Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the Gener... more Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the General Evolution Research Group, and their 37-year partnership.
Chaotic fluctuations in the voltage available at different nodes in the electric power grid have ... more Chaotic fluctuations in the voltage available at different nodes in the electric power grid have been observed in connection with unwanted events, such as voltage collapse and power outage. The overall peak power in the grid also fluctuates unpredictably. In this report we consider techniques for controlling, or at least containing, the fluctuations after their appearance, as well as the problem of prediction of the peak load. In other reports, we will consider avoidance stretegies to prevent the appearance of these unwanted events. In all cases, the strategies we consider are based on the theory of complex dynamical systems: chaotic attractors and their bifurcations.
Following a visit in the Winter of 2007 to the Alhambra — a Moorish palace built around 1350 CE i... more Following a visit in the Winter of 2007 to the Alhambra — a Moorish palace built around 1350 CE in southern Spain — I have been digging up the roots of the Alhambra Theorem, which says that Moorish craftsmen in the Middle Ages knew that there were exactly 17 crystallographic groups in two dimensions. This fundamental result of modern group theory emerged into the mathematical literature only in 1891. Digging back from the Alhambra, we found an Ur source for the motifs of the Alhambra in the earliest wall-paintings of Çatal Hüyük, around 10 KYA (thousand years ago). Digging deeper led to the cave paintings of Upper Paleolithic Europe, beginning around 35 KYA. There appears to be a straight line of development from the Upper Paleolithic caves to the Neolithic villages of Anatolia, and on to the Alhambra. In this article we tell the story of the geometric patterns found in the Neolithic village of Çatal Hüyük.
Reporting a miraculous week of entheogens and sadhus in a temple in the montane jungle of the Him... more Reporting a miraculous week of entheogens and sadhus in a temple in the montane jungle of the Himalayan foothills in 1972.
Following a visit in the Winter of 2007 to the Alhambra — a Moorish palace built around 1300 CE i... more Following a visit in the Winter of 2007 to the Alhambra — a Moorish palace built around 1300 CE in the South of Spain — I have been digging up the roots of the Alhambra Theorem, which says that Moorish craftsmen in the Middle Ages knew that there were exactly 17 crystallographic groups. This fundamental result of modern group theory emerged into the mathematical literature only in 1891. Digging back from the Alhambra, we found an Ur source for the motifs of the Alhambra in the earliest Turkish carpets of Catal Hüyük, around 10 KYA (thousand years ago). Digging deeper led to the cave paintings of Upper Paleolithic Europe, beginning around 35 KYA. In this article we tell the story of the earliest geometric motifs we have found. This reveals the birth of geometric thinking in the ambiance of psychedelic shamanism -- religion, art, and mathematics were born together in the youth of our species.
This is a further progress report on the computer simulation of a mathematical model for a morphi... more This is a further progress report on the computer simulation of a mathematical model for a morphic field. The model is a two-dimensional lattice of oscillators derived from the d'Alembertian wave equation by spatial discretization. The communication is between two clamped objects inserted into the field. A change of shape in one of them sets off a transient wave which perturbs the boundary field of the other one after a brief delay. Unlike radio propagation, this is a static monopole transmission. In this second simulation, we clamp the field at the edges of a rectangular region. This note is an explanation of the companion video, which is a record of the experiment.
Complex dynamical systems models have been used (and misused) in service of the sustainability (o... more Complex dynamical systems models have been used (and misused) in service of the sustainability (or tragedy) of a commons. Their misuse results from the widespread ignorance of chaos theory. Here we consider this problem in general, and study the special case of the tragedy of the oceans in detail. We go on then to relate the mathematical model for sheries, due to Beverton and Holt in the 1940s, to the chaos revolution that followed. Finally, a potential role of education in commons management is proposed, in which participative simulation using NetLogo might be an integral part.
... Third Vienna Architecture Congress, November 3,1995: Conventional Thoughts on the Chaos of Eu... more ... Third Vienna Architecture Congress, November 3,1995: Conventional Thoughts on the Chaos of Europe The Mathematics of Chaos and the Urban Revolution by Ralph H ...
