Definitions

# Geometric style

Style of vase painting that flourished in Athens circa 1000–700 BC. Vases decorated in this style feature horizontal bands filled with geometric patterns such as zigzags, triangles, and swastikas in dark paint on a light ground. The rhythmic effect is similar to that of basketry. The abstract motifs developed into stylized animal and human forms in such narrative scenes as funerals, dances, and boxing matches. Small bronze and clay figurines, elaborately decorated fibulas, and limestone seals were also produced. The patterns remained popular and influenced much later Greek art.

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In mathematics, the geometric-harmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the geometric mean of g0 = x and h0 = y and call it g1, i.e. g1 is the square root of xy. We then form the harmonic mean of x and y and call it h1, i.e. h1 is the reciprocal of the arithmetic mean of the reciprocals of x and y.

Now we can iterate this operation with g1 taking the place of x and h1 taking the place of y. In this way, two sequences (gn) and (hn) are defined:

$g_\left\{n+1\right\} = sqrt\left\{g_n h_n\right\}$

and

$h_\left\{n+1\right\} = frac\left\{2\right\}\left\{frac\left\{1\right\}\left\{g_n\right\} + frac\left\{1\right\}\left\{h_n\right\}\right\}$

Both of these sequences converge to the same number, which we call the geometric-harmonic mean M(x, y) of x and y.

M(x, y) is a number between the geometric and harmonic mean of x and y; in particular it is between x and y. If r > 0, then M(rx, ry) = r M(x, y).

If AG(x, y) is the arithmetic-geometric mean, then we also have

$M\left(x,y\right) = frac\left\{1\right\}\left\{AG\left(frac\left\{1\right\}\left\{x\right\},frac\left\{1\right\}\left\{y\right\}\right)\right\}$