May 14th, 2013 Posted by Jon Jaroker Filed in: Mathematics

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