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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. 

Hyperbolic Spiral: Equation: r=a/θ. It is also know as the reciprocal spiral and is the opposite of an Archimedian spiral. It begins at an infinite distance from the pole in the center (for θ starting from zero r = a/θ starts from infinity), and it winds faster and faster around as it approaches the pole; the distance from any point to the pole, following the curve, is infinite. 

The speed of light.

light speed is, well, a lot: about 670,616,629 mph. If you could travel at the speed of light, you could go around the Earth 7.5 times in one second

Insanity….

“In 1972, Edward Lorenz gave a presentation titled: “Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” The term “butterfly effect” was born. Lorenz, who was born on this day in 1917, was a pioneer of chaos theory, which analyzes how subtle changes in the initial conditions of a system can lead to widely differing outcomes. This 2013 Physics Today article examines Lorenz and the birth of chaos theory.” - Physics Today

Chaos at fifty by Adilson E. Motter and David K. Campbell is an article that lays out the discovery of chaos. I found it a very satisfying and informative read about dynamical systems, the butterfly effect, bifurcations, predictability and fractals. You should probably go check it out by clicking on the bolded title. 👍

“Chaos sets itself apart from other great revolutions in the physical sciences. In contrast to, say, relativity or quantum mechanics, chaos is not a theory of any particular physical phenomenon. Rather, it is a paradigm shift of all science, which provides a collection of concepts and methods to analyze a novel behavior that can arise in a wide range of disciplines.” - Chaos at fifty

Image above: “The Lorenz attractor, as revealed by the never-repeating trajectory of a single chaotic orbit. The spheres shown here represent iterations of the so-called Lorenz equations, calculated using the original parameters in Edward Lorenz’s seminal work. (Spheres are colored according to the iteration count.) From certain angles, the two lobes of the attractor resemble a butterfly, a coincidence that helped earn sensitive dependence on initial conditions its nickname—the butterfly effect. An animated visualization of the attractor is available here. (Image courtesy of Stefan Ganev.)”

The best vintage cars, hot rods, and kustoms

Perfect loop of the Eigenvalues of two vectors  

Hot Wheels CUSTOM ‘67 CAMARO “Rat Fink”

The cosmic calendar - from the very beginning of time itself (January the 1st) till now. Scaled down to a year. Dinosaurs would have died one day ago on this scale - check it out.

Old but gold 💛