dynamic geometry

Chaos Theory.

Chaos is the science of surprises, of the nonlinear and the unpredictable. It teaches us to expect the unexpected. While most traditional science deals with supposedly predictable phenomena like gravity, electricity, or chemical reactions, Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on. These phenomena are often described by Fractal Mathematics, which captures the Infinite Complexity of Nature. Many natural objects exhibit Fractal Properties, including landscapes, clouds, trees, organs, rivers etc, and many of the systems in which we live exhibit Complex, Chaotic behavior. Recognizing the Chaotic, Fractal Nature of our world can give us new insight, power, and wisdom. 

Principles of Chaos

The Butterfly Effect: This effect grants the power to cause a hurricane in China to a butterfly flapping its wings in New Mexico. It may take a very long time, but the connection is real. If the butterfly had not flapped its wings at just the right point in space/time, the hurricane would not have happened. A more rigorous way to express this is that small changes in the initial conditions lead to drastic changes in the results. Our lives are an ongoing demonstration of this principle. 

Unpredictability: Because we can never know all the initial conditions of a complex system in sufficient (i.e. perfect) detail, we cannot hope to predict the ultimate fate of a complex system. Even slight errors in measuring the state of a system will be amplified dramatically, rendering any prediction useless. Since it is impossible to measure the effects of all the butterflies (etc) in the world, accurate long-range weather prediction will always remain impossible.

Order / Disorder: Chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways.

Mixing: Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed. 

Fractals: A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.

Maryam Mirzakhani

There are moments when we do not agree with what is happening around us… 

Maryam Mirzakhani (born May 3, 1977) is an Iranian-American mathematician and a professor of mathematics at Stanford University.

On 13 August 2014, Mirzakhani became both the first woman and the first Iranian honored with the Fields Medal, the most prestigious award in mathematics. The award committee cited her work in “the dynamics and geometry of Riemann surfaces and their moduli spaces”.

Her research topics include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry.


Elastic spheres can walk on water. Incited by public fascination and engineering application, water-skipping of rigid stones and spheres has received considerable study. While these objects can be coaxed to ricochet, elastic spheres demonstrate superior water-skipping ability, but little is known about the effect of large material compliance on water impact physics. Here we show that upon water impact, very compliant spheres naturally assume a disk-like geometry and dynamic orientation that are favourable for water-skipping. Experiments and numerical modelling reveal that the initial spherical shape evolves as elastic waves propagate through the material. We find that the skipping dynamics are governed by the wave propagation speed and by the ratio of material shear modulus to hydrodynamic pressure. With these insights, we explain why softer spheres skip more easily than stiffer ones. Our results advance understanding of fluid-elastic body interaction during water impact, which could benefit inflatable craft modelling and, more playfully, design of elastic aquatic toys.”

Star Tetrahedron.

The Star Tetrahedron is the geometric and energetic representation of the human body, heaven and earth, male and female, also called the Merkaba.

Many believe that the Merkaba represents the shape of our energy system in upper dimensions. By meditating on the Merkaba, you can more easily connect with your Higher Self and this can help to align your spiritual, mental, emotional and physical bodies. It can assist in the connection between the physical and ethereal bodies, allow us to see the psychological patterns and programs that may limit us, and is a constant reminder to remember our true, loving and divine nature. By meditating on the Merkaba Star Tetrahedron we are able to merge with Source, the Divine, All that is.

Unfold Human Consciousness.

A 3D map of Spiral Dynamics, an anthropologic theory on the evolution of mankind. We can witness an unfolding development, starting from its core 100,000 years ago, through the various stages of existence we have been through until today, and to our postulated manifestating future. The formation of our development rotates around the ideals of its time while it slowly unfolds into new states of being. The previous stages move towards the centre never losing their influence and adding to the grand scheme we call human consciousness.


An Expanding Bubble in Space

by NASA on The Commons
Astronomers, using the Wide Field Planetary Camera 2 on board NASA’s Hubble Space Telescope in October and November 1997 and April 1999, imaged the Bubble Nebula (NGC 7635) with unprecedented clarity. For the first time, they were able to understand the geometry and dynamics of this very complicated system. Earlier pictures taken of the nebula with the Wide Field Planetary Camera 1 left many issues unanswered, as the data could not be fully calibrated for scientific use. In addition, those data never imaged the enigmatic inner structure presented here. The remarkably spherical “Bubble” marks the boundary between an intense wind of particles from the star and the more quiescent interior of the nebula. Research Team: Donald Walter (South Carolina State University), Paul Scowen, Jeff Hester, Brian Moore (Arizona State University), Reggie Dufour, Patrick Hartigan and Brent Buckalew (Rice University). Image # : PR00-04 Date: January 13, 2000

First woman to win Nobel Prize of mathematics!

Maryam Mirzakhani, an Iranian born mathematician became the first woman to win Field’s Medal, widely considered as Nobel Prize of mathematics, in the year 2014. She is a Harvard-educated mathematician and professor at Stanford University in California.

She was honored for her work in the  fields of geometry and dynamical systems, particularly in understanding the symmetry of curved surfaces such as spheres.