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Metallic ratios: Difference between revisions
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used a bigger font to show the superscripts better (so as to aid readability), indented the formulas and centered the HTML table.
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'''Metallic ratios''' are the real roots of the general form equation:
<big> x
where the integer '''b''' determines which specific one it is.
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Using the quadratic equation:
<big> ( -b ± √(b
Substitute in (from the top equation) '''1''' for '''a''', '''-1''' for '''c''', and recognising that -b is negated we get:
<big> ( b ± √(b
We only want the real root:
<big> ( b + √(b
When we set '''b''' to '''1''', we get an irrational number: the '''Golden ratio'''.
<big> ( 1 + √(1
With '''b''' set to '''2''', we get a different irrational number: the '''Silver ratio'''.
<big> ( 2 + √(2
When the ratio '''b''' is '''3''', it is commonly referred to as the '''Bronze''' ratio, '''4''' and '''5'''
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'''Metallic ratios''' where '''b''' > '''0''' are also defined by the irrational continued fractions:
<big> [b;b,b,b,b,b,b,b,b,b,b,b,b,b,b,b,b...] </big>
So, The first
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|+ Metallic ratios
!Name!!'''b'''!!Equation!!Value!!Continued fraction!!OEIS link
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