The golden ratio (1.61803 … ) is greatly hyped, partly for its beautiful mathematical properties but also for nonsensical reasons. Distinguishing between the two requires understanding that mathematics is about structures and relationships, not just numbers in isolation. When the golden ratio truly appears (not just some number in its approximate neighborhood), we can find patterns that account for it.
Two good references for properties of the golden ratio are Clement Falbo, “The Golden Ratio: A mmenntrary Viewpoint,” College Mathematics Journal 36, no. 2 (March 2005): 123-134, and George Markowsky, review of The Golden Ratio, by Mario Livio, Notices of the AMS 52, no. 3 (March 2005): 344-347.
More videos from the Mathematical Impressions series.