Soma cube

The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933[1] during a lecture on quantum mechanics conducted by Werner Heisenberg. Seven pieces made out of unit cubes must be assembled into a 3×3×3 cube. The pieces can also be used to make a variety of other 3D shapes. The pieces of […]

The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933[1] during a lecture on quantum mechanics conducted by Werner Heisenberg. Seven pieces made out of unit cubes must be assembled into a 3×3×3 cube. The pieces can also be used to make a variety of other 3D shapes.

The pieces of the Soma cube consist of all possible combinations of three or four unit cubes, joined at their faces, such that at least one inside corner is formed. There is one combination of three cubes that satisfies this condition, and six combinations of four cubes that satisfy this condition, of which two are mirror images of each other (see Chirality). Thus, 3 + (6 × 4) is 27, which is exactly the number of cells in a 3×3×3 cube.

The Soma cube has been discussed in detail by Martin Gardner and John Horton Conway, and the book Winning Ways for your Mathematical Plays contains a detailed analysis of the Soma cube problem. There are 240 distinct solutions of the Soma cube puzzle, excluding rotations and reflections: these are easily generated by a simple recursive backtracking search computer program similar to that used for the eight queens puzzle.

The seven Soma pieces are six polycubes of order four and one of order three:

  • Soma-ra.svg Piece 1, or “V”.
  • Soma-l.svg Piece 2, or “L”: a row of three blocks with one added below the left side.
  • Soma-t.svg Piece 3, or “T”: a row of three blocks with one added below the center.
  • Soma-s.svg Piece 4, or “Z”: bent triomino with block placed on outside of clockwise side.
  • Soma-rscrew.svg Piece 5, or “A”: unit cube placed on top of clockwise side. Chiral in 3D.
  • Soma-lscrew.svg Piece 6, or “B”: unit cube placed on top of anticlockwise side. Chiral in 3D.
  • Soma-branch.svg Piece 7, or “P”: unit cube placed on bend. Not chiral in 3D.[2]

Dunbar’s number

Dunbar’s number is a suggested cognitive limit to the number of people with whom one can maintain stable social relationships. These are relationships in which an individualknows who each person is and how each person relates to every other person.[1][2][3][4][5][6] This number was first proposed in the 1990s by British anthropologist Robin Dunbar, who found […]

Dunbar’s number is a suggested cognitive limit to the number of people with whom one can maintain stable social relationships. These are relationships in which an individualknows who each person is and how each person relates to every other person.[1][2][3][4][5][6] This number was first proposed in the 1990s by British anthropologist Robin Dunbar, who found a correlation between primate brain size and average social group size.[7] By using the average human brain size and extrapolating from the results of primates, he proposed that humans can only comfortably maintain 150 stable relationships.[8] Proponents assert that numbers larger than this generally require more restrictive rules, laws, and enforced norms to maintain a stable, cohesive group. It has been proposed to lie between 100 and 250, with a commonly used value of 150.[9][10] 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, nor people just generally known with a lack of persistent social relationship, a number which might be much higher and likely depends on long-term memory size.

Dunbar 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 schoolfriends, with whom a person would want to reacquaint himself if they met again.[11]

Dunbar has argued that 150 would be the mean group size only for communities with a very high incentive to remain together. For a group of this size to remain cohesive, Dunbar speculated that as much as 42% of the group’s time would have to be devoted to social grooming. Correspondingly, only groups under intense survival pressure.

Dunbar, in Grooming, Gossip, and the Evolution of Language, proposes furthermore that language may have arisen as a “cheap” means of social grooming, allowing early humans to maintain social cohesion efficiently. Without language, Dunbar speculates, humans would have to expend nearly half their time on social grooming, which would have made productive, cooperative effort nearly impossible. Language may have allowed societies to remain cohesive, while reducing the need for physical and social intimacy.[12]

Dunbar’s number has since become of interest in anthropology, evolutionary psychology,[13] statistics, and business management. For example, developers of social software are interested in it, as they need to know the size of social networks their software needs to take into account; and in the modern military, operational psychologists seek such data to support or refute policies related to maintaining or improving unit cohesion and morale. A recent study has suggested that Dunbar’s number is applicable to online social networks[14][15] and communication networks (mobile phone).[16]

