Mathematics and Functional Art

Mathematics and Functional Art

annarita ruberto does a great job of explaining the Euler spiral and gives us some stories behind the names and people who first described it. Did you know that these curves are used in looping roller coasters, railway lines and in the physics of diffraction grating? Don’t miss checking out the links at the bottom of the post. 

This post was chosen for #SoG+CuratorsChoice by moderator Rajini Rao who appreciates form and function in the natural world. 

Originally shared by annarita ruberto

Euler Spirals

An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids or Cornu spirals.

A Brief History

The first to take care of the curve was the great mathematician Leonhard Euler in 1744, solving a problem proposed by Jakob Bernoulli, and for this reason the curve is often known as the Euler spiral. More than a century later, in 1874, the French physicist Marie Alfred Cornu, used the curve in his research about the phenomena of diffraction, and for this reason it is sometimes called the Cornu spiral. In the early 1900s, the Italian mathematician Ernesto Cesaro finally gave her the name of clothoid, in honor of Clotho.

**(Greek: Κλωθώ) is the youngest of the Three Fates or Moirai – including her sisters Lachesis and Atropos, in ancient Greek mythology. Clotho was responsible for spinning the thread of human life. She also made major decisions, such as when a person was born, thus in effect controlling people’s lives. This power enabled her not only to choose who was born, but also to decide when gods or mortals were to be saved or put to death.(http://en.wikipedia.org/wiki/Clotho)

The animation from Matt Henderson (http://mi.eng.cam.ac.uk/~mh521/)

Draw a straight line, and then continue it for the same length but deflected by an angle. If you continue doing this you will eventually return to roughly where you started, having drawn out an approximation to a circle. But what happens if you increase the angle of deflection by a fixed amount at each step? The curve will spiral in on itself as the deflection increases, and then spiral out when the deflection exceeds a half-turn. These spiral flourishes are called Euler spirals.

Related links:

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cornu.html#c1

http://mathworld.wolfram.com/CornuSpiral.html

http://www.2dcurves.com/spiral/spirale.html

#mathematics #euler_spiral #cornu_spiral #math #educational_resources