The geometry of a square pyramid comprised of spheres ...

**ZookieMonster** · 13th July 2011, 03:07

Level 1_________ 1 ball

Level 2_________ 4
Level 3_________ 9

Level 4_________ 16
Level 5_________ 25
Level 6_________ 36

Level 7_________ 49
Level 8_________ 64
Level 9_________ 81
Level 10________ 100 balls

Total balls = 1+4+9+16+25+36+49+64+81+100 = 385 balls

In this modeling, the perfect human square pyramidal organization of 10 levels would have 385 human units.

Suppose the first level represents the apex ... the alpha human ... the head honcho
Suppose the next two levels represent the privileged classes (the blue bloods; the golden spoons; the associates)
Suppose the next three levels represent the minion classes (silver spoons; soldiers; securators; subprimators; sucklings)
Suppose the remaining four classes represent the slave classes (middle class; lower middle class; poor class)

Now, let's assume the capstone is the sigma of levels 1,2,3 ... then the population is 1+4+9 = 14 human units
Let's further assume that the ziggurat is the sigma of levels 4,5,6 ... then the population is 16+25+36 = 77 human units
Finally, assume that the base is the sigma of levels 7,8,9,10 ... then the population is 49+64+81+100 = 294 human units

To wit, if the perfect human pyramid of 10 levels comes even close to modeling the real organizational human pyramid ... then the base alone has a 294 to 91 or 3-to-1 advantage over the capstone and the ziggurat combined.

Extrapolating down to the smallest pyramid possible (e.g. three levels) ... capstone, ziggurat, base ... we have 1+4+9 = 14 human units. IOW, the base in this smallest pyramid has a 9 to (1+4) ratio or approx. 2-1 advantage over the capstone and the ziggurat combined. We note that the ratio favors the base increasingly as the height of the pyramid increases. At three levels, a 2 to 1 ratio ... at ten levels, a 3 to 1 ratio. The capstone would be further susceptible if the ziggurat ever decided to side with the base in any degree.

Question begs: how much of the ziggurat needs to be awakened so that it sides with the base (assuming that virtually all the base will be unified against the capstone once the grocery shelves run bare)? (Remember, in the worst case of the ten level pyramid, the base still has a 3-to-1 advantage over the sum of the capstone and the ziggurat.)

Anyways, thought I'd put that out to help illustrate the mathematics of the square pyramid.

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**Chicodoodoo** · 13th July 2011, 04:56

Just some help visualizing the layout of the stack:

http://en.wikipedia.org/wiki/Square_pyramidal_number

**Akhenaten** · 13th July 2011, 06:17

OK Zook - I agree that the base 4 levels of the social pyramid as modeled have a NUMERICAL advantage over the upper levels (294 to 91 human units) HOWEVER many of the 294 human units in the base levels may be obligated, bound, or related by blood to the upper 91 units and therein lies the problem. The stability/rigidity of the pyramidical structure is based not only on the geometric form but on the interrelations between the human units at the various levels. In other words the pyramidical structure is vertically and horizontally integrated. In order to get a clearer idea of the dynamics involved a more detailed analysis of the vertical integration is needed. Moreover the degree of obligation and therefor connection of the human units in the lower levels to the upper levels would have to be weighted, assigning some exponential or other value to the units in the upper levels. For example, human units occupying higher levels in the upper levels would presumably be able to exercise much more control/influence over those at the lower levels. Thus, this model needs to be somewhat refined IMO. But it is headed in a great direction. A picture is worth a thousand words. Thanks for doing this! Akh

**ZookieMonster** · 13th July 2011, 07:34

@Chico Thanks ... excellent stuff. Shoulda went to Wiki first myself and used the shortcut formula: (n)(n+1)(2n+1)/6 ... woulda saved me the brute method

@Ahks Yup ... the static pyramid modeling suffers from not taking into account the dynamics of human interactions and shifting allegiances. I'm banking on Mother Hubbard's cupboard pushing the allegiances laterally and downward.

**Chicodoodoo** · 13th July 2011, 16:53

I should also mention that when I searched for information on arranging balls in the square pyramid shape, using Google, this thread showed up in the first page of the results! The original post couldn't have been more than a few hours old. Big Brother is definitely watching what we're up to here.