# L@̂点񌻏ۂƓ񕪂̈΁ipCjU

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ČfF@Ԃ̍ɊւbIȈ̌_@in 2007.09.16 Sunday

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Helix
A helix constructed in brick .
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 special kind of space curve , i.e. a smooth curve in three-space. As a mental image of a helix one may take the spring (although the spring is not a curve, and so is technically not a helix, it does give a convenient mental picture). A helix is characterised by the fact that the tangent line at any point makes a constant angle with a fixed line. A filled in helix, for example a spiral staircase, 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 ἕɃǃ.
Types

Helices can be either right-handed or left-handed. With the line of sight being the helical axis, if clockwise movement of the helix corresponds to axial movement away from the observer, then it is called a right-handed helix. If anti-clockwise movement corresponds to axial movement away from the observer, 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 through a mirror, and vice versa.

Most hardware screws 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.

A double helix typically consists geometrically of two congruent helices with the same axis, differing by a translation along the axis, which may or may not be half-way.[2]

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 of a helix would be the Corkscrew roller coaster at Cedar Point amusement park.

A circular helix has constant band curvature and constant torsion . The pitch of a helix is the width of one complete helix turn, measured along the helix axis.

A curve is called a general helix or cylindrical helix[3] 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[4] .
[edit ] Mathematics

In mathematics , a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a helix[5] :

x(t) = \cos(t),\,
y(t) = \sin(t),\,
z(t) = t.\,

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

As the parameter t increases, the point (x(t),y(t),z(t)) traces a right-handed helix of pitch 2 about the z-axis, in a right-handed coordinate system.

In cylindrical coordinates (r, , h), the same helix is parametrised by:

r(t) = 1,\,
\theta(t) = t,\,
h(t) = t.\,

The above example is an example of circular helix of radius 1 and pitch 2.

Circular helix of radius a and pitch 2b 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 ).

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 of the x, y or z components.

The length of a circular helix of radius a and pitch 2b 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}.

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}.
[edit ] 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 .

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