Tiger got to hunt,
Bird got to fly,
Man got to sit and wonder 'WHY, WHY, WHY',
Tiger got to sleep,
Bird got to land,
Man got to tell himself he UNDERSTAND
My contribution to "UNDERSTAND" is my PhD dissertation, "Hierarchy Theory - Some Common Properties of Competitively-Selected Systems", System Science Department, Binghamton University, NY, 1996. If you wish to pursue further research in this area please contact me at email@example.com. A few copies of my dissertation are available.
The OPTIMAL SPAN HYPOTHESIS is at the heart of my dissertation. Using Hierarchy Theory, Information Theory, and Graph Theory, I proved that Optimal Span is about the same, between five and nine, for virtually all complex structures that have been competitively selected.
- The products of Natural Selection (Darwinian evolution) and
- The products of Artificial Selection (Human inventions that competed for acceptance by human society)
You may download my PowerPoint Show that should run on any Windows PC here:
Most complex structures are compositional or control hierarchies:
- An example of a compositional hierarchy is written language. A word is composed of characters. A simple sentence is composed of words. A paragraph is composed of simple sentences, and so on.
- An example of a control hierarchy is a management structure, where a manager controls a number of foremen or team leaders, and they, in turn, control a number of workers.
Hierarchy (from Greek:ἱερός — hieros, ‘sacred’, and ἄρχω — arkho, ‘rule’) originally denoted the holy rule ranking of nine orders of angels, from God to Seraphims to Cherubims and so ondown to the Archangels and plain old Angels at the lowest level. Kind of like the organization of God’s Corporation!
The seminal book on this topic is Hierarchy Theory: The Challenge of Complex Systems[ Pattee, 1973 ]. This book includes chapters by distinguished academics, including:
- Herbert A. Simon (Nobel laureate) on “The Organization of Complex Systems”.
- James Bonner “Hierarchical Control Programs in Biological Development”
- Howard H. Pattee “The Physical Basis and Origin of Hierarchical Control” and “Postscript: Unsolved Problems and Potential Applications of Hierarchy Theories”
- Richard Levins “The Limits of Complexity”
- Clifford Grobstein “Hierarchical Order and Neogenesis”.
There are two kinds of hierarchy. A few well-known examples will set the stage for more detailed examination of modern Hierarchy Theory:
1 -Management Structure (Control Hierarchy)
2 -Software Package (Control Hierarchy)
3 – Written Language (Containment Hierarchy)
4 – “Chinese boxes” (Containment Hierarchy)
Traversing a Hierarchy
Folding” a “String”
The above “parade” was described with the Chief Exec at the head of it, but you could just as well turn it around and have the lowest-level employees lead and the Chief Exec at the rear. When military hierarchies go to war, the lowest-level soldiers are usually at the front and the highest-level Generals well behind.
A more practical example is the text you are reading right now! It was transmitted over the Internet as a string of “bits” – “1″ and “0″ symbols. Each group of eight bits denotes a particular character. Some of the characters are the familiar numbers and upper and lower-case letters of our alphabet and others are special characters, such as the space that demarks a word (and is counted as a character that belongs to the word), punctuation characters such as a period or comma or question mark, and special control characters that denote things like new paragraph and so on.
You could say the string of 1′s and 0′s is folded every eight bits to form a Character. The string is folded again at each Space Character to form Words. Each group of Words is folded yet again at each comma or period symbol that denotes a Simple Sentence. Each group of Simple Sentences is again folded to make Paragraphs, and so on.
You could lay out a written document as a tree structure, similar to a Management hierarchy. The Characters would be at the bottom, the Words at the next level up, the Simple Sentences next, the Paragraphs next, and so on up to the whole Section, Chapter, and Book.
With all these different types of hierarchical structures, each with its own purpose and use, you might think there is no common property they share other than their hierarchical nature. You might expect a particular Span of Control that is best for Management Structures in Corporations and a significantly different Span of Containment that is best in Written Language.
The simple equation for Optimal Span derived by [ Glickstein, 1996 ] is:
So= 1 + De
(Where D is the degree of the nodes and e is the Natural Number 2.71828459)
In the examples above, where the hierarchical structure may be described as a single-dimensional folded string where each node has two closest neighbors, the degree of the nodes is, D = 2, so the equation reduces to:
So= 1 + De = 1 + 2 x 2.71828459 = 6.43659
“Take home message”: OPTIMAL SPAN, So = ~ 6.4
Also see Quantifying Brooks Mythical Man-Month (Knol) , [Glickstein, 2003 ] and [ Meijer, 2006 ] for the applicability of Optimal Span to Management Structures.
