- The Geometry -

So across the wisdom paths of the world's great spiritual traditions, prayer and related mystic practices are upheld as the direct way to transcend the limited individual self and identity with the primordial source undergirding all of reality. But what exactly is the transformative process catalyzed by sincere prayer that can open one to experience this infinite ground of being, truth, and beauty? How does the humble act of prayer beckon forth a revelation of the unseen cosmic creator? From Outward to Inward One perspective is that prayer operates by progressively shifting the locus of our awareness and identity from the outer, sensory-based consciousness to more subtle levels of mindfulness and being. In our normal waking state, we tend to be solely focused on the physical world around us through the limited gateways of the body's senses. We falsely take our separate selves encased in these bodily vehicles to be our entire reality.  But as most wisdom traditions assert, this ...

Fractal geometry and patterns of self-similar beauty underpin the universe across scales. Examples of fractal-generated shapes abound in the realm of nature and within our imagination as geometrical constructs lending proof that principles of space-filling beauty underpin the universe from the smallest atoms to the largest galaxies.  The Mandelbrot Set, one of the most iconic fractal structures, arises from a deceptively simple equation plotting points in the complex plane that correspond to the recursive formula Z → Z^2 + C. As this formula iterates infinitely through feedback loops exploring ever-finer scales, it generates visually striking tendrils and buds of intricate, spiraling shapes that seem to flow organically outward toward mathematical infinity. Perhaps even more astounding is that these geometrical patterns bear uncanny resemblance to natural forms we observe all around us - from trees and forests to rivers and coastlines to clouds, mountain ranges, spiral galaxies, an...

Sonoran Desert. These Hohokam Indian petroglyphs can be found in Arizona's Saguaro National Park atop of Signal Hill. United States.

    Transcription excerpt: Isolated systems tend to evolve towards a single equilibrium, a special state that has been the focus of many-body research for centuries. But when we look around us we don't see simple periodic patterns everywhere, the world is a bit more complex than this and behind this complexity is the fact that the dynamics of a system maybe the product of multiple different interacting forces, have multiple attractor states and be able to change between different attractors over time. Before we get into the theory lets take a few examples to try and illustrate the nature of nonlinear dynamic systems. A classical example given of this is a double pendulum; a simple pendulum with out a joint will follow the periodic and deterministic motion characteristic of linear systems with a single equilibrium that we discussed in the previous section. Now if we take this pendulum and put a joint in the middle of its arm so that it has two limbs instead of one, now the dyn...

  An Introduction to Optical Vortices and Topological Fluids of Light - https://youtu.be/JAobQq8aEWk?si=EXxCtuuemnVKXWvi   Hydrodynamic whirlpools have fascinated scientists for centuries, seeking to understand their individual structure, stability, and the ways in which they interact with one another. Who hasn’t marveled at tornadoes or watched as soap bubbles get sucked into the vortex of a bathtub drain? To reduce ideas to their essence, such fluid vortices are often considered in a two-dimensional setting where they amount to current swirling around a singularity. These, in turn, bear a striking resemblance to cross-sections of optical vortices that can be created with lasers, but with the propagation axis now treated as time. The vortex center is a then a dark spot about which the phase of light rotates like a barber shop sign. Such engineered light can therefore be interpreted as a two-dimensional, compressible fluid, and the vortices it harbors exhibit all ...