Spiral

See also: Spiral (homonymy)

In Mathematical S, the spiral is a Courbe which starts in a central point then moves away from there more and more, at the same time as it turns around.

The term spiral refers in general to a Courbe planes. When a spiral develops in three dimensions, one speaks rather about Hélice.

Etymology

In Latin will spira or in Greek will speira , this word indicates a rolling up.

In the language running, and in particular in drawing and structures the adjectives of them spiral and spiral indicates all the forms evoking the mathematical spiral (staircase in spiral…) or including/understanding a succession of convolutions.

Spirals with two dimensions

A spiral with two Dimension S easily describes using polar Coordonnées: the ray R is given by a function continues and Monotone of the angle θ . The circle is then a degenerated case.

There exist several types of spirals with two dimensions. Here most important:

  • the Spiral of Archimedes: r=a+b \ cdot \ theta ~

  • the hyperbolic Spiral : r= \ frac {has} {\ theta} ~
  • the Spirale logarithmic curve: r=a \ cdot b^ \ theta ~
  • the Spiral of Fermat: r= \ sqrt {\ theta}
  • the Spiral of Nielsen
  • the Spiral of Dimensions
  • the Spiral of Galileo
  • the spiral of Fibonacci: particular case of the spiral logarithmic curve built with a Continuation of Fibonacci

A spiral has necessarily an infinity of distinct whorls; the polar ray can grow indefinitely with the angle, or tend towards a finished limit; in this last case, the spiral is asymptotic with a circle and even sometimes on a line (spiral hyperbolic).

Spirals in multiple centers

The progressive distance of a spiral depends on the number of the centers which were used for forming it. There are spirals:
  • in 2 centers which are located on the same line,
  • in 3 centers which are located at the three tops of an equilateral triangle,
  • in 4 centers which are located at the four tops of the angles of a square.

Construction

A simple process makes it possible to trace of a continuous motion a relatively regular spiral, for example for the decoration of gardens: It is enough to roll up a chalk line attached to a stake planted in the indicated center of the spiral. By then unrolling the chalk line around the stake by keeping it tended, a point maintained vertical and attached at the end of this chalk line makes it possible to trace on the ground a spiral with the fur as the chalk line is held. In this process, the whorls of the traced line are obviously all the more isolated, the central stake is larger.

Construction of a false spiral (or approximate spiral) on a sheet of paper

With a compass and a rule:
  1. To trace a right which divides the sheet in two equal parts.
  2. To place two points A1 and A2 on the line around the center of the sheet. The distance between a1 point and the point A2 parameter concentration of the curve. The shorter this distance is, the more the spiral will be concentrated.
  3. To prick the compass on a1 point. To draw aside from a1A2 distance.
  4. To trace the half-circle A2 origin. To note A3 the point resulting from the intersection between the half-circle and our line.
  5. To prick the compass on a2 point. To draw aside from a2A3 distance.
  6. To trace the half-circle of A3 origin. Noter A4 the point resulting from the intersection enters the half-circle and our line.
  7. To prick the compass on a3 point. To draw aside from a3A4 distance.
  8. To trace the half-circle of A4 origin. Noter A5 the point resulting from the intersection enters the half-circle and our line.
  9. etc

The result is a spiral built starting from half-circles of which the double ray each time.

Several alternatives are possible and combinable:

  • the ray of each half-circle does not double but is rather increased by a fixed value;
  • to trace quadrants instead of half-circles (requires two right perpendiculars).

Examples

In Botanical, the spiral is present in the provision of seed of the Tournesol, or in the insertion point of the sheets on the stem (the plane angle passing by the axis of the stem and two points which follow one another is the divergence, value characteristics of the species).

Culture

  • the spiral is a frequent Motif in decoration (planks, jewels, fabrics, drawings, tattooings, tilings, etc)
  • Regarder a spiral which turns causes a visual effect, which fascine and is considered to facilitate hypnosis. It is a topic often exploited in the cartoons.
  • As a cartoon, the eyes of a character drawn in spiral evoke - according to the context - the confusion of the character, the fact that it is sounded, insane, etc

See too

Related bonds

External bonds

  • Spiral (page of the Mathcurve.com site)

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