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3. Symmetrograms

The shortcoming of the "Brownian frieze" method is that two different numbers may have the same graph due to this restrictive second step of Algorithm A that performs "mod 8" operation. Besides,  b(x) are long graphs that sometimes look "fuzzy" and occasionally not very aesthetic.

The next algorithm is also based on continued fraction expansion and partly removes the above shortcomings although the output graphics is quite different than in the previous section..

Algorithm B.

    1. Take the continued fraction expansion of  x. Cut the sequence {ak} up to the (m+1)-th term to obtain a0 a1, ..., am;

    2. Create a sequence of pairs  {(ak, am-k), k = 0, 1,..., m};

    3. Rotate each point Ak = (ak, am-k) counterclockwise about the origin, and draw this part of polygonal line g(x, m) = A0A1...Am.

The graph g(x, m)  is obviously a symmetric figure with respect to the y-axis, the reason to call it symmetrogram of x. It can be used as another "portrait" of this number. Also, as we will see, the shape of g(x, m) heavily depends on m.

Like in the previous section, we will supply some examples. Note that the numbers with the greater diverse in the sequence {ak} will have more interesting symmetrograms than these with finite or uniform expansions. Let us start with numbers containing p:  The figure below shows symmetrograms g(p, 31), g(p2, 12), g(Öp,18) and g(1/p,25).

E-numbers are peculiar as in the case of Brownian walk graphs. Here we display the sequence g(e,50), g(e2,65), g(e3,30)  and  g(e1/4,70) (figure below).

The first two subpictures of the next figure represents g(Log (2), 17),  g(Log (3), 9). Next two subfigures show the dependence of a symmetrogram on m - the length of the sequence {ak}. These contains  the graphs g(g, 18) and g(g,32) (g » 0.577216..., the Euler gamma).

The next row shows symmetrograms of numbers obtained by the Bessel function of the first kind, Jn(x), i.e. g(J4(2.1),26), g(J4(8),12), g(J2(9),12)  and  g(J4(e),39).

Symmetrograms, obtained by Algorithm B are also numerically very sensitive. This will be confirmed by the leftmost two subframes of the next figure that represents numbers cos (1) and  cos (1+ 10-10) for the same m = 27. The rightmost subframes are portraits of sin(1) (8 points) and sin2(1) (39 points).

The following four figures show g(Arcsin (1), 12)g(sh(sin (1)), 20) ,   g(ch(sh (1)), 12) and g(ch(sin (1)), 23),

while the hyperbolic functions produce the symmetrograms g (sh(1),12), g(ch(1),15), g(sh(1),15), and g(sh(0.35),17) shown on the figure below.

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