By the time you get to the frontiers of math, the words to describe the concepts don’t really exist yet. Communicating these ideas is a bit like trying to explain a vacuum cleaner to someone who has never seen one, except you’re only allowed to use words that are four letters long or shorter.
Work by Jason Padgett, a man with Acquired Savant Syndrome who now sees all of reality as mathematical fractals describable by equations.
The beauty of numbers and their connection to the pure geometry of space time and the universe is shown in his fractal diagrams…He is currently studying how all fractals arise from limits and how E=MC2 is itself a fractal. When he first started drawing he had no traditional math training and could only draw what he saw as maths. Eventually a physicist saw his drawings and helped him get traditional mathematics training to be able to describe in equations the complex geometry of his drawings. He is currently a student studying mathematics in Washington state where he is learning traditional mathematics so he can better describe what he sees in a more traditional form. Many of the captions were written before he had any traditional maths training. His drawing of E=MC^2 is based on the structure of space time at the quantum level and is based on the concept that there is a physical limit to observation which is the Planck length. It shows how at the smallest level, the structure of space time is a fractal…So sit back and enjoy the beauty of naturally occurring mathematics in pure geometric form connecting E=MC2 (energy) to art. All are HAND DRAWN using only a pencil, ruler and compass.
SACRED GEOMETRY: An Introduction
What is meant by “sacred geometry”? Well, in its simplest terms it is the geometry which underlays all creation. There are repeating geometric forms which can be seen in all existence from the atomic to the cosmic. They range from the simple and familiar such as circles, squares, triangles, spheres, cubes to the more complex such as hexagons, pentagons, spirals, toroids, fractals, helix to fourth dimensional forms such as the hypercube and the hypersphere. These forms make up all of our visual reality and their repetition and their combinations speak to the nature of reality and the underlying symmetry and order of the universe which may be at first indiscernible to the naked eye.
Once we have learned to recognize these forms and to understand a bit about the mathematical relationship between them a whole new world dawns for us. You’ll recognize these patterns everywhere. You will see them in the arrangement of atoms within a crystal. In the forms of the virus and cell. In flowers, seeds and leaves. In the structure of an insects eye. You will see them in the cream in your coffee and in the shape of geological structures on the broad face of the Earth. You will see them in clouds and weather patterns. You will see them in the structure of planets, their orbits in galaxies and in the fourth dimensional shape of the universe itself.
The shapes are a language. They speak of relationships and patterns and those patterns are meaningful no matter what the scale. The spiral in your coffee cup has the same relationship as the spiral of the galaxy. You see these patterns are not “things” as we are accustomed to think of things that exist in our three dimensional realm. A baseball is a “thing” it approximates a sphere but it is not a sphere. A sphere is an ideal that exists independently of the crude world of our perception. However, because a baseball approximates a sphere we can use what we know of the ideal of a sphere to predict how a baseball will act in three dimensional space and in the fourth dimension of time. This is the world of ideals and their relationship to the outward world of forms.
☯ Samsaran ॐ
“To understand is to perceive patterns.”
— Isaiah Berlin
“The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
— G. H. Hardy
AXIOM & SIMULATION by Mark Dorf
Photographic series where natural landscapes are seen with quantified digital eye, reduced to nodes, polygons and lines:
AXIOM & SIMULATION examines the ways in which humans quantify and explore our surroundings by comparing artistic, scientific, and digital realism. As a developed global culture, we are constantly transforming physical space and objects into abstract non-physical thought to gain a greater understanding of composition and the inner workings of our surroundings. These transformations often take the form of mathematical or scientific interpretation. As a result of these changes, we can misinerpret or even lose all reference to the source: when the calculated representation is compared to its real counterpart, an arbitrary and disconnected relationship is created in which there is very little or no physical or visual connection resulting in questions of definition. Take for example a three-dimensional rendering of a mountainside. While observing the rendering, it holds a similar form to what we see in nature but has no physical connection to reality– it is merely a file on a computer that has no mass and only holds likeness to a memory. When translating the rendering into binary code, we see just 1’s and 0’s – a file creating the representation from a language composed of only two elements that have no grounding in the natural world. After all of these transformations, a new reality is created – one without an original referent, a copy with no absolute source. When observing these simulations and interpretations of our landscape within a single context or picture plane, ideas of accuracy, futility, and original experience arise.
