Metallic MOS: Difference between revisions
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First we’ll document some behavior of continued fractions. Then we’ll get into applications. | First we’ll document some behavior of continued fractions. Then we’ll get into applications. | ||
The metallic means are all irrational numbers. Therefore their continued fractions are infinite, with a periodic pattern at the end. An advantage of continued fractions over the decimal system is that we can easily determine whether a number is rational by whether a periodic pattern at the end is required. Decimals, on the other hand, sometimes require periodic patterns at the end even when the number is rational, such as | The metallic means are all irrational numbers. Therefore their continued fractions are infinite, with a periodic pattern at the end. An advantage of continued fractions over the decimal system is that we can easily determine whether a number is rational by whether a periodic pattern at the end is required. Decimals, on the other hand, sometimes require periodic patterns at the end even when the number is rational, such as <span><math>\frac 13</math></span> which is <span><math>0.\overline{3}</math></span>, or <span><math>\frac 17</math></span> which is <span><math>0.\overline{142857}</math></span>; as continued fractions these two values are, respectively, <span><math>[0; 3]</math></span> and <span><math>[0; 7]</math></span>. | ||
The golden mean has the continued fraction [1; < | The golden mean has the continued fraction <span><math>[1; \overline{1}]</math></span>. The larger a term in a continued fraction, the closer the approximation of the value at that point; by this conception of irrationality, the golden mean is sometimes said to be the most irrational number possible, eluding close approximation by any ratio as much as possible at every turn. | ||
The silver mean follows it closely with continued fraction [2; < | The silver mean follows it closely with continued fraction <span><math>[2; \overline{2}]</math></span>, and the bronze mean with <span><math>[3; \overline{3}]</math></span>. | ||
A handy way to quickly find the reciprocal of a number is to prepend its continued fraction with a 0. So, we can find the golden mean’s isotope 0.618034 is [0; < | A handy way to quickly find the reciprocal of a number is to prepend its continued fraction with a 0. So, we can find the golden mean’s isotope <span><math>0.618034</math></span> is <span><math>[0; \overline{1}]</math></span>. | ||
Actually any isotope’s continued fraction is found by simply depleting the initial term. For example, the silver mean’s isotopes are [1; < | Actually any isotope’s continued fraction is found by simply depleting the initial term. For example, the silver mean’s isotopes are <span><math>[1; \overline{2}]</math></span> and <span><math>[0; \overline{2}]</math></span>. The initial term of a continued fraction, the one to the left of the semicolon, carries the same information as the digit of a decimal just to the left of the decimal point, i.e. any number starting with <span><math>[0;]</math></span> is between 0 and 1. | ||
The continued fraction for any abstract generator should start with 0, then, because it must be less than the period, which is 1 (if not less than 0.5, as we’ve been preferring, because of the generator complement effect). | The continued fraction for any abstract generator should start with 0, then, because it must be less than the period, which is 1 (if not less than 0.5, as we’ve been preferring, because of the generator complement effect). | ||
Noble generators start with other numbers but then settle on all 1’s; for example, our earlier example of 0.419821 is [0; 2, 2, < | Noble generators start with other numbers but then settle on all 1’s; for example, our earlier example of <span><math>0.419821</math></span> is <span><math>[0; 2, 2, \overline{1}]</math></span>. | ||
Crossing nobles with beyond golden cases results in continued fractions which can start with anything but eventually settle on all 2’s, 3’s, or n if we base our noble on the n< | Crossing nobles with beyond golden cases results in continued fractions which can start with anything but eventually settle on all 2’s, 3’s, or <span><math>n</math></span> if we base our noble on the <span><math>n</math></span>th metallic mean. For example, our earlier example <span><math>0.226541</math></span> is <span><math>[0; 4, \overline{2}]</math></span> | ||
=== Application: <span><math>L{:}s</math></span> sequences=== | === Application: <span><math>L{:}s</math></span> sequences=== | ||