helix

American Heritage Dictionary:

he·lix

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helix

circular helix in a right circular cylinder
(Academy Artworks)

(hē'lĭks)

n., pl., -lix·es, or hel·i·ces (hĕl'ĭ-sēz', hē'lĭ-).

Mathematics. A three-dimensional curve that lies on a cylinder or cone, so that its angle to a plane perpendicular to the axis is constant.
A spiral form or structure.
Anatomy. The folded rim of skin and cartilage around most of the outer ear.
Architecture. A volute on a Corinthian or Ionic capital.

[Latin, from Greek.]

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helix

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pronounced hee-liks and meaning 'a spiral or coiled curve', has the plural form helices, pronounced hel-i-seez, or occasionally helixes.

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Oxford Companion to Australian Literature:

Helix

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A literary journal founded in Canberra by Les Harrop in 1978, later transferred with him to Victoria. Dedicated to publishing Australian and overseas writing in the same journal, Helix produced an Ezra Pound special issue in 1982. A book review supplement to Helix (9, 10) was published as a separate issue, Fresh Flounder (1981). Helix ceased regular publication in 1986; its editor, David Brooks, then began to work with The Phoenix Review, which took over something of the role of Helix, although its future in 1993 also appears uncertain. The Phoenix Review is published twice a year.

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Next:	Hemisphere: An Asian-Australian Magazine, Henry Bournes Higgins, Henry Halloran

McGraw-Hill Science & Technology Encyclopedia:

Helix

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Any nonplanar curve all of whose tangents make the same angle with a fixed line. Other characteristic properties are that all principal normals are parallel to a plane and that the ratio of torsion to curvature is constant. If a helix has constant curvature (and hence constant torsion), it is a circular helix; it lies on a circular cylinder whose elements it cuts at a constant angle. See also Analytic geometry; Differential geometry.

McGraw-Hill Dictionary of Architecture & Construction:

helix

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1. Any spiral, particularly a small volute or twist under the abacus of the Corinthian capital.
2. The volute of an Ionic capital.

helix, 1: H

Dictionary of Cultural Literacy: Science:

helix

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(hee-liks)

In geometry, a three-dimensional spiral shape, resembling a spring.

Oxford Dictionary of Biochemistry:

helix

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(pl. helices)

a coiled or spiral structure, e.g. the thread of a bolt or a coil (tubular) spring. See also alpha helix, beta helix, gamma helix.
the curve traced on the surface of a cylinder or cone by a point crossing its right sections at a constant angle.
a space curve with turns of constant angle to the base and constant distance from the axis.

—helical adj.; helicity n.

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Saunders Veterinary Dictionary:

helix

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1. a coiled structure.
2. the free margin of the pinna of the ear.

α-h., alpha-h. — the folding arrangement of parts of protein molecules in which a single polypeptide chain forms a right-handed helix.
h. destabilizing proteins — proteins that bind in a cooperative manner to DNA single-strands during DNA replication and help open up the replication fork. Called also single-strand DNA binding proteins.
double h. — the native state of deoxyribonucleic acid (DNA), in which two antiparallel chains with complementary nucleotide sequences are wound around each other. The DNA molecule consists of two sugar-phosphate strands with the nucleotide base pairs stacked between them. The orientation of the two strands is antiparallel, i.e. 5′→3′ directions are opposite. Called also Watson–Crick helix.

Random House Word Menu:

categories related to 'helix'

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Random House Word Menu by Stephen Glazier

For a list of words related to helix, see:

Ears - helix: curved fold along rim of external ear
Geometric Shapes and Mathematically Defined Forms - helix: three-dimensional curve whose spiral arms maintain constant angle relative to base
Shapes - helix: spiral; curve traced on cylinder or cone by point moving at constant oblique angle across right sections

Rhymes:

helices

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See words rhyming with "helices."

Bradford's Crossword Solver's Dictionary:

helix

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See crossword solutions for the clue Helix.

Wikipedia on Answers.com:

Helix

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This article is about the shape. For other uses, see Helix (disambiguation).

The helix (cos t, sin t, t) from t = 0 to 4π with arrowheads showing direction of increasing t.

A helical coil spring

Crystal structure of a folded molecular helix reported by Lehn and coworkers in Helv. Chim. Acta., 2003, 86, 1598-1624.

A natural left-handed helix, made by a climber plant.

