A helix (pl: helixes or helices) is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line 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.[1] 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 ἕλιξ, "twisted, curved".[2]
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Types [link]
Helices can be either right-handed or left-handed. With the line of sight along the helix's axis, if a clockwise screwing motion ملف:Elica-conica.gifmoves the helix away from the observer, then it is called a right-handed helix; if towards the observer then it is a left-handed helix. Handedness (or chirality) is a property of the helix, not of the perspective: a right-handed helix cannot be turned or flipped to look like a left-handed one unless it is viewed in a mirror, and vice versa.
Most hardware screw threads are right-handed helices. The alpha helix in biology as well as the A and B forms of DNA are also right-handed helices. The Z form of DNA is left-handed.
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.[3]
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 and constant torsion.
A curve is called a general helix or cylindrical helix[4] 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.[5]
Mathematical description [link]
In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a helix:[6]
- Failed to parse (Missing texvc executable; please see math/README to configure.): x(t) = \cos(t),\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): y(t) = \sin(t),\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): z(t) = t.\,
As the parameter t increases, the point (x(t),y(t),z(t)) traces a right-handed helix of pitch 2π and radius 1 about the z-axis, in a right-handed coordinate system.
In cylindrical coordinates (r, θ, h), the same helix is parametrised by:
- Failed to parse (Missing texvc executable; please see math/README to configure.): r(t) = 1,\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \theta(t) = t,\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): h(t) = t.\,
A circular helix of radius a and pitch 2πb is described by the following parametrisation:
- Failed to parse (Missing texvc executable; please see math/README to configure.): x(t) = a\cos(t),\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): y(t) = a\sin(t),\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): z(t) = bt.\,
Another way of mathematically constructing a helix is to plot a complex valued exponential function (exi) taking imaginary arguments (see Euler's formula).[vague]
Except for rotations, translations, and changes of scale, all right-handed helices are equivalent to the helix defined above. The equivalent left-handed helix can be constructed in a number of ways, the simplest being to negate any one of the x, y or z components.
Arc length, curvature and torsion [link]
The length of a circular helix of radius a and pitch 2πb expressed in rectangular coordinates as
- Failed to parse (Missing texvc executable; please see math/README to configure.): t\mapsto (a\cos t, a\sin t, bt), t\in [0,T]
equals Failed to parse (Missing texvc executable; please see math/README to configure.): T\cdot \sqrt{a^2+b^2} , its curvature is Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{|a|}{a^2+b^2}
and its torsion is Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{b}{a^2+b^2}.
Examples [link]
In music, pitch space 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.
See also [link]
- Alpha helix
- Boerdijk–Coxeter helix
- Collagen
- Double helix
- Helical symmetry
- Helicoid
- Helix angle
- Seashell surface
- Solenoid
- Spiral
- Superhelix
- Triple helix
References [link]
- ^ Weisstein, Eric W., "Helicoid" from MathWorld.
- ^ ἕλιξ, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
- ^ "Double Helix" by Sándor Kabai, Wolfram Demonstrations Project.
- ^ O'Neill, B. Elementary Differential Geometry, 1961 pg 72
- ^ O'Neill, B. Elementary Differential Geometry, 1961 pg 74
- ^ Weisstein, Eric W., "Helix" from MathWorld.
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Helix (comics)
Helix was a short-lived, science fiction and science fantasy imprint of DC Comics, launched in 1996 and discontinued in 1998. In early promotional materials prior to the release of the first title, the imprint was called Matrix instead of Helix. It was renamed because of the then-upcoming film, The Matrix. It featured a handful of ongoing monthly series, several limited series, and one short graphic novel.
Despite the involvement of successful science fiction/fantasy novelists Michael Moorcock, Lucius Shepard and Christopher Hinz, and established comics creators Howard Chaykin, Elaine Lee, Matt Howarth, Warren Ellis, Walt Simonson, and Garth Ennis, sales of the comic books were low, and most of the ongoing titles were cancelled after 12 or fewer issues. Ellis' Transmetropolitan was switched to the Vertigo imprint, where it continued for several more years before reaching its planned conclusion. Ennis' Bloody Mary mini-series and Michael Moorcock's Multiverse were later collected in Vertigo-label paperbacks and Moeller's Sheva's War was released as a graphic novel paperback by Dark Horse.
Helix (band)
Helix is a Canadian hard rock/heavy metal band. They formed in 1974, and are best known for their 1984 single "Rock You". The original lineup was formed by drummer Bruce Arnold, and consisted of lead vocalist Brian Vollmer, guitarists Ron Watson and Rick "Minstrel" Trembley, keyboardist Don Simmons, and bassist Keith "Bert" Zurbrigg. However, their most well known lineup, and the one that recorded "Rock You", was the 80s version of the band: Vollmer on vocals, accompanied by guitarists Brent "The Doctor" Doerner and Paul Hackman, bassist Daryl Gray, and drummer Greg "Fritz" Hinz. The history of the band has been marked by many lineup changes, with Vollmer being the sole constant member and only remaining member of the original lineup. Although Hackman was killed in a tour bus accident in 1992, the surviving members of the 80s lineup reunited in 2009 for an album and have continued to tour since 2011.
Helix have toured with bands such as Kiss, Aerosmith, Rush, Mötley Crüe, Alice Cooper, Whitesnake, Night Ranger, Heart, Quiet Riot, W.A.S.P., Ian Gillan and Motörhead among others.
