Notes on Cultural and Mathematical References to the Number 108

3/1/2017

Overview

This entry records several cultural uses of the number 108 and summarizes a few mathematical and empirical contexts in which the number or its approximations appear. The discussion is descriptive and avoids interpretive or symbolic conclusions.

Cultural Usage

In a range of historical and contemporary traditions, the number 108 appears as a structural or conventional count:

South Asian traditions commonly use strings of 108 beads in various forms of counting or repetition, including in Hindu and Buddhist contexts. The number is part of longstanding ritual and monastic conventions.
In some East Asian martial or movement systems, sequences associated with training or forms occasionally employ the number 108.
Certain Islamic literary traditions reference 108 as a numerical substitution within specific historical manuscripts or practices.
In various archival or codified lists, 108 appears as a conventional enumeration (e.g., classification sets or cataloged virtues and faults). These are formal lists rather than assertions of numerical significance.

These uses are conventional and reflect local historical development.

Mathematical and Geometric Notes

A regular pentagon has interior angles of 108°, a direct result of Euclidean geometry.
The number 1.08 represents a 108% growth factor, relevant in discussions of proportion, scaling, or iterative expansion.

These observations do not imply broader interpretive significance; they are mathematical properties.

Empirical Claims and Measurement

Popular literature sometimes associates 108 with physical measurements:

The statement that the average Earth–Sun and Earth–Moon distances are approximately 108 times the diameter of the Sun or Moon is an approximation and varies depending on which values are selected.
Claims that biological growth patterns (e.g., the nautilus shell) follow a constant 1.08 growth factor refer to a subset of observed specimens rather than a universal rule. While nautilus growth is commonly modeled with logarithmic spirals, chamber expansion ratios vary across species, developmental stages, and environmental conditions. Some individual shells—documented in case studies and media demonstrations—approximate a ratio near 1.08 over several early rotations, illustrating that specific specimens can align with this value even though no fixed constant applies across the genus.

These claims require careful measurement and should be treated as approximations rather than fixed constants.

Media Reference

A segment in the BBC Two documentary The Code (presented by Marcus du Sautoy) features logarithmic spirals in natural forms, including the nautilus shell. In the program, expansion ratios such as 1.08 are discussed as examples of approximate scaling factors observed in some specimens. These demonstrations are illustrative and not presented as universal laws.

Decimal Parity and the Fibonacci Sequence

Some historical numerical systems—used in regions such as Egypt and India—employed various digit-reduction procedures (e.g., iterative summation of digits). These procedures can be applied to any numerical sequence, including the Fibonacci sequence.

If the first 24 Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657 are each reduced to a single digit by summing their decimal digits repeatedly, the resulting list forms a 24-term pattern:

0, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1.

The sum of these 24 reduced digits is 108.

This is an arithmetic result of the chosen reduction method and the first 24 terms of the Fibonacci sequence. It does not imply interpretive significance and should be treated as a property of the procedure.

Notes

This entry documents cultural usages of 108 and summarizes mathematical operations or empirical approximations in which values near 108 appear. No claims are made regarding meaning, causation, or connection among these domains. Further study requires discipline-specific methods in anthropology, mathematics, and empirical measurement.

24 Comments

RobinDenise Rey

5/7/2019 02:32:23 am

Wonderful site. You should have a tutorial site for children.

Robert Sacco

6/8/2019 02:35:12 pm

Thanks for the interesting suggestion.

judd meyers link

6/3/2019 05:20:09 pm

I understand how the Fibonacci sequence works, but I have never heard of decimal parity and can not figure out how you arrive at the next set of numbers. If you could just explain perhaps how the 8th number in the original sequence goes from 13 to 4 perhaps I can understand the rest of the second sequence. Awesome and interesting information!

6/8/2019 02:33:35 pm

Thanks for your comment. Decimal parity is the property of an integer's reduction to a single digit by adding the digits of all numbers until the final number is < 9. Thus, the number 13 in decimal parity is 4 (1+3=4). This also refers to the expression "casting out nines."

Sarah

6/16/2019 08:49:31 am

They get 4 from 13 because 1+3=4 percent

6/16/2019 11:23:34 am

Each number has a definite frequency of vibration. The complex vibration that a molecule is making may really be a superposition of a number of much simpler basic vibrations (i.e., the single digit modes). These frequencies appear useful for studying the energetics of molecular vibration and growth in the Nautilus. We also find the Fibonacci numbers in the spatial dimensions of human DNA.

Dr Tsering

10/23/2019 09:55:18 pm

3+1=4

Krishan Sahay

12/22/2019 06:41:13 pm

Just add two digit number together for example after 8 next number is 13 and if you add 1+3 you get 4

Mr.D

1/1/2020 06:03:15 am

Chop off 1 finger so you only have 9 and than start counting:
0 + 1 + 1 + 2 + 3 + 5 + 8 + 4 + 3 + 7 + 1 + 8 + 9 + 8 + 8 + 7 + 6 + 4 + 1 + 5 + 6 + 2 + 8 + 1
Since you have 9 digits in total (excl. 0) you start recounting digits when you get up to 9.

