Your web-browser is very outdated, and as such, this website may not display properly. Please consider upgrading to a modern, faster and more secure browser. Click here to do so.

Mathmajik

Thinking Too Much
Jul 20 '14

spring-of-mathematics:

Fractal Patterns:

  1. Pitagoras Tree - Image: Fractal Tree by Ankur Pawar &  30/60/90 tree by Matt Cavanagh.
  2. Sierpinski Triangle - Image: Sierpinski Pattern by Matt Cavanagh.
  3. Recursive Box by Erik Wrebholt.
Jul 20 '14

The Mathematics of Brain Imaging from the Genetics Imaging Centre:

Computationally intense and innovative projects concentrate on the mathematical and engineering aspects of brain imaging. One broad area for algorithm development focuses on measuring patterns of brain connections in the living brain with diffusion imaging, while others areas focus on surface modeling and warping.

Diffusion Weighted Imaging - Using computer science algorithms and non-linear curve matching techniques to find the optimal path along tract representations of white matter integrity and map them to a common space for population studies.

Population mapping of white matter fiber tracts is an active area of study. Tractography methods can often lead to an abundance erroneous fibers and ill-defined regions of interest, making comparisons across large groups of individuals difficult. There are under development several complementary techniques for comparing white matter fiber tracts across individuals to better monitor developmental changes across time or to trace white regions altered with disease.

A novel automated tract clustering algorithm is used to label anatomically meaningful tracts from whole-brain tractography. The individual results from multiple atlases are fused and reliably extracted the 18 major tracts from diffusion weighted images. To perform population studies, we used a point-wise correspondence method to match and compare white matter tracts across subjects.

Read more: http://igc.ini.usc.edu/mathematics/

Jul 19 '14
Jul 19 '14
curiosamathematica:

The Google trend for the search query “quadratic formula”.
It repeats in the same pattern every year. Down in summer, up in September, down again in December and up again in spring time before going down again in the summer. And so it goes on forever.

curiosamathematica:

The Google trend for the search query “quadratic formula”.

It repeats in the same pattern every year. Down in summer, up in September, down again in December and up again in spring time before going down again in the summer. And so it goes on forever.

Jul 19 '14

Artist Leo Villareal - A study in Cellular Automata

Back in the 1970′s the mathematician John Conway first began exploring cellular automata, a fancy name for patterns that were generated by fairly simple mathematical rules applied over and over and over and over and over…. In Conway’s game, cells on a grid were occupied in whatever initial pattern you liked, and the rules for the next step stated roughly that
if too many or too few cells around a space were also occupied (over- or under-crowding), then the cell was emptied or left empty, and if a space was not over- or under-crowded, then it would become or stay occupied.

After Martin Gardner published an article in Scientific American about it the game became known as Conway’s Game of Life. Why that title? Because no matter what initial pattern you started with the patterns that emerged after a few rounds were sometimes eerily life-like. You could swear that you were watching a small society of ants or other small creatures moving around in patterns that you’d expect would require far more sophisticated rules than the actual ones used.

If you Google Conway’s Game of Life you’ll see many, many programs to run it as well as catalogues of patterns that emerge and persist and change, and this idea of “emergent behavior”—simple rules giving rise to sophisticated and nearly-impossible-to-predict patterns—has become an important theme in many parts of science. The artist Leo Villareal has now brought the idea to art. Villareal’s light sculptures are like a souped-up and beautifully tricked out Life. Made up of small LED lights operated by simple rules, Villareal has created art that has been featured in museums and galleries on the East Coast, West Coast, and other places in between. The reviews, of course, don’t really do this kind of art justice, but there are a number of videos (this, this, and this to start) available on YouTube that give some sense of Villareal’s work.

Source: blogs.shu.edu

Jul 19 '14

FUNDAMENTAL REGIONS IN MATHEMATICAL PATTERNS

From N. Simakoff: “Islamic Designs”.
Design from decorative painting on Bukhara architacture.

When you have a group acting on a set you can think of the whole set or design as made up of actions by the group on one little piece. This is how the computer works when it uses Terrazzo.

Read more: www.dartmouth.edu/~matc/math5.pattern

Jul 18 '14
mortisia:

Greek mathematician Constantin Caratheodory (Κωνσταντίνος Καραθεοδωρή) (1873-1950) (left) sitting with his father, brother-in-law George Streit, and his sister Julia. At Carlsbad, Czechoslovakia 1898. Original | Edit

mortisia:

Greek mathematician Constantin Caratheodory (Κωνσταντίνος Καραθεοδωρή) (1873-1950) (left) sitting with his father, brother-in-law George Streit, and his sister Julia. At Carlsbad, Czechoslovakia 1898. Original | Edit

Jul 17 '14
"Using the chain rule is like peeling an onion. You have to deal with every layer at a time and if it’s too big you’ll start crying."
Calculus professor (via mathprofessorquotes)
Jul 16 '14

(Source: beesandbombs)

Jul 16 '14

spring-of-mathematics:

Chaos Theory and Starling Flocks in Nature.

Chaos theory have many applications in meteorology, sociology, physics, engineering,etc…..Also, Chaotic behavior can also be observed in many natural systems. In a scientific context, the word chaos has a slightly different meaning than it does in its general usage as a state of confusion, lacking any order. Chaos, with reference to chaos theory, refers to an apparent lack of order in a system that nevertheless obeys particular laws or rules. Chaotic behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.

Starling flocks in Nature: When the starlings changes direction, speed, each of the other birds in the flock responds to the change and they do so nearly simultaneously regardless of the size of the flock.  In essence, information moves across the flock very quickly and with nearly no degradation. The researchers describe it as a high signal-to-noise ratio. The starlings are capable of extraordinary collective responses. These masses of birds move so synchronously, swiftly, and gracefully. (Shared from the article by Andrea Alfano)
The flock’ s movement is based on evasive maneuvers. There is safety in numbers, so the individual starlings do not scatter, but rather are able to move as an intelligent cloud, fainting away from a diving raptor, thousands of birds changing direction almost simultaneously and move in union. See more at: The incredible science behind starling murmurations by Jaymi Heimbuch & A Darwinian Dance by Grainger Hunt.


Image & Source: I shared at Fig.1:Logistic map  - Fig.2: Wildlife by Alan MacKenzie Photography - Fig.3: Murmurations.

Fig.4: Bifurcation diagram of the logistic map. Logistic systems bifurcate as their rates of change increase. - Fig.5: A Darwinian Dance - Fig.6: Starling Murmuration by midlander1231 on Flickr.- Fig.7: Starling, United Kingdom by John O Neill.

Jul 15 '14
Jul 15 '14
fieldemergence:

Mandelbulb variation. 3D fractals are very new. A few years ago the famous Mandelbrot set was projected in 3D with some additional calculations like folding coordinates in an imaginary 4th D plane. This resulted in Mandelbulb, an organic like structure. Mandelbox or The Amazing Box derived from this. These fractals are truly amazing. One cannot imagine the things that can be found in them. Art, architecture, landscapes, anything. The universe is most probably a natural fractal.

fieldemergence:

Mandelbulb variation. 3D fractals are very new. A few years ago the famous Mandelbrot set was projected in 3D with some additional calculations like folding coordinates in an imaginary 4th D plane. This resulted in Mandelbulb, an organic like structure. Mandelbox or The Amazing Box derived from this. These fractals are truly amazing. One cannot imagine the things that can be found in them. Art, architecture, landscapes, anything. The universe is most probably a natural fractal.

Jul 14 '14
Jul 14 '14
xysciences:

The Cubli, a self balancing cube.

xysciences:

The Cubli, a self balancing cube.

Map