Archive for October, 2015

Mowing the Lawn in Spirals

Like a circle in a spiral / Like a wheel within a wheel / Never ending or beginning / On an ever-spinning reel.    The Windmills Of Your Mind

Broadly speaking, a spiral curve originates at a central point and gets further away (or closer) as it revolves around the point. Spirals abound in nature, being found at all scales from the whorls at our finger-tips to vast rotating spiral galaxies. The seeds in a sunflower are arranged in spiral segments. In the technical world, the grooves of a gramophone record and the coils of a watch balance-spring are spiral in form.

Left: Archimedean spiral. Centre: Fermat spiral. Right: Hyperbolic spiral.

Left: Archimedean spiral. Centre: Fermat spiral. Right: Hyperbolic spiral.

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A Few Wild Functions

Sine Function: {\mathbf{y=\sin x}}

The function {y=\sin x} is beautifully behaved, oscillating regularly along the entire real line {\mathbb{R}} (it is also well-behaved for complex {x} but we won’t consider that here).

The sine function, the essence of good behaviour.

The sine function, the essence of good behaviour.

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It’s a Small – Networked – World

Networks are everywhere in the modern world. They may be physical constructs, like the transport system or power grid, or more abstract entities like family trees or the World Wide Web. A network is a collection of nodes linked together, like cities connected by roads or people genetically related to each other. Such a system of nodes and links is what mathematicians call a graph [TM078; or search for “thatsmaths” at irishtimes.com ].

Detail of a Twitter communications network. Image from: https://dhs.stanford.edu/gephi-workshop/twitter-network-gallery/

Detail of a Twitter communications network.
Image from: https://dhs.stanford.edu/gephi-workshop/twitter-network-gallery/

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Which Way did the Bicycle Go?

“A bicycle, certainly, but not the bicycle,” said Holmes.

In Conan-Doyle’s short story The Adventure of the Priory School  Sherlock Holmes solved a mystery by deducing the direction of travel of a bicycle. His logic has been minutely examined in many studies, and it seems that in this case his reasoning fell below its normal level of brilliance.

As front wheel moves along the positive {x}-axis the back wheel, initially at {(0,a)}, follows a tractrix curve.

As front wheel moves along the positive x-axis the back wheel, initially at (0,a), follows a tractrix curve (see below).

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New Tricks: No Clicks

The readable surface of a Compact Disc has a spiral track over 5 km in length.

The readable surface of a Compact Disc has a spiral track over 5 km in length.

The quality of music recordings on compact discs or CDs is excellent. In the age of vinyl records, irritating clicks resulting from surface scratches were almost impossible to avoid. Modern recording media are largely free from this shortcoming. But this is curious: there are many reasons why CD music can be contaminated: dirt on the disc surface, flaws in the plastic substrate, errors in burning on the recording, scratches and fingerprints, and so on [TM077; or search for “thatsmaths” at irishtimes.com ]

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