 FACTOID # 10: The total number of state executions in 2005 was 60: 19 in Texas and 41 elsewhere. The racial split was 19 Black and 41 White.

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Encyclopedia > Logarithmic spiral  Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral  The arms of spiral galaxies often have the shape of a logarithmic spiral, here the Whirlpool Galaxy

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In polar coordinates (r, θ) the curve can be written as This article describes some of the common coordinate systems that appear in elementary mathematics. ... $r = ae^{btheta},$

or $theta = frac{1}{b} ln(r/a),$

hence the name "logarithmic". In parametric form, the curve is Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... $x(t) = a cos(t)e^{bt},$ $y(t) = a sin(t)e^{bt},$

with real numbers a and b. The parameter b controls how tightly and in which "direction" it is wrapped. For b >0 the spiral expands with increasing θ, and for b <0 it contracts with increasing θ. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

The spiral is nicely parametrized in the complex plane: exp(zt), given a z with Im(z)≠0 and Re(z)≠0. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... The exponential function is one of the most important functions in mathematics. ...

In differential geometric terms the spiral can be defined as a curve c(t) having a constant angle α between the radius or path vector and the tangential vector In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... $arccos frac{langle mathbf{c}(t), mathbf{c}'(t) rangle}{|mathbf{c}(t)||mathbf{c}'(t)|} = alpha$

If α = 0 the logarithmic spiral degenerates into a straight line. If α = ± π / 2 the logarithmic spiral degenerates into a circle. A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... Results from FactBites:

 Cams (1137 words) The logarithmic spiral is a mathematical curve which has the unique property of maintaining a constant angle between the radius and the tangent to the curve at any point on the curve (figure 1). A logarithmic spiral cam (a "constant angle cam") ensures that the line between the axle and the point of contact (the "line of force") is at a constant angle to the abutting surface, independent of how the cam is oriented. The mathematical equation for a logarithmic spiral is R=beaØ.
 Logarithmic spiral - Wikipedia, the free encyclopedia (847 words) The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant. Logarithmic spirals are self-similar in that they are self-congruent under all similarity transformations (scaling them gives the same result as rotating them). Hawks approach their prey in a logarithmic spiral: their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch.
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