Dear Yuri! This is my suggestion, only.There is a famous phrase: "in the beginning was the word." In Ancient Egypt was the first massive stones location experience on the ground. Egyptians observed the best stability stones. That is, in Ancient Egypt, "in the beginning was the case." And then already formed mathematical experience, which helped speed up the construction of the pyramids.

The term pyramid was first used in mathematics to describe 3-dimensional geometrical figure similar to pyramids in Egypt. This object had some magic meaning.

Recently this term was also used to describe financial pyramids as networks of certain type. In both cases, pyramid is a structure, where the mass of upper level is less than the middle, while the lower level has the highest mass.

If geometrical pyramid stays on a solid ground, probably there is no problem about its sustainability. Consider a pyramid as a network where the first node is connected with N lower nodes, each of lower with the next N, etc. In school I read a book about some applied mathematical problems. A case was described. In city X with population M there was an ad: “You can buy a bicycle with value of $100 for only for $20. You pay $20 and distribute 4 similar tickets promising a bike if you find 4 other people willing to buy it”. Clearly, this network has a natural limit for its growth. Because the sum 1+4+4x4+4x4x4+… will sometimes exceed M, and the people attracted to it at the last stage will pay $20 but stay without a bike, because at that moment there is no more population to attract.

That time I thought that it was funny. But I was surprised when such pyramids emerged in reality in 1990s in Russia after economic transition started. There is well known financial pyramid MMM based on similar principle. In network marketing there were similar structures. They were selling, for example, Herbalife. If a buyer found several more clients, the good became cheaper for him.

I want to hear further examples as well as theoretical discussions. Any hierarchical structure is a pyramid.

Dear Yuri! This is my suggestion, only.There is a famous phrase: "in the beginning was the word." In Ancient Egypt was the first massive stones location experience on the ground. Egyptians observed the best stability stones. That is, in Ancient Egypt, "in the beginning was the case." And then already formed mathematical experience, which helped speed up the construction of the pyramids.

Dear @Yuri, thanks for sharing the question. As it is very well known, pyramids are very stable structures, while financial pyramids are frauds, non-stable financial structures! We have had in Serbia 2 well know financial pyramids Jugoskandik and Dafiment. They were closely related with Government !

Multilevel marketing companies are also types of fraud pyramids.

Dear All,

I think the answer is YES, and many recently constructed buildings are shaped in design similar to pyramid, taking into account wind effect for example.

Regards, Emad

I don't know the word 'sustainable' refers to what in the question. 'Sustainable' has a definition as following: 

For example, "Green building (also known as green construction or sustainable building) refers to both a structure and the using of processes that are environmentally responsible and resource-efficient throughout a building's life-cycle." (from  https://en.wikipedia.org/wiki/Green_building). In this sense, I don't think that pyramid is a sustainable construction. A stable construction doesn't mean that it is a sustainable construction.

In my field, we have a term called 'sustainable agriculture'. It means: "In simplest terms, sustainable agriculture is the production of food, fiber, or other plant or animal products using farming techniques that protect the environment, public health, human communities, and animal welfare. This form of agriculture enables us to produce healthful food without compromising future generations' ability to do the same."  (http://www.sustainabletable.org/246/sustainable-agriculture-the-basics).

Dear Vanessa! Thank you for the sisters butterflies... Vladimir

Dear Ljubomir, you just confirmed that financial pyramids existed in different countries. Dear Yuan-Yeu, I am pretty aware what does sustainability mean in environmental economics. But I want to expand this concept to other structures created by humans. Only those that take into account interaction with all elements can survive for a long time.

I gave a hint about hierarchies, but nobody elaborated. Contrary to democratic principle in politics (all are equal), many structures (especially military and business) are organized as hierarchical. But informal structures (like mafia) are often similar. One boss - several bosses of 1st order answer to him - each controls group of bosses of 2nd order, etc. The last level boss interacts with population (or part of it). Ideal structure to rob a country. The problem is at the last level, where the pyramid structure has a border: it interacts with a structure of different type. Here the model can be a predator-prey game; see http://www.scholarpedia.org/article/Predator-prey_model . In general, there are cycles as solutions. But if robbery is too high, the population (of prey) may not survive.

In our interaction with nature we often have similar relations. Landlord exploits land worker, land-worker exploits nature. At some stage nature cannot replicate itself; it already happens; see http://steadystate.org/act/help-promote-supply-shock/

I think that static and dynamic pyramid are different objects w.r.t. structural stability.

It is clear that static pyramid is stable when we put it on its basis (not on top!) on some plane in gravity field. Good for architecture.

But the concept of pyramid is widely used for other applications. We have already talked about this example: http://www.fcac-acfc.gc.ca/Eng/resources/educationalPrograms/ft-of/Pages/fraud-1-5.aspx . Here it is more general:  https://creativesystemsthinking.wordpress.com/2014/10/26/capitalism-is-a-pyramid-scheme/ .

Consider now a dynamic pyramid. First make a grid with nodes, like in computer modelling of continuous media, and make interaction between neighboring nodes. If a signal starts propagating from the top, there will be no problem until it reaches the bottom. Then it either has to be absorbed or reflected - in both cases structural change. In the worst case this energy can destroy the bottom.

If the signal starts from the bottom, it is a bit better. Such scheme is used for cups in sport. On the 1st step 2^N teams participate, then 2^(N-1) winners compete again - in the end 1 winner of a cup. This is peaceful. But the focus of energy signal of small intensity emerging from the bottom can give very high amplification on the top.

So we see an asymmetry in all structures of pyramid type. What structures are symmetric? Perhaps ring (1-dimensional object in R2) or sphere (2-dimensional object in R3).