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Visual Math April 26, 2010

Posted by zach in Math, Science & Culture.
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Look at the mountains,  it may look as if it would be impossible to use a mathematical model to show the relationships between all the peaks and valleys. In the 1970’s the mathematician Benoit Mandelbrot discovered a formula that would allow a model to be made for natural objects such as mountains and branching structures such as limbs off a tree.  Fractals use geometric shapes that can continually be split into smaller fragments.    With an infinite number of points both real and imaginary art can be made using math.  Lets look at all the cool aspects of visual math fractals opened up.

Fractal showing the surface of the mountains.

Many many iterations

This first image is of a fractal with only five iterations. In the second image is the number of iterations when it is increased to 75, giving the image a much greater level of complexity.

The image below explores the edges around a Mandelbrot set.

Now lets explore how fractals can been shown in plants.

The branching structures of the California Oak tree and the smaller sprouts keep coming off the  Romanesco Broccoli.  The fractal concept is displayed in the broccoli when one closely examines how the whole is made up of multiple, smaller, repeating  parts.

Patterns that resemble fractals are not limited to the biological world.

Fractals can also be seen in architecture, this is most apparent in European Gothic Cathedrals.  The image below has a central spire with multiple other spires, repeatedly surrounding the central one, in varying sizes.

Snowflakes

Kock snowflake starts with an equilateral triangle, each iteration replaces the middle thrid of the line segment, with a part of line segments that form an equilateral bump.

Sierpinski triangle

Nature by Numbers March 31, 2010

Posted by isotopeeffect in Biology, Math.
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Nature by Numbers from Cristóbal Vila on Vimeo.

The math behind the movie – the Fibonacci sequence, the Golden Ratio, Voronoi tilings, Delaunay triangulation.

For about 1100 pages on similar topics, see On Growth and Form, written in 1945 (2nd edition) by D’Arcy Wentworth Thompson, a pioneering work in mathematical biology. The color PDF is a (free) 75 MB download.

Early computers February 8, 2010

Posted by isotopeeffect in Math.
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What did early computers look like?

Well, they looked a bit like this.

From around the beginning of the eighteenth century, when laborious numerical calculations began to be a regular part of scientific activity particularly in fields such as navigation and astronomy, the need arose for a person or group of persons to carry out these calculations carefully and as far as possible without error. Over time, this became a respected and important job.

A recent book by David Alan Grier – whose grandmother was one of these “human computers” – documents this history.

Brute-force numerical solution of equations is required when an analytical solution is not possible. An example of such a case is the “three-body problem” in mechanics, in which the trajectories of a group of objects (such as planets interacting through gravitational forces) cannot be solved analytically if the number of objects is greater than two. Although Edmund Halley had already predicted (in 1705, using Newtonian calculus) the return of the comet named after him, a much more accurate calculation was carried out in 1757 by Clairaut, Lalande, and Lepaute, who broke the orbit down into a series of small steps and incorporated the comet’s gravitational interactions with the orbits of the then-known major planets Jupiter and Saturn. (The primary source of error in their calculations would have been the neglect of the gravitational effects of Uranus and Neptune, which were unknown at the time.) This is one of the earliest known examples of parallel computing, still used today to make large numerically complex problems tractable in a reasonable amount of time by distributing the effort over multiple “computational nodes”.

Initially the profession of “computer” was mainly the province of men, and largely closed to women (although Lepaute was female), but over time this situation gradually changed. The Harvard astronomer Edward Charles Pickering (known, for example, for the “Pickering series” due to ionized helium, first observed in the spectrum of the hot O-type star ζ Puppis) employed a group of computers known as “Pickering’s Harem”, one of whom, Henrietta Swan Leavitt, went on to discover the relationship between the period and luminosity of Cepheid variables, without which work Edwin Hubble would not have been able to find the linear relationship between galactic redshift and distance that led to the understanding that we live in an expanding universe.

The prevalence of women in Pickering’s group was most likely a consequence of Harvard’s pay scale at the time, in which women were remunerated at half the standard rate for men.

The last days of the human computer came at the time of the Manhattan Project, where the complex calculations involved in the design of the atomic bomb were mostly accomplished by hand (carried out by the scientists’ wives, an interesting window into the gender hierarchy of intellectual work at the time). The ever-puckish Richard Feynman, who was responsible for overseeing the laboratory of human computers and who with Nicholas Metropolis was charged with installation of the first electronic computers using IBM punch cards, staged a showdown between human computers and punchcard-programmed machines. For two days the humans held their own; on the third day, the humans began to slow down, and the tireless machines pulled ahead.

There is an intriguing postscript to the era of human computers. The first six people charged with setting up programs on the ENIAC, one of the world’s first general-purpose computers (that is, not set up to solve a single, specific problem), were drawn from the job marketplace of human computers, and thus the world’s first professional computer programmers were women, specifically Kay McNulty, Betty Snyder, Marlyn Wescoff, Ruth Lichterman, Betty Jean Jennings, and Fran Bilas.

There’s a review of the Grier book here, and a very nice Scientific American article on the subject of the origins of computing here.

Welcome! January 21, 2010

Posted by Dr. O in Biology, Chemistry, Math, Physics, Science Education.
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Welcome to the Marian University, School of Mathematics & Science’s MU SCIENCE BLOG!

This blog is powered by Marian University science students including those majoring or minoring in biology, chemistry, math, or physics.

Please check out what the students have to say and ENJOY!