# What is a Helix?

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A helix is a type of space curve, i.e. a smooth curve in three-dimensional space. A curve (with endpoints) is a continuous function whose domain is the unit interval (0,1). In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most common cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a planar curve) or the 3-dimensional space (space curve). Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, whose Elements are the earliest known systematic discussion of geometry. Helix is characterized by the fact that the tangent line (a line that touches a circle at exactly one point and which makes a right angle with the circle’s radius) at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for example, a spiral ramp – is called a ‘helicoid’ (an object of spiral or helical shape). Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word ‘helix’ comes from the Greek word "twisted, curved". In mathematics, a helix is a curve in 3-dimensional space.

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### What are the Different Types of Helix?

Helices can either be right or left-handed depending on the turn of the curve around the center axis. The orientation of a helix can be determined by looking at it along its length.

• Right-handed Helix: If the movement away from the observer is clockwise, then the helix is right-handed. Most hardware screw threads (a screw thread, often shortened to thread, is a helical structure used to convert between rotational and linear movement or force) are right-handed helices. The alpha helix in biology as well as the A (A-DNA is one of the many possible double helical structures of DNA) and B (double helix refers to the structure formed by double-stranded molecules of nucleic acids such as DNA and RNA) forms of DNA are also right-handed helices.
• Alpha-helix: In 1951, two biologists, Linus Pauling and Robert B. Corey of the California Institute of Technology, were eager to solve the mystery of the shape of DNA. They demonstrated that accurate coiling of the protein keratin allowed hydrogen bonds to form and stabilize the structure. The Helix they discovered was an alpha helix, which is an example of a right-handed helix. They postulated that the helical structure helps to maintain the shape of the molecule and gives it a much more stable structure than when it is unwound. Polypeptides are proteins which are made up of chains of amino acids. Within proteins, each turn of the alpha helix takes up about 3.6 amino acids of the polypeptide chain. The helical shape is maintained through hydrogen bonds forming between the amino group of one amino acid and the oxygen of the third amino acid beyond it in the chain.
• Left-handed Helix: If the movement is in the anti-clockwise direction, then a left-handed helix is being observed. The Z form (Z-DNA is one of the many possible double helical structures of DNA) of DNA is left-handed.

The handedness, or chirality, cannot be changed by looking at it from a different perspective, but is an inherent property of the helix.

### What is the Pitch of a Helix?

The pitch of a helix is the width of one complete helix turn, measured parallel to the axis of the helix.

• A double helix consists of two (typically congruent) helices with the same axis, differing by a translation along the axis, which may or may not measure half the pitch. For instance, DNA does not usually exist as a single molecule, but instead as a pair of molecules that are held tightly together. These two long strands entwine like vines, in the shape of a double helix. The nucleotide repeats contain both the segment of the backbone of the molecule, which holds the chain together, and a base, which interacts with the other DNA strand in the helix. In a double helix the direction of the nucleotides in one strand is opposite to their direction in the other strand; the strands are anti-parallel. The two strands of DNA in a double helix can therefore be pulled apart like a zipper, either by a mechanical force or high temperature. As a result of this complementarily, all the information in the double-stranded sequence of a DNA helix is duplicated on each strand, which is vital in DNA replication. As DNA polymerases can only extend a DNA strand in a 5’ to 3’ direction, different mechanisms are used to copy the anti-parallel strands of the double helix.
• Triple helix: In biology, there are also complex proteins that have helical arrangements, including collagen, which is a triple helix. Collagen is made up of three polypeptide chains, each an alpha helix, all wound around each other.
• A conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of the angle indicating direction from the axis. An example is the Corkscrew roller coaster at Cedar Point amusement park.
• A circular helix, (i.e. one with constant radius) has constant band curvature (curvature refers to any of a number of loosely related concepts in different areas of geometry) and constant torsion (torsion of a curve measures how sharply it is twisting).
• A curve is called a general helix or cylindrical helix, if its tangent makes a constant angle with a fixed line in space. A curve is a general helix if and only if the ratio of curvature to torsion is constant.

### What are the Examples of Helix?

In music, pitch space (pitch spaces model relationships between pitches) is often modeled with helices or double helices, most often extending out of a circle such as the circle of fifths, so as to represent octave equivalency. The circle of fifths (or circle of fourths) shows the relationships among the twelve tones of the chromatic scale (a musical scale with twelve pitches), their corresponding key signatures (a key signature is a series of sharp or flat symbols), and the associated major and minor keys. In music, an octave is the interval between one musical pitch and another with half or double its frequency.

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