Spiral

A spiral or helix is a curve drawn by a point or an axis and moved away from this, depending on the viewers perspective / R or approaches.

• 3.1 Prehistory and antiquity
• 3.2 Special case of triple and multiple spirals
• 3.3 Spiral motifs in the Middle Ages
• 3.4 Renaissance, Baroque, Art Nouveau

Spiral or screw

The spiral is sometimes confused with the screw. While the prototypical spiral is a structure in the plane, such as the groove of a record, or the arms of a spiral galaxy, both the screw and the twist drill is a spatial structure along the yard of a cylinder. The divisions to a vortex is ultimately unclear.

Spiral types

Level spirals

One can best describe these spirals mathematically as equations in the coordinate plane polar coordinate system, where r is plotted as a function of Φ; Φ generally runs to infinity rather than just to 2π. Below each is a formula for r ( Φ ) and the length s of the spiral from Φ = 0 are given.

The Archimedean spiral is created, for example, when winding a uniformly thick carpet.

The required material during winding length (especially decorative forging spirals ) can be estimated with the help of Clacksons spiral formula ( the so indicates the angle in number of orbits in full angle ) to 1 by the Taylorreihennäherung order of the arc length.

The logarithmic spiral is created, for example, the growth of snail shells.

The hyperbolic spiral can be seen for example in the vertical view through a spiral staircase.

There are many other well-known mathematical functions that produce flat spirals.

Logarithmic spiral

Fermat's spiral

Hyperbolic spiral

Lituus Spiral

Spiral of Theodorus ( root worm)

Clothoid ( Cornu spiral)

Three-dimensional spirals

In a circular cylinder you can put an Archimedean spiral to the ground and fit a screw as a curve in the mantle. The superposition of the curve is referred to as the coil and screw spiral conical or conical spiral space.

Now bends to the cylinder to a torus, a different room arises spiral ( free after MC Escher, see the links below ).

The cone shell this drawing underlying reminiscent of the Ouroboros motif, and can act as a torus with ever-increasing ring diameter be considered.

In an Archimedean spiral, the distance from the center increases linearly for increasing angle of her round. If this distance is projected as the angular distance to a pole on a spherical surface, creates an Archimedean spiral ball. It has a line of finite length, and not identical to the line drawn which is similar by its method of construction of the logarithmic spiral.

Rhumb line

In the arts

Unlike in nature, and in most geometric constructs are used in the art also inward and outward before flipped double spirals.

Prehistory and antiquity

Spirals appear already in prehistoric and early historic times as common ornamental motif on stone and pottery. Examples can be found in the Pottery Neolithic period, but also in the early civilizations of Egypt, Crete and China. In Europe, spiral designs from the Bronze Age to the early Iron Age Celts and Germanic tribes are common, later they appear on Iberian ceramics. The spiral symbolizes the idea of ​​infinity, but can also be protective or tribal sign.

Special case of triple and multiple spirals

In the church of Vallstena a Gotland picture stone was found, its central part is decorated with a four-fold spiral ornament. The character that is much older than triple spiral and begin to occur in the passage tomb of Newgrange in Ireland is on Gotland than 4 -, 6 - and 7-fold combination to be found. Also stylized animal heads and more realistic images of people and animals are sometimes united with this geometric motif. It is in the spiral, as well as in the vortex, probably a sun symbol or representation of a multiplicity of gods. Color supported the flat but finely carved ornament and emphasized the representation. The spiral motif comes in various forms and composition before on the older rocks that originated between 400 and 600 AD. Spiral motifs appear but both earlier and later on in different archaeological contexts. In the British Isles they are of Christ's birth and spread of several centuries younger late Celtic art, they can be studied in early Christian manuscripts. This art is the image stones closer in time; it has therefore been suggested that a certain relationship exists.

Spiral motifs in the Middle Ages

In the European art of the Middle Ages ( Romanesque and Gothic ) spiral motifs are rather rare, though - especially in Gothic tracery - Geometric Games ( ludi geometrici ) häufg were. In contrast, take mazes, plaited bands, tendrils and other convoluted, but very often also - unlike the Spriralen - overlapping decorative motifs in number. The face of the inner arch portal of the church of San Pedro de Gaíllos ( province of Segovia, Spain) shows - in addition to rosettes and vertebrae - small spirals. At door hardware usually found more - there but they evolve from straight bands.

Renaissance, Baroque, Art Nouveau

During the Renaissance, the spiral found its way into the arabesque and grotesque, in architecture it is found in the volute and the scrollwork and Mannerism in the characteristic Figura Serpentinata. Late highlights experience spirals in the volutes of the Baroque and Art Nouveau (eg Gustav Klimt).

Social sciences

In the demo copy of the metaphor " spiral of silence " by Elisabeth Noelle - Neumann has been used to explain a specific mutual building up of social reactions and fight at the same time: In public opinion, certain minority views were so strongly represented that the majority hesitate at all to express, on the minority would always dictatorial, and the majority is always mute, etc. Empirically, this relationship very difficult to verify.

Generally, in any mechanism that causes an escalation of the state, spoken by a spiral, like " spiral of violence ". In systems science show harmonic oscillators, the exponential increase ( escalate ), logarithmic spirals in their phase diagrams ( escalation ). Therefore, this term is mathematically correct than the term used synonymously " vicious circle " which does not include escalation of the states.

In nature

Many plants and some animals have in their plan spiral structures such as the Schneckenhaus.Fossile examples are the ammonites. The " arrangement " of these biologically generated spirals, which are mostly based on logarithmic spirals, in turn, takes place in most cases as the Fibonacci sequence.