Helices are curves with a constant slope around a central axis, with the helix angle determining properties. Mechanical engineers use helix angles in designs, such as screws and conveyors, while springs store energy. Helices are found in biology, including the DNA double helix. Mathematically, a helix is the trace of a circle with increasing z-dimension.
A helix is a curve of constant slope surrounding a central axis, much like a spiral staircase. The helix angle is the tangent of the curve to the axis. Propellers 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 show an angle that determines how much biting or new material will be picked up by the screw with each turn. A greater force is required to rotate a device with a greater helix angle. Likewise, a screw with a larger angle will hold more securely.
Screw conveyors use screw conveyors to transport many granular or paste materials. Archimedes, the ancient Greek philosopher, is credited with inventing the cochlea. He used a large wooden screw carved into a hollowed out tree trunk. By turning the screw, the water could be moved uphill for irrigation purposes.
Springs illustrate another useful feature of propellers. In addition to the ability to transport matter and bind materials, propellers store energy in their geometric design that can be put to use. The energy of a pogo stick, automotive shock absorbers, or bed springs comes from the compression and subsequent expansion of the coil. The angle of the helix, as well as the material of construction, determines the force required to compress the spring.
Many examples of helices are found in biology. The turn of a pea plant’s tendrils follows a fixed helix angle, although the diameter of the turn can vary. Likewise, seashells and many other shells exhibit a fixed angle. Some have an increasing diameter with each addition of growth, while others have a fixed diameter that creates 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 angle of the double helix is so regular that the structure of the molecule has been identified using crystallographic techniques.
Mathematically, a helix is simply the trace of a circle with increasing z-dimension. The Cartesian coordinates are given by: x = r cos t, y = r sin t, z = ct; where r is the radius and 2 πc is the pitch or vertical distance between the loops. According to Lancret’s theorem, if r/c = a constant, the curve is a helix. In practice, the math of screw design is quite complicated, as many parameters are involved.
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