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260px Close packed spheres with umbrella light camerea (http://en.wikipedia.org/wiki/Close-packing_of_spheres)

NPR says: Don't Believe Facebook; You Only Have 150 Friends and discusses Dunbar's number.

Dunbar says there are some neurological mechanisms in place to help us cope with the ever-growing amount of social connections life seems to require. Humans have the ability, for example, to facially recognize about 1,500 people. Now that would be an impressive number of Facebook friends.

Yet the problem with such a large number of "friends," Dunbar says, is that "relationships involved across very big units then become very casual — and don't have that deep meaning and sense of obligation and reciprocity that you have with your close friends."

One solution to that problem, he adds, can be seen in the modern military. Even as they create "supergroups" — battalions, regiments, divisions — most militaries are nonetheless able to maintain the sense of community felt at the 150-person company level.

"The answer has to come out of that," Dunbar says, "trying to create a greater sense of community.

Wikipedia says of Dunbar's number

Dunbar's number is a theoretical cognitive limit to the number of people with whom one can maintain stable social relationships. These are relationships in which an individual knows who each person is, and how each person relates to every other person. Proponents assert that numbers larger than this generally require more restrictive rules, laws, and enforced norms to maintain a stable, cohesive group. No precise value has been proposed for Dunbar's number. It has been proposed to lie between 100 and 230, with a commonly used value of 150. Dunbar's number states the number of people one knows and keeps social contact with, and it does not include the number of people known personally with a ceased social relationship, a number which might be much higher and likely depends on long-term memory size.

Dunbar's number was first proposed by British anthropologist Robin Dunbar, who theorized that "this limit is a direct function of relative neocortex size, and that this in turn limits group size ... the limit imposed by neocortical processing capacity is simply on the number of individuals with whom a stable inter-personal relationship can be maintained." On the periphery, the number also includes past colleagues such as high school friends with whom a person would want to reacquaint oneself if they met again.[3]

Christopher Allen writes about "The Dunbar Number as a Limit to Group Sizes", and posits various sizes are stable, and others unstable, focusing on online communities.


In 2-dimensions, one penny can be surrounded by exactly 6 pennies (of equal size) that it touches. A group of eight pennies will not be as stable as a group of seven (six plus one), since the eighth orbits the close packing of pennies. However if you can fill the second ring, then you can add 12 more pennies (for a total of 19).

Closest packing of circles, spheres, cubes, pyramids, etc, provides a certain number of linkages at degree 0, another number at degree 1, and so on. This is like the valence number of electrons around the nucleus of an atom. Some numbers are stable, others are + or - and less stable.

Does the Dunbar number correspond to any particular physical shape that is stable around 150, but falls apart if larger? This might help explain the limits and network topology of our neurology.

Number theory

Prompted by the Iranian election where the results were made up on a spreadsheet and violated Benford's first digit law I was playing with a spreadsheet, and for some reason, decided to count the number of times a set of numbers was evenly divisible by other numbers. (This of course is not Benford's law at all).

So taking, for example, the numbers from 1 to 1000, the column below "SERIES" indicates that in this set with 1000 entries, the elements are evenly divisible by two 500 times, by three 333 times, etc. which is intuitive.

Taking the cumulative of numbers from 1 to 1000 (so the series is 1, 3=1+2, 6=3+3, 10=6+4, 15=10+5 ... 500500=499500+1000), we get the column "CUMULATIVES". In this case, numbers in this series are evenly divisible by two 500 times, but now threes show up 666 times. Fours are about half of two, and eights are one-fourth of two, and fives are twice tens, but six is half threes, not a third of twos.

The column RATIO is just the ratio of the SERIES and CUMULATIVES numbers.

I am sure there a good intuitive explanation for this fact, and I am sure mathematicians would find it obvious, but it surprised me.

I have no clue if this has any deeper meaning.

Evenly Divisible by SERIES CUMULATIVES RATIO
twos 500 500 1
threes 333 666 2
fours 250 250 1
fives 200 400 2
sixes 166 333 2
sevens 142 285 2
eights 125 124 1
nines 111 222 2
tens 100 200 2
none of the above 228 72 0.31

(none of the above does not imply prime status, but prime numbers will be none of the above).

3/25/2010 - corrected cumulatives column as per J.Horwitz

David Levinson

Network Reliability in Practice

Evolving Transportation Networks

Place and Plexus

The Transportation Experience

Access to Destinations

Assessing the Benefits and Costs of Intelligent Transportation Systems

Financing Transportation Networks

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