A helix is a curve of constant slope that circles around a central axis, much like a spiral staircase. The helix angle is the tangent of the curve compared to the axis. Helices are common in nature and in mechanical devices. The angle determines many other properties of a helix.
Mechanical engineers are concerned with the helix angle of many of their designs. Threaded screws and bolts display an angle that determines how much bite or new material will be gathered in by the screw with each turn. More force is required to turn a device with a greater helix angle. Likewise, a screw with a greater angle will hold more tightly.
Screw conveyors employ helices to transport many granular or paste-type materials. Archimedes, the ancient Greek philosopher, is credited with inventing the screw conveyor. He used a large carved wood screw inside a hollowed tree trunk. By turning the screw, water could be moved uphill for irrigation purposes.
Springs illustrate another useful characteristic of helices. In addition to the ability to convey mater and bind materials, helices store energy in their geometric design that can be used. The energy of a pogo stick, automobile shocks, or bed springs comes from the compression and subsequent expansion of the coil. The helix angle, as well as the material of construction, determines the force necessary to compress the spring.
Many examples of helices are found in biology. The turn of a pea plant’s tendrils follow a fixed helix angle, although the diameter of the turn may vary. Likewise, conch seashells and many other shells exhibit a fixed angle. Some have an increasing diameter with each growth addition, while others have a fixed diameter creating a long, tube shape.
Perhaps the most famous example of helices in nature is the double helix of the deoxyribose nucleic acid (DNA) molecule. DNA is the molecular basis of genetics. The unique double helix angle is so regular that the molecule’s structure was discerned using crystallography techniques.
Mathematically, a helix is simply the trace of a circle with an increasing z-dimension. The Cartesian coordinates are given by: x = r cos t, y = r sin t, z = c t; where r is the radius and 2 π c is the pitch or vertical distance between loops. According to Lancret’s theorem, if r/c = a constant, the curve is a helix. In practice, the mathematics of screw design is quite complicated, as many parameters are involved.