Processing coding library for designing physical objects like the ones pictured above using geometric computation:
Codeable Objects is a library for Processing that enables novice coders, designers and artists to rapidly design, customize and construct an artifact using geometric computation and digital fabrication. The programming methods provided by the library allow the user to program a variety of structures and designs with simple code and geometry. When the user compiles their code, the software outputs tool paths based on their specifications, which can be used in conjunction with digital fabrication tools to build their object.
More about how this works can be found at High-Low Tech here
Wooden Clockwork Fractal Computer
Blog by Brent Thorne documenting the development of his clockwork wooden computer designed to calculate and draw fractals:
I’ve been working on this for a while now. Its a wooden computer that computes continuous self-similar fractals. I’ll post the working model of a general computer implemented in gears as soon as I get some laser cutter time to complete the counter/comparator unit.
How the hell is this supposed to work?
I could tell you that it took years and years of research and development to create a theory of computation that could be implemented in wood, but alias it would be untrue. The idea was formed after only a few reductions and one night when I couldn’t get to sleep. You see, computers are much simpler than your teachers might of taught you in school. You don’t even need the Boolean logic primitives to create a computer. These so called primitives are merely symbolic.
The most primitive computer is comprised of only two parts and from these two parts we can create all others. Those two parts are memory and a comparator. Some may claim that any practical computer must also have input and output, but that just is memory, or registers, memory again, or an ALU, nope that’s a comparator.
We can further delineate memory into two types, read-only and read-write. We need the read-write type of memory to store temporary values for comparison. For example, read-write memory could be a toggle or counter. Read-only memory is convenient for storing tables or a program, however these two examples are symbolic and not necessary for computation. An example of read-only memory is pegs in a disc, where the presents of a peg represents a symbol.
The true heart of a computer is the comparator. A comparator simply compares two values. One of those two values was read from memory previously and the other value is read at the current position in memory.
Now that we have our fundamental blocks we can start creating all the other complications that are common to modern computers.
You can find more information about the project at the blog here, including some videos of prototypes in action.
Auxin Flux Canalisation is a an algorithmic process developed by Adam Runions at the University of Calgary Algoritmic Botany group to model the morphogenesis of leaf venation. The key to the process is a simulation of the distribution and flux of auxin (a plant hormone) whose distribution contributes to many aspects of cell growth, division and specialization. The transport of auxin through a plant coordinates response to external stimuli in a way that does not require a central command system.
The beauty of the algorithm is (like flocking and other bottom-up algorithms) in the iteration of a very few simple steps. Nervous System have put together the most clear and concise description in explaining the generation of their beautiful Hyphae Lamp series. Go watch their video, it’s a great example of how an algorithm is the best representation of its own process.
Zachary Abel can’t help himself when it comes to making really cool mathematical sculptures out of everyday objects:
I think about math constantly, and I see and look for math in everything around me. Geometry in particular fascinates me, and I delight in discovering hidden patterns even in the most mundane of objects. By transforming often-overlooked household items into elaborate, mathematical sculptures, I hope to share this sense of excitement, curiosity, and beauty that a mathematical outlook has instilled in me. Maybe I’ll even be able to learn and teach some math along the way.
You can see more examples of his creative compulsion here
Breaking Wave I (2004)
Computer Generated Art.Image produced with a related series of Strange Attractor equations generated by 24 to 57 million computed points. The order of the points is recorded in each location during the computational process. This ordering becomes the basis for assigning a color to each point. The element of time is interpreted as color levels of red. The result is a two-dimensional image that has three-dimensional characteristics. A custom program was written in Microsoft VisualBasic for the computation of the points and the assignment of the color values.