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The Mandelbrot set ( / ˈmændəlbroʊt, - brɒt /) [ 1][ 2] is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers for which the function does not ...
Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions : The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.
This is a polar plot of the first 20 real values r n of the zeta function along the critical line, ζ(1/2 + it), with t running from 0 to 50. The values of r n in this range are the first 10 non-trivial Riemann zeta function zeros and the first 10 Gram points, each labeled by n.
Furthermore, the non-trivial zeros are symmetric about the real axis and the line σ = 1 / 2 and, according to the Riemann hypothesis, they all lie on the line σ = 1 / 2 . This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical ...
The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial. This quadratic polynomial has two roots, and. The golden ratio is also closely related to the polynomial. which has roots and As the root of a quadratic polynomial, the golden ratio is a constructible number.
In the most familiar acuity test, a Snellen chart is placed at a standard distance: 6 metres. At this distance, the symbols on the line representing "normal" acuity subtend an angle of five minutes of arc, and the thickness of the lines and of the spaces between the lines subtends one minute of arc. This line, designated 6/6 (or 20/20), is the ...
~2.7: The surface of Broccoli: San-Hoon Kim used a direct scanning method and a cross section analysis of a broccoli to conclude that the fractal dimension of it is ~2.7. [49] Measured ~2.8: Surface of human brain: Measured with segmented three-dimensional high-resolution magnetic resonance images [50] Measured and calculated ~2.8: Cauliflower