Exponents

In our first encounter with exponents, we are taught that exponentiation is notation that simply means “multiply the number by itself some number of times.” For instance, 35 means the same thing as 3×3×3×3×3. For most people, this is a perfectly adequate definition. To a mathematician, it begs for exploration.

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Monday Mandelbrot Madness XVIII

Seahorses - click to enlarge.

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Monday Mandelbrot Madness XXVII

Circus Elephants II - click to enlarge.

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Monday Mandelbrot Madness XXVI

Julia in Purple - click to enlarge.

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The Hard Way

Here’s an interesting problem from a homework assignment in a class on stochastic processes: A woman standing at an intersection wants to cross the street. She will only cross if she feels that she can get all the way to the other side before a car arrives. Cars arrive randomly at a particular rate, but she can see them far enough off to determine if she has time to cross. On average, how long will the woman need to wait before she can cross the street?1Ross, Sheldon M., Introduction to Probability Models, Academic Press, 2010.

While the original problem provides more details (e.g. an exact description of the rate at which cars arrive, and a parameter that describes how cautious the woman is as a function of how long she requires the gap between cars to be), it is possible to learn something about mathematical thinking without going into those details.

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Monday Mandelbrot Madness XXV

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Spring Has Sprung

Today is the vernal equinox. More exactly, the vernal equinox will occur about half an hour before midnight in Greenwich, England; or at about 4:30 PM PDT (my time zone). At the moment of the equinox, the sun’s position in the sky will be the origin of the coordinate system that we use to track celestial objects.

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Welcome to Mercury

More than six and a half years ago, NASA launched the MESSENGER probe. Since its launch, it has traveled around the inner solar system, flying by Earth once, Venus twice, and Mercury three times. Yesterday, MESSENGER finished its long journey, and settled into orbit around Mercury, where it will continue to collect data and send it home for the next year (or, hopefully, longer).

This is the first time that any human launched probe has ever been put into orbit around Mercury, and is a phenomenal engineering and scientific achievement. Aside from the indomitable human desire to explore the universe, the sheer scale of this project boggles the mind, and cannot fail to impress.

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Monday Mandelbrot Madness XXIV

Overview - click to enlarge.

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The Mandelbrot Set—Part VI: Julia Sets

The Mandelbrot Set Series:

This is the sixth part in a series on Mandelbrot set fractals. Up until now, we have looked at fractals that are generated by examining the limit behaviour of points on the complex plane when a particular function is applied over and over again. A question naturally arises: what happens if we change the function?

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