... RH Abraham, MS#89, Webometry #3, Preprint 1. Introduction. ... today permit a full exploratio... more ... RH Abraham, MS#89, Webometry #3, Preprint 1. Introduction. ... today permit a full exploration of the potential of this new space making them an expression of the values that we are attempting to define as we reinvent our society according to the new artistic and scientific givens. ...
... Dedicated to Courtney Sale Ross and Anders Holst Abstract. Certainly one of the larger bifurc... more ... Dedicated to Courtney Sale Ross and Anders Holst Abstract. Certainly one of the larger bifurcations of world cultural history must be the shift from medieval to modern science. According to many historians, the hinge point of that shift was Gali-leo's early works, beginning about ...
Complex dynamical systems theory is an evolution of nonlinear dynamics, developed for modeling an... more Complex dynamical systems theory is an evolution of nonlinear dynamics, developed for modeling and simulation of biological systems. Here, we speculate on the potential of this strategy for the emerging theory of social systems, for general evolution theory, and the implications for the future of our own planetary society.
Complex dynamical systems theory and system dynamics diverged at some point in the recent past, a... more Complex dynamical systems theory and system dynamics diverged at some point in the recent past, and should reunite. This is a concise introduction to the basic concepts of complex dynamical systems, in the context of the new mathematical theories of chaos and bifurcation.
Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the Gener... more Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the General Evolution Research Group, and their 37-year partnership.
Chaotic fluctuations in the voltage available at different nodes in the electric power grid have ... more Chaotic fluctuations in the voltage available at different nodes in the electric power grid have been observed in connection with unwanted events, such as voltage collapse and power outage. The overall peak power in the grid also fluctuates unpredictably. In this report we consider techniques for controlling, or at least containing, the fluctuations after their appearance, as well as the problem of prediction of the peak load. In other reports, we will consider avoidance stretegies to prevent the appearance of these unwanted events. In all cases, the strategies we consider are based on the theory of complex dynamical systems: chaotic attractors and their bifurcations.
Following a visit in the Winter of 2007 to the Alhambra — a Moorish palace built around 1350 CE i... more Following a visit in the Winter of 2007 to the Alhambra — a Moorish palace built around 1350 CE in southern Spain — I have been digging up the roots of the Alhambra Theorem, which says that Moorish craftsmen in the Middle Ages knew that there were exactly 17 crystallographic groups in two dimensions. This fundamental result of modern group theory emerged into the mathematical literature only in 1891. Digging back from the Alhambra, we found an Ur source for the motifs of the Alhambra in the earliest wall-paintings of Çatal Hüyük, around 10 KYA (thousand years ago). Digging deeper led to the cave paintings of Upper Paleolithic Europe, beginning around 35 KYA. There appears to be a straight line of development from the Upper Paleolithic caves to the Neolithic villages of Anatolia, and on to the Alhambra. In this article we tell the story of the geometric patterns found in the Neolithic village of Çatal Hüyük.
Reporting a miraculous week of entheogens and sadhus in a temple in the montane jungle of the Him... more Reporting a miraculous week of entheogens and sadhus in a temple in the montane jungle of the Himalayan foothills in 1972.
Following a visit in the Winter of 2007 to the Alhambra — a Moorish palace built around 1300 CE i... more Following a visit in the Winter of 2007 to the Alhambra — a Moorish palace built around 1300 CE in the South of Spain — I have been digging up the roots of the Alhambra Theorem, which says that Moorish craftsmen in the Middle Ages knew that there were exactly 17 crystallographic groups. This fundamental result of modern group theory emerged into the mathematical literature only in 1891. Digging back from the Alhambra, we found an Ur source for the motifs of the Alhambra in the earliest Turkish carpets of Catal Hüyük, around 10 KYA (thousand years ago). Digging deeper led to the cave paintings of Upper Paleolithic Europe, beginning around 35 KYA. In this article we tell the story of the earliest geometric motifs we have found. This reveals the birth of geometric thinking in the ambiance of psychedelic shamanism -- religion, art, and mathematics were born together in the youth of our species.
