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.
https://wn.com/Helix
Phelix
Phelix is a high-speed stream cipher with a built-in single-pass message authentication code (MAC) functionality, submitted in 2004 to the eSTREAM contest by Doug Whiting, Bruce Schneier, Stefan Lucks, and Frédéric Muller. The cipher uses only the operations of addition modulo 232, exclusive or, and rotation by a fixed number of bits. Phelix uses a 256-bit key and a 128-bit nonce, claiming a design strength of 128 bits. Concerns have been raised over the ability to recover the secret key if the cipher is used incorrectly.
Performance
Phelix is optimised for 32-bit platforms. The authors state that it can achieve up to eight cycles per byte on modern x86-based processors.
FPGA Hardware performance figures published in the paper "Review of stream cipher candidates from a low resource hardware perspective" are as follows:
Helix
Phelix is a slightly modified form of an earlier cipher, Helix, published in 2003 by Niels Ferguson, Doug Whiting, Bruce Schneier, John Kelsey, Stefan Lucks, and Tadayoshi Kohno; Phelix adds 128 bits to the internal state.
Helix (newspaper)
After a series of organizational meetings held at the Free University of Seattle involving a large and eclectic group including Paul Dorpat, Tom Robbins, Lorenzo Milam and others from KRAB-FM, John Ullman of the Seattle Folklore Society, Unitarian minister Paul Sawyer, and many others, the Helix first appeared on March 23, 1967. A member of both the Underground Press Syndicate and the Liberation News Service, it published a total of 125 issues (sometimes as a weekly, sometimes as a biweekly) before folding on June 11, 1970. The first issue was produced by Paul Dorpat with $200 in borrowed capital, out of a rented storefront on Roosevelt Way NE. After being turned down by the first printers they approached, they found a printer in Ken Monson, communications director of the International Association of Machinists local, who had recently acquired a printing press. 1500 copies were printed of the first issue. By the fourth biweekly issue sales had reached 11,000 copies. After the first two issues a "split-font" rainbow effect was sometimes used to print psychedelically colorful front covers; issues averaged 24 pages, with illustrations and graphics clipped from old magazines and having little to do with the adjoining copy crammed into the interior pages.
Movement
Movement may refer to:
Society and the arts
Music
She Wolf
She Wolf (Spanish: Loba) is the eighth studio album by Colombian singer-songwriter Shakira. It was released on 9 October 2009, by Epic Records and Sony Music Latin. As executive producers, Shakira and Amanda Ghost enlisted collaborators including The Neptunes, John Hill, Wyclef Jean, Lukas Burton, Future Cut, Jerry Duplessis and Timbaland. Musically, the record shifts from her traditional Latin pop and pop rock musical styles, instead exploring electropop, with influences of folk and world music. The lyrical themes of the album mostly focus on love and relationships and were based on the conversations Shakira had with her friends.
She Wolf reached number one on the charts of Argentina, Ireland, Italy, Mexico and Switzerland. It also charted inside the top five in Spain, Germany and the United Kingdom. It debuted at number fifteen on the Billboard 200. She Wolf was certified double-platinum in Colombia and Mexico, platinum in Italy and Spain, and gold in numerous countries including France and the United Kingdom.
Movement (Holly Herndon album)
Movement is the first studio album by American electronic musician Holly Herndon, released by RVNG Intl. on 12 November 2012. A music video for the album's title track was uploaded to the RVNG Intl. YouTube account on 29 November 2012
Track listing
All songs written and composed by Holly Herndon.