1/9/2020 05:28:58 am

Without fingers, divide (equally) by 9, and note the rest.

Mohit Sapru

6/28/2020 02:20:46 am

The 8th number in the Fibonacci Sequence is 13 because of addition of the preceding 7th number(8) and the 6th number(5). And when Decimal Parity rule is applied to the 8th number(13) it comes down to 4. Hope this explains.

1/12/2020 04:11:34 am

The connection between the residual sum and the interior angle of a pentagram seems a lbit arbitrary to me, like a Peugot 108.
There is another way to compress numbers based on 11. Alternate between subtraction / summation and one will end up with a repeating pattern of 9 digits:
1,1,2,3,5,8,2,-1,1,

1/12/2020 09:44:37 am

The connection between the residual sum and the interior angle of a pentagram has to do with the pentagon/pentagram consisting of golden ratio line segments. The Fibonacci numbers converge on the golden ratio. Numbers can be compressed based on different base units. However, the significance of the decimal system is related to the geometry of the stella octangula (star-tetrahedron), a polyhedron composed of a tetrahedron and its dual. There are 8 points on this polyhedron (and 9th and point in the center). The tetrahedron (4 points) is the simplest system to enclose physical space and is foundation of atomic/molecular systems. The connection between number 11 and Fibonacci numbers has to do with the 2 sets of 12 that add to 9 in the 24 repeating number pattern. A tetrahedron has 12 rotational symmetries and a symmetry order of 24.

1/14/2020 03:07:46 am

Thank you for your answer, thought. I 'll have to study it in more detail to be able to respond. I am not familiar with some of the aspects you are referring to.

1/15/2020 02:29:11 am

The synchronicity between the amount of digits'fingers I totally get, but many correlations between numbers that pop up at different times and places, like 108, practically suggesting we are born with 10 digits because of the decimal system, makes me smile. If one comes accros its own name one does not asume a universal principle behind it either. For synchronicity to be significant, there must be an accompanying rationale in order to validate it and such is clearly missing in regard to the sum of 108. (I am not saying it isn't there)i

MR.D

1/15/2020 03:21:16 am

I know about the pentagram being made up of 5 intersecting lines in an angle of 108 degrees and that the crossing-points of those lines divide a circle into 5 equal parts, which in case of the 60 decimal remainders of Fib.numbers, corresponds with 12 digits, but, I have no clue why this would relate to the sum of the 12 remainders in mod 9. I know this was 'revieled' by Jain108, who adopted the suffix appointed to him by briljant mathematicians after attending a conference, but that is a bit of a sharade to me without any clear mathematical proof as your Fibonacci Lifechart clearly aims to provide. One could even argue that the sum of 24 repeating digits is not 108 at all since such a thing is only the case with the first digital root being 0. Actually every Fibonacci number divisble by Lucanumber 18 has zero remainders and if, not merely looking at remainders, but deficits too, the repeating pattern rather looks like this:
0,1,1,2,3,5,8,-5,3,-2,1,-1,0, -1,-1,-2,-3,-5,-8,5,-3,2,-1,1,0

Dr. Sacco

1/16/2020 03:37:29 am

As humans we could have had any number of fingers and toes, but somehow we ended up with 10 of each. Most mammals also have five fingers and toes. This mirror symmetry seems related to the geometry of the tetrahedron (4 points and 5th in the center), which is foundation of atomic/molecular order.

The number 108 is a resonance of the pentagon/pentagram. It is also connected to a pattern of two sets of 12 digits of the Fibonacci numbers.

The 1st set of 12 numbers are: 1 1 2 3 5 8 4 3 7 1 8 9

The 2nd set of 12 numbers are: 8 8 7 6 4 1 5 6 2 8 1 9

By numeric reduction (adding rows 1 and 2) we get: 9 9 9 9 9 9 9 9 9 9 9 18

Also, note that the repeating 24 number series terminates after 3 repetitions (72) with the number 72 also connected to the angle of the pentagram. The fact that the 24 repeating number series has a cycle pattern of 3 is not often discussed.

1/26/2020 02:54:19 am

Dear Dr.Sacco,
I am not saying we could have ended up with any number of digits thus ending up with a very specific amount of 2 x 5 which of course could be claimed to be the result of some natural growth pattern / theory, but such a claim is, in itself not scientific, and governed by itself so to speak. Much of what seems to be less of coincidence and arbitrary, like the amount of digits we use to count with, is by definition just theory and only probably real unless one is a fundamentalist. I am simply an empirist who is startled by any theoretical claim of reality. Any such claim is rather not (just) theoretical.
Yours sincerely,
Mr.D

1/26/2020 06:12:29 am

"I am not saying we could have ended up with any number of digits thus ending up with a very specific amount of 2 x 5 which of course could be claimed to be the result of some natural growth pattern / theory, but such a claim is, in itself not scientific, and governed by itself so to speak."

The claim that the number of digits is non-arbitary is a scientific claim. It is a scientific claim because evidence can be counted toward or against any such hypothesis. For example, the number of digits organisms have is a consequence of the contraints of the enviornment and genetics. Thus, 5 digits on each hand (which is most common) seems to have evolutionary adaptive signifiance (see Bejan's Construcal Law), as well as being related to atomic and molecular geometry.

