It took one of the world's most powerful supercomputers five days to model a simple childhood past time: Andreas Bastian Researchers at the Lawrence Berkeley National Laboratory and at the University of California Berkeley have mathematically described the evolution of a cluster of bubbles. The research was published May 10, in the journal Science. Bubbles and foams have been notoriously difficult to model mathematically.
Mathematics can be defined as the study of abstract patterns. Numbers, of course, are one of the first examples of an abstraction: But for mathematicians, patterns of even greater abstraction are found all around us.
Symmetries form another familiar example of patterns.
The human visual system is especially sensitive to vertical mirror symmetry, as found approximately in the faces and bodies of most animals. Geometric symmetries like this are only the tip of the mathematical iceberg. Mathematicians find symmetries everywhere, for instance in number systems, or in equations and their solutions.
Different symmetries of an object can be combined: It turns out that the rules for combining symmetries the group laws of abstract algebra are the same no matter whether the symmetries are geometric or not.
Furthermore, these rules turn out to be similar Bubble clusters for essays the rules of ordinary arithmetic. Mathematics and Art Mathematics can be related to art in many ways.
One can study art mathematically, looking for symmetries or other relations in the construction of a painting or sculpture. Conversely, mathematical algorithms can be used to help create art: Famously, perspective drawing has a mathematical basis, and is a good example of how different the human brain is from a digital computer.
It is trivial for a computer to apply the rules of perspective to project a three-dimensional model world to a two-dimensional drawing, while human artists often have difficulty applying these rules.
On the other hand, we effortlessly use our visual system to reconstruct a three-dimensional model of the world around us from the two-dimensional images presented on our retinas. We thereby solve a very difficult even ill-posed problem that the best computers still have trouble with.
Clearly, it is of great evolutionary advantage to be able to build accurate three-dimensional mental models of what we see, and we devote significant brain-power to doing this. The opposite task, converting this three-dimensional model back into a two-dimensional drawing, is mathematically simple, but gives little survival advantage, and not surprisingly is a difficult task for us, requiring conscious effort.
Mathematics in the Natural Sciences Perhaps the Pythagoreans were the first to suggest that "at its deepest level, reality is mathematical in nature. The flourishing of science in Europe during the Renaissance was made possible by increasing knowledge of mathematics.
Galileo echoed Pythagoras when he observed that the laws of nature are "written in the language of mathematics. Eugene Wigner described this as "the unreasonable effectiveness of mathematics in the natural sciences," in his famous essay.
Optimization Problems Perhaps one reason for this effectiveness of mathematics is that many laws of physics can be expressed in terms of minimizing free energy or minimizing action. These optimization problems have mathematical solutions.
In general, surface energies become more important than bulk energies at small scales: A small bug can easily walk on water, because the force of surface tension outweighs gravity at that small scale. Thus problems about real-world materials, especially those concerning structure at small scales, can often be cast in the form of optimizing some feature of shape.
The system minimizes some energy depending on the shape of a surface or sometimes a curve describing the material's structure.
Mathematically, we obtain an optimization problem for some geometric energy. A classical example is the soap bubble which minimizes its area while enclosing a fixed volume; this leads to the study of the more general constant-mean-curvature surfaces found in bubble-clusters and foams.
Biological cell membranes, on the other hand, are more complicated bilayer surfaces, and seem to minimize an elastic bending energy known as the Willmore energy.
My own mathematical research concerns geometric optimization problems like these.The median bubble score of the 16 people who had lived in at the age of 10 was To give you an idea how low is, a score of 12 puts one at the 3rd percentile of the entire sample.
A look at trypophobia, a condition where a person has a fear of clusters of small holes. Included is detail on what triggers the condition and treatments.
Nov 02, · Originally published in March as a guest blog on the No More Marking site. monstermanfilm.com I’m working with three clusters of schools focusing on Year 6 writing, two clusters focusing on Year 2 writing and one cluster focusing on Year 4 writing.
Brainstorm Web & Template Brainstorming is an activity that stimulates the mind and produces multiple ideas around the topic.
There are many variations to brainstorming activities. Living in bubbles is the natural state of affairs for human beings.
People seek out similarities in their marriages, workplaces, neighborhoods, and peer groups. and the rich cluster in another. In industry, clusters of bubbles belong to a class of materials called foams. From ocean froth, to soapy detergents, fire retardants, and bicycle helmets, foams play an large role in our day to day lives.