Philip Lieberman argues that since band societies of approximately 30-50 people are bounded by nutritional limitations to what group sizes can be fed without at least rudimentary agriculture, big human brains consuming more nutrients than ape brains, group sizes of approximately 150 cannot have been selected for in paleolithic humans.[20]Brains much smaller than human or even mammalian brains are also known to be able to support social relationships, including social insects with hierachies where each individual knows its place (such as the paper wasp with its societies of approximately 80 individuals [21]) and computer-simulated virtual autonomous agents with simple reaction programming emulating what is referred to in primatology as “ape politics”.[22]

Thermoeconomics

Thermoeconomics, also referred to as biophysical economics, is a school of heterodox economics that applies the laws of thermodynamics to economic theory.[1] The term “thermoeconomics” was coined in 1962 by American engineer Myron Tribus,[2][3][4]Thermoeconomics can be thought of as the statistical physics of economic value. Thermoeconomics is based on the proposition that the role of […]

Thermoeconomics, also referred to as biophysical economics, is a school of heterodox economics that applies the laws of thermodynamics to economic theory.[1] The term “thermoeconomics” was coined in 1962 by American engineer Myron Tribus,[2][3][4]Thermoeconomics can be thought of as the statistical physics of economic value.

Thermoeconomics is based on the proposition that the role of energy in biological evolution should be defined and understood not through the second law of thermodynamics but in terms of such economic criteria as productivity, efficiency, and especially the costs and benefits (or profitability) of the various mechanisms for capturing and utilizing available energy to build biomass and do work.[6][7]

Thermodynamics

Thermoeconomists maintain that human economic systems can be modeled as thermodynamic systems. Then, based on this premise, theoretical economic analogs of the first and second laws of thermodynamics are developed.[8] In addition, the thermodynamic quantity exergy, i.e. measure of the useful work energy of a system, is one measure of value.

System dynamics

System dynamics is a computer-aided approach to policy analysis and design. It applies to dynamic problems arising in complex social, managerial, economic, or ecological systems — literally any dynamic systems characterized by interdependence, mutual interaction, information feedback, and circular causality. The three most commonly used software packages are listed below in alphabetical order.  Additional tools […]

System dynamics is a computer-aided approach to policy analysis and design. It applies to dynamic problems arising in complex social, managerial, economic, or ecological systems — literally any dynamic systems characterized by interdependence, mutual interaction, information feedback, and circular causality.

The three most commonly used software packages are listed below in alphabetical order.  Additional tools that support model construction are noted at the end.

iThink® and STELLA® are two names for one model development platform published by isee systems. The software is available in different configurations under a commercial license for Windows and Macintosh computers. Educational licenses and a free runtime version of the software are available.

Powersim Studio is available in a number of different configurations from Powersim Software. The software is available under  commercial license and runs under Windows. Educational licenses and options for publishing standalone model packages are available. A new free version, Studio Express is now available.

Vensim® is available in a number of different configurations from Ventana Systems, Inc. The software is available under a commercial license and runs on Windows and the Macintosh. Educational licenses, including a configuration of the software that is free for educational use, and a free runtime version of the software are available.

See Also: There are a number of other products that can be used to construct models. These include: Anylogic, Goldsim, Berkely Madonna, Sysdea and SimGua under related methodologies and MyStrategy under pedagogical tools.

Simantics System Dynamics is a ready-to-use system dynamics modelling and simulation software application for understanding different organizations, markets and other complex systems and their dynamic behavior.

ASCEND is a free open-source software program for solving small to very large mathematical models. ASCEND can solve systems of non-linear equations, linear and nonlinear optimisation problems, and dynamic systems expressed in the form of differential/algebraic equations.

2-adic metric

Published on Aug 13, 2015
An exploration of infinite sums, from convergent to divergent, including a brief introduction to the 2-adic metric, all themed on that cycle between discovery and invention in math.

Published on Aug 13, 2015
An exploration of infinite sums, from convergent to divergent, including a brief introduction to the 2-adic metric, all themed on that cycle between discovery and invention in math.

The voting paradox

The voting paradox (also known as Condorcet’s paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i.e., not transitive), even if the preferences of individual voters are not cyclic. This is paradoxical, because it means that majority […]

The voting paradox (also known as Condorcet’s paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i.e., not transitive), even if the preferences of individual voters are not cyclic. This is paradoxical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.

Thus an expectation that transitivity on the part of all individuals’ preferences should result in transitivity of societal preferences is an example of a fallacy of composition.