mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a conchospiral a specific type of

space spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

(a '' conical spiral''), whose floor projection is a

logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...

. Conchospirals are used in biology for modelling snail shells, and flight paths of insects and in

electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

for the construction of

antennas In radio engineering, an antenna or aerial is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies a ...

Parameterization

cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...

, the conchospiral is described by the parametric equations: :

r=\mu^t a

\theta=t

z=\mu^t c.

The projection of a conchospiral on the

(r,\theta)

plane is a

. The parameter

\mu

controls the opening angle of the projected spiral, while the parameter

c

controls the slope of the cone on which the curve lies.

History

The name "conchospiral" was given to these curves by 19th-century German mineralogist

Georg Amadeus Carl Friedrich Naumann Georg Amadeus Carl Friedrich Naumann (30 May 1797 – 26 November 1873), also known as Karl Friedrich Naumann, was a German mineralogist and geologist. The crater Naumann on the Moon is named after him. Life Naumann was born at Dresden, the s ...

, in his study of the shapes of sea shells.

Applications

The conchospiral has been used in the design for

radio antenna In radio engineering, an antenna or aerial is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies an ...

s. In this application, it has the advantage of producing a radio beam in a single direction, towards the apex of the cone.

References

External links

*{{mathworld, title=Concho-Spiral, urlname=Concho-Spiral Spirals