Metallic MOS: Difference between revisions

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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 1/3 which is 0.\overline{3}, or 1/7 which is 0.\overline{142857}; as continued fractions these two values are, respectively, [0; 3] and [0; 7].

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 1/3 which is 0.3, or 1/7 which is 0.142857; as continued fractions these two values are, respectively, [0; 3] and [0; 7].

The golden mean has the continued fraction [1; \overline{1}]. 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 golden mean has the continued fraction [1; 1]. 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; \overline{2}], and the bronze mean with [3; \overline{3}].

The silver mean follows it closely with continued fraction [2; 2], and the bronze mean with [3; 3].

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; \overline{1}].

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; 1].

Actually any isotope’s continued fraction is found by simply depleting the initial term. For example, the silver mean’s isotopes are [1; \overline{2}] and [0; \overline{2}]. 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 [0;] is between 0 and 1.

Actually any isotope’s continued fraction is found by simply depleting the initial term. For example, the silver mean’s isotopes are [1; 2] and [0; 2]. 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 [0;] 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).

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, \overline{1}].

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, 1].

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 nth metallic mean. For example, our earlier example 0.226541 is [0; 4, \overline{2}]

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 nth metallic mean. For example, our earlier example 0.226541 is [0; 4, 2]

=== Application: L:s sequences ===

@@ Line 477: / Line 477: @@
 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 1/3 which is 0.\overline{3}, or 1/7 which is 0.\overline{142857}; as continued fractions these two values are, respectively, [0; 3] and [0; 7].
+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 1/3 which is 0.<span style="text-decoration:overline">3</span>, or 1/7 which is 0.<span style="text-decoration:overline">142857</span>; as continued fractions these two values are, respectively, [0; 3] and [0; 7].
-The golden mean has the continued fraction [1; \overline{1}]. 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 golden mean has the continued fraction [1; <span style="text-decoration:overline">1</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; \overline{2}], and the bronze mean with [3; \overline{3}].
+The silver mean follows it closely with continued fraction [2; <span style="text-decoration:overline">2</span>], and the bronze mean with [3; <span style="text-decoration:overline">3</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; \overline{1}].
+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; <span style="text-decoration:overline">1</span>].
-Actually any isotope’s continued fraction is found by simply depleting the initial term. For example, the silver mean’s isotopes are [1; \overline{2}] and [0; \overline{2}]. 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 [0;] is between 0 and 1.
+Actually any isotope’s continued fraction is found by simply depleting the initial term. For example, the silver mean’s isotopes are [1; <span style="text-decoration:overline">2</span>] and [0; <span style="text-decoration:overline">2</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 [0;] 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).
-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, \overline{1}].
+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, <span style="text-decoration:overline">1</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<sup>th</sup> metallic mean. For example, our earlier example 0.226541 is [0; 4, \overline{2}]
+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<sup>th</sup> metallic mean. For example, our earlier example 0.226541 is [0; 4, <span style="text-decoration:overline">2</span>]
 === Application: L:s sequences ===