ADDITIONAL FIBONACCI AND LUCAS OCCURRENCES
A thing of beauty is a joy forever:
Its loveliness increases; it will never
Pass into nothingness.
–John Keats (1795–1821)
Fibonacci and Lucas numbers appear in many other unexpected places. For example, index cards come in size 
or 
; most rugs come in five different sizes: 
, 
, 
, 
, or 
. In the first two cases, the dimensions are adjacent Fibonacci numbers; in the third and fourth cases, the ratio 4:6 = 6:9 = 2:3, a Fibonacci ratio; and in the last case, 9:12 = 3:4, the ratio of two adjacent Lucas numbers. We will study such ratios in Chapter 16.
4.1 FIBONACCI OCCURRENCES
Before turning to our first example, we will make a few formal definitions. ...
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