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The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. The term Archimedean spiral is sometimes used to refer to the more general class of spirals of this type (see below), in contrast to Archimedes' spiral (the specific arithmetic spiral of ...
On Spirals (Greek: Περὶ ἑλίκων) is a treatise by Archimedes, written around 225 BC. [1] Notably, Archimedes employed the Archimedean spiral in this book to square the circle and trisect an angle.
An Archimedean point (Latin: Punctum Archimedis) is a hypothetical viewpoint from which certain objective truths can perfectly be perceived (also known as a God's-eye view) or a reliable starting point from which one may reason.
For <, spiral-ring pattern; =, regular spiral; >, loose spiral. R is the distance of spiral starting point (0, R) to the center. R is the distance of spiral starting point (0, R) to the center. The calculated x and y have to be rotated backward by ( − θ {\displaystyle -\theta } ) for plotting.
Today this is known as the Archimedean property of real numbers. [ 74 ] Archimedes gives the value of the square root of 3 as lying between 265 / 153 (approximately 1.7320261) and 1351 / 780 (approximately 1.7320512) in Measurement of a Circle .
An Archimedean spiral is, for example, generated while coiling a carpet. [5] A hyperbolic spiral appears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times called reciproke spiral, because it is the image of an Archimedean spiral with a circle-inversion (see below). [6]
The example shows trisection of any angle θ > 3π / 4 by a ruler with length equal to the radius of the circle, giving trisected angle φ = θ / 3 . Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics .
The logarithmic spiral or the pictured Archimedean spiral provide examples of curves whose curvature is monotonic for the entire curve. This monotonicity cannot happen for a simple closed curve (by the four-vertex theorem , there are at least four vertices where the curvature reaches an extreme point) [ 1 ] but for such curves the theorem can ...