Another interesting property of the logarithmic spiral is revealed if you roll it along a horizontal line. This animation shows the curves traced by points on the spiral, and note that the very centre follows the path of a straight line. The angle between this line and the horizontal is called the pitch of the spiral, and for our spiral galaxy the pitch is around 12 degrees. [more] [code]
Love and Math – a beautiful read on the whimsy of math and how it serves as an equalizer for humanity
Awesome. To be seen in full screen on high definition. But to be accurate, I have to say that all the things featured in this video actually are Applied Mathematics, a crucial branch of Pure Mathematics, but not “just” Mathematics.
By Yann Pineill & Nicolas Lefaucheux
Seen in Gizmodo
The physics of beauty requires math. The sunflower has spirals of 21, 34, 55, 89, and - in very large sunflowers - 144 seeds. Each number is the sum of the two preceding numbers. This pattern seems to be everywhere: in pine needles and mollusk shells, in parrot beaks and spiral galaxies. After the fourteenth number, every number divided by the next highest number results in a sum that is the length-to-width ratio of what we call the golden mean, the basis for the Egyptian pyramids and the Greek Parthenon, for much of our art and even our music. In our own spiral-shaped inner ear’s cochlea, musical notes vibrate at a similar ratio.
The patterns of beauty repeat themselves, over and over. Yet the physics of beauty is enhanced by a self, a unique, self-organizing system. Scientists now know that a single flower is more responsive, more individual, than they had ever dreamed. Plants react to the world. Plants have ways of seeing, touching, tasting, smelling, and hearing.
Rooted in soil, a flower is always on the move. Sunflowers are famous for turning toward the sun, east in the morning, west in the afternoon. Light-sensitive cells in the stem “see” sunlight, and the stem’s growth orients the flower. Certain cells in a plant see the red end of the spectrum. Other cells see blue and green. Plants even see wavelengths we cannot see, such as ultraviolet.
Most plants respond to touch. The Venus’s-flytrap snaps shut. Stroking the tendril of a climbing pea will cause it to coil. Brushed by the wind, a seedling will thicken and shorten its growth. Touching a plant in various ways, at various times, can cause it to close its leaf pores, delay flower reproduction, increase metabolism, or produce more chlorophyll.
Plants are touchy-feely. They taste the world around them. Sunflowers use their roots to “taste” the surrounding soil as they search for nutrients. The roots of a sunflower can reach down eight feet, nibbling, evaluating, growing toward the best sources of food. The leaves of some plants can taste a caterpillar’s saliva. They “sniff” the compounds sent out by nearby damaged plants. Research suggests that some seeds taste or smell smoke, which triggers germination.
The right sound wave may also trigger germination. Sunflowers, like pea plants, seem to increase their growth when they hear sounds similar to but louder than the human speaking voice.
In other ways, flowers and pollinators find each other through sound. A tropical vine, pollinated by bats, uses a concave petal to reflect the bat’s sonar signal. The bat calls to the flower. The flower responds.
Sharman Apt Russell | Anatomy of A Rose: Exploring the Secret Life of Flowers [x]
Five great mathematicians and their contributions, in minimalist posters – the best thing since those minimalist posters celebrating pioneering women in science and philosophy’s major movements distilled in minimalist graphics.
Pair with 17 equations that changed the world.
String Theory is probably the best candidate for a Theory of Everything, including the so far elusive Quantum Gravity. I have to say, that as viewer of the the matter from the outside, I’ve gone from a deep skepticism, mostly because of the lack of empirical validation, to believe that is the theory with most likely to thrive. The absence of reasonable alternatives, the internal consistency of the theory itself, and the historical background of other theories that emerged from mere theoretical considerations (eg the Standard Model), is behind of this personal evolution.
We’ll see, but meanwhile, this site does a good job of popularizing the theory itself, as well as an excellent review of almost all Theoretical Physics. From the page:
This site provides a brief and entertaining introduction to string theory for the general public. Topics include quantum gravity, string physics, current research, future prospects, history and news. Kindly supported by The Royal Society and Oxford Physics.
If you like math check this out (via TED Blog)
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.
The Art Of Alexandra Roozen
Hand-drawn abstract monochrome art with algorithmic, patterned and digital feel.
More work can be found at the artist’s website here
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.