[Added 4 April 2013: The Meijer, 2006 link no longer works. His .pdf document is available at http://repository.tudelft.nl/assets/uuid:843020de-2248-468a-bf19-15b4447b5bce/dep_meijer_20061114.pdf ]
Examples of Competitively-Selected Optimal Span
Management Span of ControlManagement experts have long recommended that Management Span of Control be in the range of five or six for employees whose work requires considerable interaction. Depending upon the level of interaction, experts recommend up to nine employees per department.This recommendation comes from experience with organizations with different Spans of Control. The most successful tend to have Spans in the recommended range, five to nine,an example of competitive-selection.
When the lowest level consists of service-type employees, whose interaction with each other is less complex, there may be a dozen or two or more in a department, but there will usually be one or more foremen or team leaders to reduce the effective Management Span of Control to the range five to nine.Corporate hierarchies usually have about the same range of first-level departments reporting to the next level up and so on.
Say you had a budget for 49 employees and had to organize them to make most effective use of your human resources. Which of the following seems most reasonable?
(A) you have ONE manager and 48 workers, which is a BROAD hierarchy. Management experts would say a Management Span of Control of 48 is way too much for anyone to handle!
(B) you have a third-level chief executive, three executive-level managers, each with three department managers, totaling THIRTEEN managers in a three-level management hierarchy and only 36 workers, which is a TALL hierarchy with an average Management Span of Control of only 3.3. Management experts would say this is way too inefficient with too many managers!
(C) you have a second-level manager and six department managers, totaling SEVEN managers and 42 workers in a MODERATE hierarchy with an average Management Span of Control of about 6.5. Management experts would say this is about right for most organizations where the workers have to interact with each other. Optimal Span theory supports this common-sense belief!
Human Span of Absolute JudgementEvolution and Natural Selection have produced the human brain and nervous system and our senses of vision, hearing, and taste. It turns out that these senses are generally limited to five to nine gradations that can be reliably distinguished. It is also the case that we can remember about five to nine chunks of information at any one time. This is another example of competitive-selection, where, over the eons of evolutionary development, biological organisms competed and those that best fit the environment were selected to survive and reproduce.
George A Miller wrote a classic paper titled The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information [ Miller, 1956 ]. He showed that human senses of sight, hearing, and taste were generally limited to five to nine gradations that could be reliably distinguished.
Why do we “chunk” things in groups of about seven – seven days of the week, seven seas, seven sins, etc? The presentation I gave to the Philosophy Club in The Villages, FL, 14 March 2014 provides the theoretical answer. You may download a PowerPoint Show that should run on any Windows computer here:https://sites.google.com/site/iraclass/my-forms/PhiloMAGICALsevenMar2014.ppsx?attredirects=0&d=1
This is an easy-to-understand version of a more technical presentation I made to the Science-Technology Club in February, see: http://tvpclub.blogspot.com/2014/02/optimal-span-amazing-intersection-of.html
PERSECUTED BY THE NUMBER SEVEN !
George A Miller's classic paper appeared way back in 1956 the Psychological Review with the intriguing title: The Magical Number Seven, Plus or Minus Two – Some Limits on Our Capacity for Processing Information. That paper was extremely important and influential and is still available online. George A. Miller contains a strange plea:
"My problem is that I have been persecuted by an integer seven plus or minus two …Miller’s paper continues as follows:
"For seven years this number has followed me around, has intruded in my most private data, and has assaulted me from the pages of our most public journals. This number assumes a variety of disguises, being sometimes a little larger and sometimes a little smaller than usual, but never changing so much as to be unrecognizable.
"The persistence with which this number plagues me is far more than a random accident …
"There is, to quote a famous senator, a design behind it, some pattern governing its appearances. Either there really is something unusual about the number or else I am suffering from delusions of persecution.Miller’s paper is well worth reading and is available on the Internet at this link [Miller, 1956]"
Miller presents the results of twenty experiments where human subjects were tested to determine what he calls our "Span of Absolute Judgment", that is, how many levels of a given stimulus we can reliably distinguish. Most of the results are in the range of five to nine, but some are as low as three or as high as fifteen. For example, our ears can distinguish five or six tones of pitch or about five levels of loudness. Our eyes can distinguish about nine different positions of a pointer in an interval. Using a vibrator placed on a person's chest, he or she can distinguish about four to seven different level of intensity, location, or duration, etc. The average Span of Absolute Judgment is 6.4 for Miller's twenty one-dimensional stimuli.
Miller also presents data for what he calls our "Span of Immediate Memory", that is, how many randomly presented items we can reliably remember. For example, we can remember about nine binary items, such as a series of "1" and "0", or about eight digits, or about six letters of the alphabet, or about five mono-syllabic words randomly selected out of a set of 1000.
