anchor.diff:                \phi^4 &= 3\phi + 2\\
                 \phi^5 &= 5\phi + 3
               \end{align}
  -          </p>
  -          <p>We know from the proof above,
  -            <span class="M M--B">
  +          p
  +            | We know from the proof above,
  +            span.M.M--B.
                 \begin{align}
                   \phi^{k+1} &= a_{k+1}\phi + b_{k+1}\\
                   &= (a_k+b_k)\phi + a_k
                 \end{align}
  -            </span></p>
  -          <p>By corresponding parts of these congruent equations&mdash;and by looking at
  +          p.
  +            By corresponding parts of these congruent equations&mdash;and by looking at
               the coefficients in the examples above&mdash;we can see that
  -            <span class="M">a_{k+1}=a_k+b_k</span>, and <span class="M">b_{k+1}=a_k</span>,
  -            or rather, subtracting 1 from each index, <span class="M">b_k=a_{k-1}</span>.
  -            Therefore, <span class="M">a_{k+1}=a_k+a_{k-1}</span>, in other words,
  -            <span class="M">a_k=a_{k-1}+a_{k-2}</span>.</p>
  -          <p>The coefficients of <span class="M">\phi</span> as its power increases form a
  -            <em>sequence</em>, defined recursively. We use the symbol <var>f</var> for this
  -            special sequence.</p>
  -          <p class="M M--B">
  +            #[span.M a_{k+1}=a_k+b_k], and #[span.M b_{k+1}=a_k],
  +            or rather, subtracting 1 from each index, #[span.M b_k=a_{k-1}].
  +            Therefore, #[span.M a_{k+1}=a_k+a_{k-1}], in other words,
  +            #[span.M a_k=a_{k-1}+a_{k-2}].
  +          p.
  +            The coefficients of #[span.M \phi] as its power increases form a
  +            #[em sequence], defined recursively. We use the symbol #[var f] for this
  +            special sequence.
  +          p.M.M--B.
               f_n = f_{n-1} + f_{n-2} :\\
               f_0 = 0 \text{ and } f_1 = 1
  -          </p>
  -          <p>This sequence is called the <dfn id="fibonacci-sequence">Fibonacci Sequence</dfn>.
  +          p.
  +            This sequence is called the #[dfn#fibonacci-sequence Fibonacci Sequence].
               In layman's terms,
               each term in the Fibonacci Sequence is the sum of the previous two terms. The first
               few terms are listed here:
  -            <span class="M M--B">\bigl(0,1,1,2,3,5,8,13,21,34,55,89,144\dots\bigr)</span></p>
  -          <p>Let's do a little algebraic expansion to see something pretty cool.</p>
  -          <p class="M M--B">
  +            #[span.M.M--B \bigl(0,1,1,2,3,5,8,13,21,34,55,89,144\dots\bigr)]
  +          p Let's do a little algebraic expansion to see something pretty cool.
  +          p.M.M--B.
               \begin{align}
                 f_n &= f_{n-1} + f_{n-2}\\
                 &= \left(f_{n-2}+f_{n-3}\right) + f_{n-2} = 2f_{n-2} + f_{n-3}\\
  
positive.code:                 &= 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{\dots}}}}\\
                  \phi &=[1;1,1,1,\dots]
                \end{align}
              </p>
              <p class="M B !w:1o2">
                \begin{align}
                  \Phi = \frac{1}{\phi} &= \frac{1}{1+\Phi}\\
                  &= \frac{1}{1+\frac{1}{1+\Phi}}\\
  
negative.code: doctype html
  html(lang="en" xmlns="http://www.w3.org/1999/xhtml")
    head
      title Mathematical Background &in; Site Design
      meta(charset="utf-8")
      link(rel="stylesheet" href="../../styles/chharvey.css")
    body.o-GoldenContainer
      main.o-GoldenContainer__Content
        h1 Mathematical Background
        section#definitions
          h2 Definition of #[span.M \phi] and #[span.M \Phi]
          p.
            Suppose we have a line segment #[span.M \overline{PR}]
            with a point #[span.M Q] on it. Let #[span.M a=m\,\overline{PQ}] and
            #[span.M b=m\,\overline{QR}], and let #[span.M Q] be in a position
            such that:
            #[span.M.M--B \frac{a+b}{a} = \frac{a}{b}]
          p: img(src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/44/Golden_ratio_line.svg/220px-Golden_ratio_line.svg.png" alt="A line segment divided into two shorter segments, whose proportions form the Golden Ratio.")
          p
            | Set #[span.M \phi] to this value, that is,
            | #[span.M \phi = \frac{a}{b}]. We call this ratio the
            | #[dfn#golden-ratio Golden Ratio].
            | Then we have
            span.M.M--B.
              \begin{align}
                \frac{a+b}{a} &= \frac{a}{b}\\
                1 + \frac{b}{a} &= \frac{a}{b}\\
                1 + \frac{1}{\phi} &= \phi\\
                \phi + 1 &= \phi^{2}
              \end{align}
          p To find the value of #[span.M \phi], we use the quadratic formula.
          p.M.M--B.
            \phi^{2} - \phi - 1 = 0\\
            \phi = \frac{1 \pm \sqrt{5}}{2}
          p.
            Since we are dealing with a ratio of two #[em lengths], we are concerned
            only with the positive root. Thus we define
            #[span.M.M--B \mathbf{\phi = \frac{1+\sqrt{5}}{2}} \approx 1.61803]
          small.Note.Info.
            Let it be noted that sometimes mathematicians have a use for the
            negative root #[span.M \psi = \frac{1-\sqrt{5}}{2}], however we do not
            use the value of #[span.M \psi] on this site.
          p.
            Define #[span.M \Phi] as the multiplicative inverse (the reciprocal)
            of #[span.M \phi].
          p.M.M--B.
              \mathbf{\Phi = \frac{1}{\phi} = \frac{\sqrt{5}-1}{2}}\\
              = \frac{\sqrt{5}+1}{2}-1 = \phi - 1 \approx 0.61803
        section
          h2 Properties of #[span.M \phi] and #[span.M \Phi]
          p.
            There is an aesthetic symmetry between #[span.M \phi] and #[span.M \Phi].
            Look at some of their algebraic representations.
          section
            h3 Squares
            p.M.M--B.
              \begin{align}
                \phi^2 &= 1 + \phi\\
                \Phi^2 &= 1 - \Phi
              \end{align}
          section
            h3 Doubles
            p.M.M--B.
              \begin{align}
                2\phi &= \sqrt{5}+1\\
                2\Phi &= \sqrt{5}-1
              \end{align}
          section
            h3 In Terms of Themselves
            p.M.M--B.
              \begin{align}
                \phi &= \frac{1}{\phi} + 1\\
                \Phi &= \frac{1}{\Phi} - 1
              \end{align}
          section
            h3 In Terms of Each Other
            p.M.M--B.
              \begin{align}
                \phi &= \Phi + 1 = \frac{1}{\Phi}\\
                \Phi &= \phi - 1 = \frac{1}{\phi}
              \end{align}
          section
            h3 Continued Fractions
            div.Flex-Container
              p.M.B(class="!w:1o2").
                \begin{align}
                  \phi &= 1 + \frac{1}{\phi}\\
                  &= 1 + \frac{1}{1 + \frac{1}{\phi}}\\
  