The Ubiquitous Helix

October 3, 2016

A helix is a curve in three-dimensional space that's described by the simple set of parametric equations,

x = a cos(t)
y = a sin(t)
z = bt

In this case, a is the radius of the virtual cylinder on which the helix is wound, and the slope of the curve at any point is b/a. A right-hand helix is shown in the figure, and it's obvious how the equations can be modified to produce a left-hand helix.

A right-hand helix generated by the parametric equations shown. (Created using Gnumeric and Gnuplot.)

Probably the most familiar helix is on a machine bolt. In this case, the helix is an inclined plane, one of the simple machines that include the lever and the pulley. Renaissance mechanics considered the screw to be a simple machine, but the helical thread, itself, is an inclined plane.

A machine bolt. At this angle, it's not possible to determine which of the many cruciform head types this is. In any case, I would use a socket wrench, since it's easier. (Via Wikimedia Commons.)

Another common helical artifact is the coil spring. Springs can act in compression (see figure), tension, or torsion. While it's difficult to observe this, springs function by torsion of their wire. When designing a spring, the important material property is the shear modulus, not Young's modulus.

A compression spring. (Wikimedia Commons image by Frank Müller, modified.)

Nature was making use of the mechanical properties of helices long before humans. Cucumbers, squash, and similar plants, have helical tendrils that are attached to objects to elevate the leaves to regions with more sunlight. These tendrils are unusual in having a right hand helix on one end, and a left hand helix on the other, a fact noted in 1865 by Charles Darwin.

A tendril of an unidentified Cucurbitaceae-family plant. The right hand helix can be seen below the upper left hand helix. (Wikimedia Commons photo by Dirk van der Made, cropped.)

Darwin called the transitional region between these helices a "perversion." The word, perserion, in this case derives from the Latin adjective, "perversus," which means slanted, crooked, or cross-eyed. I wrote about these tendrils, and a recreation of their properties in a synthetic material, in two previous articles (Twisty Cucumbers, All Alike, September 7, 2012, and Let's Do The Twist, May 9, 2014).

A biological helix, DNA, is at the core of our being. In what's perhaps the scientific understatement of the last century, James Watson and Francis Crick wrote in their paper presenting the double helix structure of DNA,

"It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material."[1]

The two helices are composed of sugar phosphates, and these are linked together by molecules of cytosine, guanine, adenine, and thymine, Cytosine is always paired with guanine, and adenine is always paired with thymine.

The double helix structure of DNA. (Modified Wikimedia Commons image. Click for larger image)

This article is an opportunity for me to mention my favorite antenna, the axial mode (end fire) helical antenna, invented in 1946 by John Kraus (1910-2004). Kraus built his prototype antenna by winding a helix of seven turns of copper wire with a 12 centimeter circumference, this being the wavelength of his signal source.[1] Helical antennas are used in many high frequency applications, such as satellite communication (see figure), and long distance Wi-Fi communication.

An array of helical antennas. A typical helical antenna with ten spiral turns has a half-power beam width of about fifteen degrees. Helical antennas are suitable for high frequency transmission and reception in the UHF band and higher. At lower frequencies (longer wavelengths), the antenna dimensions are quite large. (Via Wikimedia Commons.)

The helix appears, also, in architecture, and not just for aesthetic reasons. Two examples of helical buildings are the Turning Torso, Malmö, Sweden, and the Cayan Tower, Dubai, United Arab Emirates, as shown in the figure. As you can imagine, construction of such a building is more complicated than construction of a traditional rectangular parallelpiped building, so why are such buildings being constructed?

Two examples of helical architecture, the Turning Torso, Malmö, Sweden, and the Cayan Tower, Dubai, United Arab Emirates. (left image by Artico2, and right image (modified) by Guilhem Vellut, both via Wikimedia Commons.)

The answer relates to the wind-loading on the structure, and a force concept known to all first-year physics students. When a force acts on a surface, the angle at which it acts is important. Maximum effect happens when the force acts perpendicular to the surface, and it decrease as the cosine of the angle with respect to the perpendicular. When wind hits a building face straight-on, there's a lot of force, but when it hits a glancing blow, there's very little force; so, a twisted surface has a reduced response to winds from any direction.

A helical design actually decreases construction cost, since less material is needed to resist the wind. It's reported that the helical design for the Shanghai Tower reduced wind load by 24%, and this allowed a $58 million saving in structural materials.[3] It's also reported that fifteen spiral buildings have been constructed in the past eleven years, and thirteen additional are under construction.[3]

My favorite helical food, although various pastry twists come in as a close second. (Via Wikimedia Commons.)

References:

1. J.D. Watson and F.H.C. Crick, "A Structure for Deoxyribose Nucleic Acid," Nature, vol. 171, no. 4356 (April 25, 1953), pp. 737-738.
2. J.D. Kraus, "Antennas," McGraw-Hill (New York, 1989), pp. 265f. (paperback edition, via Amazon).
3. Meg Miller, "Charting The Rise Of A New Trend In Skyscraper Design," Fast Company, August 25, 2016.