At the end of his paper Miller rambles:
...And finally, what about the magical number seven? What about the seven wonders of the world, the seven seas, the seven deadly sins, the seven daughters of Atlas in the Pleiades, the seven ages of man, the seven notes of the musical scale, and the seven days of the week? What about the seven-point rating scale, the seven categories for absolute judgment, the seven objects in the span of attention, and the seven digits in the span of immediate memory?
For the present, I prefer to withhold judgment.
Perhaps there is something deep and profound behind all these sevens, something just calling out for us to discover it.
But I suspect that it is only a pernicious, Pythagorean coincidence. [my bold]Well, it turns out that there IS something DEEP and PROFOUND behind "all these sevens" and I (Ira Glickstein) HAVE DISCOVERED IT. And, my insight applies not only to the span of human senses and memory, but also to the span of written language, management span of control, and even to the way the genetic "language of life" in RNA and DNA is organized. Furthermore, my discovery is not simply based on support from empirical evidence from many different domains, but has been mathematically derived from the basic Information Theory equation published in 1948 by Claude Shannon, and the adaptation of "Shannon Entropy" to the Intricacy of a biograph by Smith and Morowitz in 1982.
Glickstein’s Theory of Optimal Span
Hierarchy and Complexity
Weak Statement of Hypothesis
Strong Statement of Hypothesis
In Chapter 6 of my novel, Jim and Luke wonder about the control structure for the 1600 scepter-holders:
... After a period of silence, Luke spoke up. “Sixteen hundred people are way too many for there not to be a hierarchical structure,” he began. “If the scepter-holder system was properly designed, according to system science theory at least, there would have to be several grades above the lowest class of scepter-holder.”
He took out his read-WINs and put them on.
“Luke,” I observed, “There’s no WIN coverage in this area …”
“Right,” answered Luke, “But there are processors and software in my read-WINs that allows them to operate independently. I’ve got a program for ‘optimal span’ – you know the ‘magical number seven plus or minus two.’”>
“What the heck is that?” I asked, “And why would I care? Where are we going here?”
“Well, back about a century ago, a psychologist named Miller discovered that human perception, such as sight and smell and taste and memory and so on, is limited to five to nine gradations. He called it 'the magical number seven, plus or minus two' or, more scientifically, the 'span of human perception'."
“Another guy, an engineer named Glickstein, about sixty years ago, proved the optimal span for any structure is one plus the degree of the nodes times 2.71828459, the natural number ‘e.’ For a one-dimensional string, the degree is two and the formula comes out to be around six and a third, or a little more. He also showed with Shannon’s information theory that the range five to nine was, at least theoretically, over ninety-six percent efficient and four to twelve was over eighty percent efficient. And that’s not just for control hierarchies like a management chain, but also containment hierarchies in all types of physical systems and even software systems like …”
“You just told me how to build a clock,” I laughed, interrupting Luke. “All I want to know is what time it is! Please, tell me why I give a hoot about the range five to nine or the number six and a third or a bit more?”
“About forty years ago,” continued Luke, “A management expert rediscovered the optimal span theory and proclaimed that all management structures must adhere to it! Did you ever notice how nearly all departments at TABB have either six or seven workers to each manager? How each second-level manager has six or seven first-level managers working for him or her?”
“Yeah, come to think of it,” I replied, “That’s how it is. On the other hand, when I worked in a factory as a college summer job, we had about a dozen guys and gals in our team.”
“Well,” replied Luke, “The lowest level, like a platoon in the military, can have ten or twelve or sometimes a bit more. The theory only applies when the workers have to interact with each other in complex ways, not when they’re doing grunt work.”
“If you’d quit interrupting, I’ll tell you,” Luke said good-naturedly, “According to the optimal span program in my read-WINs, sixteen-hundred scepter-holders would break down into about two-hundred-fifty first-level ‘departments,’ each with six or seven scepter-holders and one higher-level scepter-holder ‘managing’ them. The two-hundred-fifty second-level scepter-holders would report to thirty-six third-level scepter-holders who, in turn, would report to six fourth-level scepter-holders who would report to the top dog scepter-holder if there was one.”
“Yeah,” replied Luke, “There should be thirty-six scepter-holders at the third level. What about it?”
“Well,” I began, very seriously, “We have a tradition in Judaism that there are thirty-six ‘tzadikim’ or ‘righteous ones’ for whose sake the world exists. No one knows who they are. When one dies, he, or she I guess, is replaced by another, chosen by God. They are sometimes called the ‘Lamed Vovniks’ because, according to gematria, which we discussed some months ago, the Hebrew letter Lamed stands for thirty and the letter Vuvfor six, which adds up to thirty-six.”
“So,” replied Luke with a level of interest that surprised me at the time, “There would be thirty-six especially powerful scepter-holders who would regulate the rest! And they do need regulation. I’m not one-hundred percent pleased with Stephanie’s ethics ..."