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]

Contents

1 Types
2 Mathematical description
- 2.1 Arc length, curvature and torsion
3 Examples
4 See also
5 References

Types

Helices can be either right-handed or left-handed. With the line of sight along the helix's axis, if a clockwise screwing motion moves 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

In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a helix:^[6]

$x(t) = \cos(t),\,$

$y(t) = \sin(t),\,$

$z(t) = t.\,$

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:

$r(t) = 1,\,$

$\theta(t) = t,\,$

$h(t) = t.\,$

A circular helix of radius a and pitch 2πb is described by the following parametrisation:

$x(t) = a\cos(t),\,$

$y(t) = a\sin(t),\,$

$z(t) = bt.\,$

Another way of mathematically constructing a helix is to plot a complex valued exponential function (e^xi) 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

The length of a circular helix of radius a and pitch 2πb expressed in rectangular coordinates as

$t\mapsto (a\cos t, a\sin t, bt), t\in [0,T]$

equals $T\cdot \sqrt{a^2+b^2}$ , its curvature is $\frac{|a|}{a^2+b^2}$ and its torsion is $\frac{b}{a^2+b^2}.$

Examples

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.

References

^ 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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Translations:

Helix

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Dansk (Danish)
n. - spiral, skruelinje, det ydre øres kant

Nederlands (Dutch)
spiraal, spiraalvormig ornament, schroeflijn, rand van oorschelp, huisjesslak

Français (French)
n. - spirale, hélice

Deutsch (German)
n. - Helix

Ελληνική (Greek)
n. - (ανατ., αρχιτ., μαθημ.) έλικα

Italiano (Italian)
spirale, elica

Português (Portuguese)
n. - espiral (f) (Geom.), hélix (f) (Med.), tipo de caracol (m) (Zool.), voluta (f) (Arquit.)

Русский (Russian)
спираль, винтовая поверхность

Español (Spanish)
n. - hélice

Svenska (Swedish)
n. - spiral, snirkel (arkit.), helix (anat.), trädgårdssnäcka, snäcka (tekn.), spirallinje (geom.)

中文（简体）(Chinese (Simplified))
螺旋, 耳轮, 螺旋状物

中文（繁體）(Chinese (Traditional))
n. - 螺旋, 耳輪, 螺旋狀物

한국어 (Korean)
n. - 나선, 귓바퀴, 소용돌이 장식

日本語 (Japanese)
n. - 螺旋, 耳輪

العربيه (Arabic)
‏(الاسم) لولب أو شئ لولبي مثل الخليه حلزونيه معماريه أو المحارة أو حافه الأذن الخارجيه, حلزون علم الهندسه‏

עברית (Hebrew)
n. - ‮חלזונית, קו שבלולי, בעל צורת קפיץ, מסגרת האוזן החיצונית, קישוט חלזוני, קו עקום החותך חרוט או גליל בזווית קבועה עם ציר הגוף‬

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		American Heritage Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
		Fowler's Modern English Usage. Oxford University Press. © 1999, 2004 All rights reserved. Read more
		Oxford Companion to Australian Literature. Oxford University Press. © 1994 All rights reserved. Read more
		McGraw-Hill Science & Technology Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Read more
		McGraw-Hill Dictionary of Architecture & Construction. McGraw-Hill Dictionary of Architecture and Construction. Copyright © 2003 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Dictionary of Cultural Literacy: Science. The New Dictionary of Cultural Literacy, Third Edition Edited by E.D. Hirsch, Jr., Joseph F. Kett, and James Trefil. Copyright © 2002 by Houghton Mifflin Company. Published by Houghton Mifflin. All rights reserved. Read more
		Oxford Dictionary of Biochemistry. Oxford University Press. Oxford Dictionary of Biochemistry and Molecular Biology © 1997, 2000, 2006 All rights reserved. Read more
		Saunders Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
		Random House Word Menu. © 2010 Write Brothers Inc. Word Menu is a registered trademark of the Estate of Stephen Glazier. Write Brothers Inc. All rights reserved. Read more
		Rhymes. Oxford University Press. © 2006, 2007 All rights reserved. Read more
		Bradford's Crossword Solver's Dictionary. Collins Bradford's Crossword Solver's Dictionary © Anne Bradford, 1986, 1993, 1997, 2000, 2003, 2005, 2008 HarperCollins Publishers All rights reserved. Read more
		Wikipedia on Answers.com. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Helix. Read more
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