GEOMETRIC UNIVERSE

This website is dedicated to the exploration of patterns in nature, geometry, and geometric model-making.

PATTERNS IN NATURE

They are evident at all scales of existence, from galactic spirals and starry globes to the tiny atoms and molecules that make up the tangible material of the universe. Even places almost completely devoid of matter in outer space are full of patterns - with light from distant stars continually passing through from all directions, more or less, and many other frequencies (x-rays, radio waves, etc.) of electromagnetic energy similarly passing through in waves. The space-time continuum itself is curved by gravitational forces - it has a shape.

Geometric patterns found in nature often have a universal aspect. Consider biological patterns, spiral sea shells for instance, that were refined over untold generations of evolution to become optimized. The Fibonacci number sequence is utilized extensively in natural structures, and is expressed beautifully as the Golden Section spiral or double spiral (as in sunflowers). These patterns therefore may represent a universal tendency to exist, as life forms in other places in Universe would also tend to develop optimized body architecture over thousands or millions of generations.

Reality is at minimum a 4-dimensional space-time experience, not a static thing. Therefore, as Buckminster Fuller pointed out, Universe is an ongoing process, a verb. Motion is the norm of existence, not bodies at rest. Therefore, dynamic patterns in nature are an important part of the designer's vocabulary. Movement and flow create some of the more interesting patterns, recognition of which can give one insight into Universal processes.

Universal patterns can reveal themselves almost wherever one chooses to look. The fractal branching of a tree is similar to blood vessels in the body or streams and rivers in the landscape. The pictures below show ice formations with fractal melt patterns. The second picture shows a stress fracture (white lines) that is superimposed onto a fractal melt pattern - 2 types of fractals joined by the act of freezing and thawing, with snowy highlights remaining in the ice to help create the contrast seen in the melt pattern.

Storm clouds and jet streams on Jupiter tend to resemble those on earth, as both are governed by the laws of fluid dynamics. Wherever water or gas flows, fluid dynamics tells us what kind of patterns to expect. These patterns will change based on a number of variables such as: flow rate, boundary conditions, viscosity and obstructions. A number of patterns from the field of fluid dynamics are visible in the photos below.

Streamlines and turbulent surface waves in water.

Streamlines are a function of laminar flow - layers of water moving at

different speeds due to friction from contact with rocks on stream bottom.

Geometry has many useful applications: it us used extensively in architecture, product design, engineering, applied physics, and surveying for instance. Geometry might seem to many to be overly abstract and annoying. I would submit that this is a problem of how the subject is taught, and what subject matter is emphasized. One of the best ways I have found to make the subject more accessible is through the use of geometric models.

GEOMETRIC MODELS

A relatively straight-forward way for most of us to learn geometry is to get some hands-on experience. Model making gives one ample opportunity to understand and visualize geometry

while also giving one a feeling for the structural properties of various shapes.

Waves, interference patterns, and bow-shaped shock waves in front of geese are evident behind them as a distinct set of waves, creating a relatively clear division between the turbulent wake directly behind the geese and the regular waves on the pond. In the image below, you can see a circular arc of water behind the duck (a drake, to be specific), a splash ring formed by water droplets. The transition from smooth flow over the drake's head to turbulent droplets in the neck area is visually interesting. The water drops tend to be round, like stars and planets. The roundness of so many things in Universe has to do with the efficient distribution of forces and material.

The Platonic Family

The Platonic polyhedra are like ABCDE in our alphabet. They appear universally on a microscopic scale in the form of crystals and molecules, quasi-crystals and microorganisms.

The simplest prisms from the prism family. Similar to the Platonic polyhedra, these are fundamental, universal shapes.

Tetrahedron expressed in various forms: linear (red), spherical/linear/concave (gold spherical outermost layer, red linear middle layer, silver concave or interstitial inner layer), sphere packing (white with black interstitial spaces shown with a paper model), and tensegrity (red and yellow). All of these are more or less equally valid, fundamental and Universal expressions of tetrahedral symmetry.

Nested Polyhedra Model based on Bucky Fuller's Cosmic Hierarchy of Volumetrically Interrelated Polyhedra.

To see Fuller's Cosmic Hierarchy concept in all it's glory:

http://www.rwgrayprojects.com/synergetics/plates/figs/plate09.html

TRANSFORMABLE GEOMETRIC MODELS

Models designed to inspire exploratory learning by Andrew Frank

Transformable Hypercube Model

Fractal Tetrahedron Model

For more examples of models designed for exploratory learning, along with state-of-the-art model making techniques, there is a link directly below to the complete text of my book, GEOMETRIC MODELING - An Advanced Guide to Creating Models in 3 Dimensions.

https://sites.google.com/site/geometricuniverse/geometric-modeling

You can also purchase a digital copy from amazon at the link below:

https://www.amazon.com/Geometric-Modeling-advanced-creating-dimensions-ebook/dp/B00W0MECFA

MINIMAL SURFACES MODELS