Golden Ratio In Rectangles
Two recangles reveal directly the presence of the Golden Ratio: $1\times 2\;$ and $1\times 3\;$ rectangles. Tran Quang Hung has posted both cases at the CutTheKnotMath facebook page.
$1\times 2\;$ rectangle
$1\times 3\;$ rectangle
In both cases the ratio of the blue segment to a red one is Golden. Quang Duong has observed that the two diagrams neatly overlay, showing that the same segments may be seen as belonging to either case:
The construction in a $1\times 3\;$ rectangle happend to be a part of a more complicated diagram with multiple occurences of $\varphi\;$ that has been posted by Elliot McGucken earlier in multiple places, including CutTheKnotMath facebook page.