The Cornú Spiral as a Golden Mean Construction

Abstract

The Cornú spiral, the graphical realization of the Fresnel Integrals, is shown to intrinsically contain the golden mean (φ=(1+√5)/2), which is characteristic of quasi-periodic atomic arrangements, Fibonacci sequences, biological structures and a number of mathematical objects.

Biscrolling Nanotube Sheets and Functional Guests into Yarns

“…"Archimedean" means that a sheet edge is buried deep in a scroll, and "dual Archimedean" means that sheet edges are buried in different interconnected scrolls, like in two dimensions for a Cornu spiral (Fig. 1, E and G) (16).…”

“…Observations suggest evidence for acceleration and heating along their long axis before they rapidly fade out of the chromospheric passbands, which may indicate heating to TR temperatures (15). We used coordinated observations of RBEs in the blue wing of Ha (-0.868 A ˚or -41 km/s) with the Narrowband Filter Imager of the Solar Optical Telescope (SOT) (17) onboard the Hinode solar physics mission (16), and found direct evidence of a strong correlation of RBEs with short-lived brightenings in a wide range of TR and coronal passbands observed with the Atmospheric Imaging Assembly (AIA) onboard the recently launched Solar Dynamics Observatory (SDO). Using an automated detection code, we found 2434 RBEs occurring in the active-region plage footpoints of coronal loops during a 1-hour-long time series (18) on 25 April 2010 (Fig.…”

Phase change in a diffracted wave: a Cornu spiral perspective

“…For example, this spiral is an ideal curve for keeping a constant speed in curves in railroads and highways [1], because its curvature changes linearly with the curve length. In fact, Castaño [2] elegantly showed that, if one imposes to a curve the condition that its curvature is proportional to the arc length, then the curve is a Cornu spiral. Other applications of this particular spiral go from roller-coaster loop shapes [3] to mobile robot trajectories [4].…”