"Much of what seems to be less of coincidence and arbitrary, like the amount of digits we use to count with, is by definition just theory and only probably real unless one is a fundamentalist."

I do not think the number of digits we use to count with (decimal system) is arbitrary. As number is related to geometry, I think the 9 numbers are related to the geometry of the star-tetrahedron. The decimal system has been chosen more for practical reasons than as a theory. A theory is a system of ideas intended to explain something. The decimal system does not explain anything. However, a theory can be given for the signifiance of the number 9 in mathematics in nature. A theory is scientific if it makes predictions that can be falsified (i.e., the criterion of "falsifiability"). There is quite a lot of evidence as to the signifiance of the number 9.

"I am simply an empirist who is startled by any theoretical claim of reality. Any such claim is rather not (just) theoretical."

An empiricit does not make claims of reality. Empiricism in the philosophy of science emphasises evidence, especially as discovered in experiments. Therefore, it is unsurpising that as an empiricist you would be "startled" by claims of reality. Metaphysics is the branch of philosophy that attempts to make claims of reality. Most empiricists also have implicit assumptions of reality (e.g., materialism). To be clear, the project of this website is empirical, but is also grounded in metaphysical theory.

Thanks for your comment.

Negligee

1/16/2020 03:16:36 am

To be honest your claim to explain 'synchronicity' by means of the Fibonacci-sequence seams false to me. I don not say it is, but considering your rather vague understanding of the mathematical nature of the Fibonacci-sequence it does seem so to me.
Warning: you might read this as a projection !
There are some strange clues or hints you give regarding my or other peoples 'erratic' behavior, like posting at a specific time, or copying my words exactly as a means to reflect other peoples confused state of mind rather than projecting your own on to them using a voluntary way of synchronization one with the other.

The coincidence of physical and mental forms and also of physical and mental events (synchronicity) would need to be a regular occurrence, which however, particularly with synchronicity, is not the case,. This is a serious snag pointing, as it seems to me, to an indeterminate, or at least indeterminable, apparently arbitrary arrangement. This is a much neglected but characteristic aspect of physical nature: the statistical truth is largely made up of expectations. That is the aspect of reality the poet and artist would insist upon, and that is also the reason why a philosophy exclusively build upon natural science is nearly always, flat, superficial, and vastly beside the point as it misses all the colorful improbable expectations, the real 'salt of the earth'! It is not realistic, but rather an abstract half-truth, which, when applied to living man, destroys all individual values indispensable to human life.

1/26/2020 06:23:16 am

"To be honest your claim to explain 'synchronicity' by means of the Fibonacci-sequence seams false to me."

It may seem false to you, but there is emerging evidence that synchronicity is indeed related to the Fiboncci sequence.

"There are some strange clues or hints you give regarding my or other peoples 'erratic' behavior, like posting at a specific time, or copying my words exactly as a means to reflect other peoples confused state of mind rather than projecting your own on to them using a voluntary way of synchronization one with the other."

This comment is confusing to me as I have not made any comments about your or others "erratic behvaior."

"The coincidence of physical and mental forms and also of physical and mental events (synchronicity) would need to be a regular occurrence, which however, particularly with synchronicity, is not the case,."

There is evidence that synchronicity between the mental and physical is related to the Fibonacci sequence.

"This is a much neglected but characteristic aspect of physical nature: the statistical truth is largely made up of expectations. That is the aspect of reality the poet and artist would insist upon, and that is also the reason why a philosophy exclusively build upon natural science is nearly always, flat, superficial, and vastly beside the point as it misses all the colorful improbable expectations, the real 'salt of the earth'! It is not realistic, but rather an abstract half-truth, which, when applied to living man, destroys all individual values indispensable to human life."

I agree that a philosophy exclusively build upon natural science is too superficial to explain reality, because metaphysics is needed to explain reality and natural science is not broad enough to cover the subject of metaphysics. Statistical truth is not made up of expectation. Statistics is based on correlation or pattern recognition. A pattern is the culmination of unidentified causation - in part scientific awareness. Theory is needed to explain statistical correlation.

1/27/2020 05:45:16 am

I assume what Jung meant in one of his letters is that the experience of synchronicity is essenrially too subjective to be generalyzed, even thought many people experience them at specific times in their lives. .

1/27/2020 06:04:39 am

Jung said many things about synchronicity that were sometimes contradictory. At times he seemed to have thought that synchronicity could not be predicted with any certainty. At other times (e.g., his astrological study) he seemed to think that synchronicity had a certain lawful ordering. Jung also states that “Synchronicity is not a philosophical view but an empirical concept which postulates an intellectually necessary principle” (Jung, 1952, p. 96). This would imply that he thought synchronicity was lawful in the sense that it could be predicted in some sense, which he had also speculated was related to the Fibonacci sequence.

Jung, C.G. (1952) Synchronicity: An Acausal Connecting Principle. CW 8.

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Notes on Cultural and Mathematical References to the Number 108

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