Spiral

In mathematics, a spiral is a special kind of curve that starts from a point and moves outward while turning around that point. As it turns, it gets farther and farther away, creating a shape that looks like a winding path. Spirals are found in many natural patterns and designs.

Spirals belong to a larger group called whorled patterns, which also includes shapes like concentric objects. These patterns appear in many places in nature, art, and science, making spirals an important idea to understand. They help us describe many beautiful and useful shapes in the world around us.

Two-dimensional

Main article: List of spirals

A two-dimensional, or plane, spiral can be simply described using polar coordinates. In these coordinates, the radius is a smooth, continuous function of the angle.

Examples

Some important types of two-dimensional spirals include:

• The Archimedean spiral: where the radius increases steadily with the angle.
• The hyperbolic spiral: where the radius changes in a different steady way with the angle.
• Fermat's spiral: used in nature, like in the arrangement of seeds in some flowers.
• The lituus: another spiral shape.
• The logarithmic spiral: often seen in nature, like in the shape of shells.
• The Cornu spiral or clothoid.
• The Fibonacci spiral and golden spiral.
• The Spiral of Theodorus: made of many connected right triangles.
• The involute of a circle.

An Archimedean spiral can be made by coiling something like a carpet.

A hyperbolic spiral looks like a helix when viewed from a special angle.

The name logarithmic spiral comes from its special equation. Nature often shows shapes close to this spiral.

Spirals that don’t fit the first five examples include:

A Cornu spiral has two points it approaches forever. The spiral of Theodorus is made of straight lines. The Fibonacci Spiral is made of many small circle arcs. The involute of a circle looks like an Archimedean spiral but is actually different.

Geometric properties

We can study spirals that follow a special rule involving the radius and the angle.

Polar slope angle

The angle between the spiral’s curve and the circle around its center is called the polar slope angle.

For a spiral where the radius changes with the angle following a power rule, the slope can be calculated.

In an Archimedean spiral, the polar slope is a steady value.

In a logarithmic spiral, the slope stays the same all along the curve.

Curvature

The curvature of a spiral shows how much it bends at any point.

For spirals where the radius changes with the angle following a power rule, there is a special formula for the curvature.

In an Archimedean spiral, the curvature changes in a known way.

The function that decides the radius of a spiral is usually smooth and either grows or shrinks steadily. If we choose a special kind of function, the spiral can stay within certain limits.

Helices

There are two main ways people think about spirals:

1. A curve on a flat surface that moves away from a center point as it turns around it.
2. A curve in 3D space that turns around an axis while also moving along that axis; this is called a helix.

The first definition talks about flat curves, like the grooves on a record or the arms of a spiral galaxy.

The second definition includes 3D shapes like springs, DNA strands, and other helical forms. For these, the word helix is often more useful than spiral.

In a picture, a flat spiral (like an Archimedean spiral) is shown in black, and a helix is shown in green. A special kind of spiral shape in 3D is shown in red.

Conical spirals

Main article: Conical spiral

If we add a third direction to a flat spiral, we can make a conical spiral. This new spiral lies on the surface of a cone.

Starting with an Archimedean spiral, we can create a conical spiral by adding a steady rise along the cone’s surface.

Spherical spirals

Any map of the world can help create a spherical spiral. By drawing a straight line on the map and then finding where that line would go on a globe, we get a spiral on the sphere.

One simple group of spherical spirals is called Clelia curves. These match straight lines on certain maps.

Another group is called rhumb lines or loxodromes. These are paths that a ship would follow if it kept a steady direction. They spiral around the poles forever.

In nature

The study of spirals in nature has a long history. Many shells form a special kind of spiral called a logarithmic spiral. Scientists have noticed that shells, from small ones like Helix to larger ones like Spirula, share similar shapes.

Spirals can also be found in horns, teeth, claws, and plants. In plants like sunflowers, the tiny flowers, called florets, grow in a spiral pattern. This pattern uses a special angle called the golden angle, which helps the florets fit together neatly.

Spirals in plants and animals are often called whorls, and this term is also used for spiral-shaped fingerprints.

As a symbol

The Celtic triple-spiral is actually a symbol from before the Celts. It is carved into a stone near the main entrance of the ancient Newgrange monument in County Meath, Ireland. Newgrange was built around 3200 BCE, long before the Celts arrived.

Spirals appear in many early cultures. The triskelion, a symbol of three interlocked spirals or bent legs, is found on Mycenaean vases, coins from Lycia, and on shields in Greek art.

Spirals are also common in art from before European contact in Latin and Central America. Many rock carvings in Guanajuato, Mexico, and along the Nazca Lines in Peru feature spirals.

Spirals are also used to show things like being hypnotized or feeling dizzy in cartoons and animations. They appear in nature, from the double helix of DNA to the shape of galaxies. The spiral can represent growth, purification, and spiritual ideas in many religions and philosophies.

In art

The spiral has inspired artists for a long time. One famous example is Robert Smithson's earthwork called "Spiral Jetty" at the Great Salt Lake in Utah. The idea of a spiral also appears in David Wood's Spiral Resonance Field at the Balloon Museum in Albuquerque. It is also a big part of the Nine Inch Nails album The Downward Spiral from 1994. In the anime Gurren Lagann, the spiral stands for a special way of thinking and living. The spiral is also important in the works of Mario Merz and Andy Goldsworthy. In the horror manga Uzumaki by Junji Ito, a town faces a strange curse connected to spirals.