on a clear day you can count forever

Mathematical Spirals

According to Wikipedia, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point (similar to helices [plural for helix!] which are three-dimensional). Pictured above are some of the most important spirals of mathematics.

Logarithmic Spiral: Equation: r=ae^bθ. I must admit that these are my favorite! Logarithmic spirals are self-similar, basically meaning that the spiral maintains the same shape even as it grows. There are many examples of approximate logarithmic spirals in nature: the spiral arms of galaxies, the shape of nautilus shells, the approach of an insect to a light source, and more. Additionally, the awesome Mandelbrot set features some logarithmic spirals. Fun fact: the Fibonacci spiral is an approximation of the Golden spiral which is only a special case of the Logarithmic spiral. 

Fermat’s Spiral: Equation: r= ±θ^(½). This is a type of Archimedean spiral and is also known as the parabolic spiral. Fermat’s spiral plays a role in disk phyllotaxis (the arrangement of leaves in a plant system). 

Archimedean Spiral: Equation: r=a+bθ. The Archimedean spiral has the property that the distance between each successive turning of the spiral remains constant. This kind of spiral can have two arms (like in the Fermat’s spiral image), but pictured above is the one-armed version. 

Richard Feynman [2, pp. 71–72]

Differentiating Under the Integral Sign - PDF

What do you see? Some kind of band spinning around, or something twisting?

The motion of the first band shown above is ambiguous. Its more obvious that the band on the left is spinning, whereas the band on the right is twisting. The particular dynamics are made evident by watching the two black points on the bands edges. But without these defining features the motion of the gray band is indistinguishable between the two cases.

Perhaps the band is not moving at all, and instead its just the observer’s perspective rotating around this twisted yet inert band!

Can you think of any other shape that might exhibit this kind of symmetry?

Inspired by Bees&Bombs

Mathematica code:

"Fibonacci Sequence #3" Art print

Math and Science Week!

aseantoo submitted to medievalpoc:

Bhāskarāchārya

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Bhāskarāchārya / Bhāskara II (1114–1185) was an Indian mathematician and astronomer.

Among his many achievements are the following:

1. He was the first person to explain that when you divide by zero, the result is infinity.

2. He was also the first person to note that a positive number has two square roots - a positive and a negative one.

3. He described the principles of differential calculus 500 years before Leibniz and Newton. (He definitively came up with Rolle’s theorem half a millennium before Rolle himself.)

4. He calculated the length of the rotation of the earth around the sun to 365.2588 days - he was just off by 3 minutes.

Intriguingly, his treatise on arithmetic and geometry, Līlāvatī, is named after his daughter. He addresses her as an eager student:

Oh Līlāvatī, intelligent girl, if you understand addition and subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10, and 100, as well as [the remainder of] those when subtracted from 10000.” and “Fawn-eyed child Līlāvatī, tell me, how much is the number [resulting from] 135 multiplied by 12, if you understand multiplication by separate parts and by separate digits. And tell [me], beautiful one, how much is that product divided by the same multiplier?

These invocations have led some to surmise that Līlāvatī, too, was a mathematician.

Image from here: http://mathdept.ucr.edu/pdf/iwm1.pdf

Story of her introduction to math here: http://4go10tales.blogspot.co.uk/2012/06/lilavati.html