% m211oh81.tex (prev.: *oh78.*) - The Golden Section
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\begin{center}\tcr{\rmH The Golden Section} \end{center} %\vspace{0pt}
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\item[\tcr{History:}] The number \mc{\tau = \frac{1 + \sqrt{5}}{2}} is called the \tcr{golden ratio} or \tcr{golden section};
it is discussed in detail in \tcg{Book XIII} of \tcg{Euclid's Elements}. (This book, as well as Book X, was probably written by \tcr{Theaetetus} in \tcg{390 B.C.}, a century before Euclid).

-probably already studied by the \tcg{Babylonians} and \tcg{Egyptians}:
 in the (probable) original measurements of the \tcr{Great Pyramid at Gizeh} in Egypt (built around \tcg{2600 B.C.}), the ratio of the length of the face to \mc{\frac{1}{2}} its height is \mc{186.4m : 115.2m = 1.61806\ldots}, which is very close to \mc{\tau = 1.61803\ldots}.

-important %plays a prominent role
 in the \tcr{art and architecture} of the \tcg{Greeks}.

-\tcr{Fra Luca Pacioli} \tcg{(15$^{th}$ century)} wrote a whole book about \mc{\tau} called \tcg{De devina proportione} (= Of devine proportion); it was illustrated by drawings of models made by his friend \tcr{Leonardo da Vinci} \tcg{(1452--1519)}.

-\tcr{name} comes from the \tcg{property} that if \mc{A, B, C} are points on a line such that \mc{AB:BC = AC:AB}, then we have \mc{AC = \tau AB.}   %\mc{\ra} best pleasing proportions. 

\tcb{\hspace{.75cm}\rule[-4pt]{.5pt}{10pt}\rule{10cm}{.5pt}
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\hspace*{.5cm}\hspace{-6pt}\mc{A}\hspace{-13pt}\hspace{10cm}
\hspace{-19pt}\mc{B}\hspace{-15pt}\hspace{6.180cm}
\hspace{-12pt}\mc{C}
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