This is a further progress report on the computer simulation of a mathematical model for a morphi... more This is a further progress report on the computer simulation of a mathematical model for a morphic field. The model is a two-dimensional lattice of oscillators derived from the d'Alembertian wave equation by spatial discretization. The communication is between two clamped objects inserted into the field. A change of shape in one of them sets off a transient wave which perturbs the boundary field of the other one after a brief delay. Unlike radio propagation, this is a static monopole transmission. In this second simulation, we clamp the field at the edges of a rectangular region. This note is an explanation of the companion video, which is a record of the experiment.
Complex dynamical systems models have been used (and misused) in service of the sustainability (o... more Complex dynamical systems models have been used (and misused) in service of the sustainability (or tragedy) of a commons. Their misuse results from the widespread ignorance of chaos theory. Here we consider this problem in general, and study the special case of the tragedy of the oceans in detail. We go on then to relate the mathematical model for sheries, due to Beverton and Holt in the 1940s, to the chaos revolution that followed. Finally, a potential role of education in commons management is proposed, in which participative simulation using NetLogo might be an integral part.
... Third Vienna Architecture Congress, November 3,1995: Conventional Thoughts on the Chaos of Eu... more ... Third Vienna Architecture Congress, November 3,1995: Conventional Thoughts on the Chaos of Europe The Mathematics of Chaos and the Urban Revolution by Ralph H ...
... RH Abraham, MS#89, Webometry #3, Preprint 1. Introduction. ... today permit a full exploratio... more ... RH Abraham, MS#89, Webometry #3, Preprint 1. Introduction. ... today permit a full exploration of the potential of this new space making them an expression of the values that we are attempting to define as we reinvent our society according to the new artistic and scientific givens. ...
... Dedicated to Courtney Sale Ross and Anders Holst Abstract. Certainly one of the larger bifurc... more ... Dedicated to Courtney Sale Ross and Anders Holst Abstract. Certainly one of the larger bifurcations of world cultural history must be the shift from medieval to modern science. According to many historians, the hinge point of that shift was Gali-leo's early works, beginning about ...
Complex dynamical systems theory is an evolution of nonlinear dynamics, developed for modeling an... more Complex dynamical systems theory is an evolution of nonlinear dynamics, developed for modeling and simulation of biological systems. Here, we speculate on the potential of this strategy for the emerging theory of social systems, for general evolution theory, and the implications for the future of our own planetary society.
Complex dynamical systems theory and system dynamics diverged at some point in the recent past, a... more Complex dynamical systems theory and system dynamics diverged at some point in the recent past, and should reunite. This is a concise introduction to the basic concepts of complex dynamical systems, in the context of the new mathematical theories of chaos and bifurcation.
An expanded view of Thom's catastrophe theory with old and new applications.
The older applicatio... more An expanded view of Thom's catastrophe theory with old and new applications. The older applications include some by David Fowler, Christopher Zeeman, David Ruelle, and Floris Takens. The new ones, by this author, relate to the cymatics of Hans Jenny, and various functions of neurophysiology.
In the five years since the publication of Foundations of Mechanics there have been a number of r... more In the five years since the publication of Foundations of Mechanics there have been a number of results extending our understanding of the qualitative structure of conservative mechanical systems, and I would like to take this opportunity to bring up to date the picture presented in that book, and to correct several mistakes.
Two roots of the Platonic cosmology join in a subtle bifurcation in Ancient Greece. The Yogic tra... more Two roots of the Platonic cosmology join in a subtle bifurcation in Ancient Greece. The Yogic tradition of Vedic Sanskrit, perhaps transmitted by Pythagoras, and the Homeric of Archaic Greek, are yoked by Plato.
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The older applications include some by David Fowler, Christopher Zeeman, David Ruelle,
and Floris Takens.
The new ones, by this author, relate to the cymatics of Hans Jenny,
and various functions of neurophysiology.