Expression 12: left parenthesis, "y" minus "f" left parenthesis, "x" , right parenthesis , right parenthesis equals negative StartFraction, "f" left parenthesis, "x" , right parenthesis minus "g" left parenthesis, "x" , right parenthesis Over "i" left parenthesis, "x" , right parenthesis minus "h" left parenthesis, "x" , right parenthesis , EndFraction left parenthesis, "x" minus "h" left parenthesis, "x" , right parenthesis , right parenthesisy−fx=−fx−gxix−hxx−hx
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The two equations below control straight lines which are tangent to the family of curves, and which also happen to lead to the same point as the lines of intersection.
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Expression 14: "y" equals StartFraction, 1 Over tan left parenthesis, arcsin left parenthesis, "A" minus 1 , right parenthesis , right parenthesis , EndFraction "x" plus StartFraction, 1 Over "A" minus 1 , EndFractiony=1tanarcsinA−1x+1A−1
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Expression 15: "y" equals negative StartFraction, 1 Over tan left parenthesis, arcsin left parenthesis, "A" minus 1 , right parenthesis , right parenthesis , EndFraction "x" plus left parenthesis, StartFraction, "A" squared minus "A" cubed plus 2 "A" Over 2 left parenthesis, "A" minus 1 , right parenthesis StartRoot, 1 minus StartNestedFraction, left parenthesis, "A" squared minus 2 , right parenthesis squared NestedOver 4 , EndNestedFraction , EndRoot , EndFraction times StartRoot, 1 minus StartFraction, left parenthesis, "A" squared minus 2 , right parenthesis squared Over 4 , EndFraction , EndRoot , right parenthesis plus StartFraction, "A" squared minus 2 Over 2 , EndFractiony=−1tanarcsinA−1x+A2−A3+2A2A−11−A2−224·1−A2−224+A2−22
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Expression 16: "F" equals left bracket, 1...1 0 , right bracketF=1...10
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Indication of the value of A. Note that it only intersects both the y-axis and the lines of intersection for the golden ratio, as mentioned in the paper.
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Expression 18: "y" equals "A"y=A
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The following equations are just machinery to allow the other equations shown to not be hilariously bulky.
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Expression 20: "f" left parenthesis, "x" , right parenthesis equals StartFraction, "A" to the 4th power minus 2 minus "A" squared minus 2 "A" Over 2 "A" plus 2 , EndFractionfx=A4−2−A2−2A2A+2
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Expression 21: "g" left parenthesis, "x" , right parenthesis equals StartFraction, "A" to the 6th power minus 4 "A" squared minus 2 "A" cubed minus "A" to the 4th power minus 2 "A" Over 2 "A" squared plus 2 "A" , EndFractiongx=A6−4A2−2A3−A4−2A2A2+2A
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Expression 22: "h" left parenthesis, "x" , right parenthesis equals StartRoot, 1 minus left parenthesis, "f" left parenthesis, "x" , right parenthesis , right parenthesis squared , EndRoothx=1−fx2
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Expression 23: "i" left parenthesis, "x" , right parenthesis equals StartRoot, "A" squared minus left parenthesis, "g" left parenthesis, "x" , right parenthesis plus 1 , right parenthesis squared , EndRootix=A2−gx+12