Colossal Gardner, ch. 15


We’re now entering the section on Symmetry, starting with the chapter on Rotations and Reflections. First, from a geometrical viewpoint, something is considered to be symmetrical if it looks the same (its physical characteristics don’t changed) if you perform a “symmetry operation” on it. That is, if you hold a mirror up to the capital letter “A” at the halfway vertical point, it’s unchanged. “A” is said to have vertical symmetry. If you hold the mirror horizontally halfway up the letter “B”, you’ll see it has horizontal symmetry. The letter “S” is the same if you rotate it 180 degrees (twofold symmetry). “H,” “I,” “X” and “O” have all three symmetries. If the letter “X” is written as a cross, it’s unchanged if you rotate it 90 degrees (fourfold symmetry). The letter “O” is considered to be the “richest” in symmetry because it’s unchanged whatever you do to it.

The entire chapter is then dedicated to giving other examples of rotations and reflections. Gardner states that because the Earth is a sphere, with a center of gravity that pulls everything on it downward, living things have developed strong vertical symmetry, but very little horizontal or rotational symmetry. Things humans have created reflect this bias towards vertical symmetry – just look at how chairs, tables, dishes, cars, planes and office buildings are designed. Most normal representational art is similar in this respect. The exception is completely non-representational art, and in some cases museum curators have hung abstract paintings upside-down and no one noticed for months. Can’t do that with the Mona Lisa without realizing the mistake pretty quickly.

Pictures that look one way when right-side up, and something else when rotated have been popular tricks for political cartoonists since the 1800’s. A couple more recent examples were used by Life magazine. In the Sept. 18, 1950 issue, a reproduced Italian poster had the face of Garibaldi, and when turned over showed the face of Stalin. The back cover of the Nov. 23, 1953 issue showed an Indian brave inspecting a stalk of corn, and upside-down was a man looking hungrily at an open can of corn. Gardner mentions Peter Newell (1862-1924), who published 2 children’s books of color plates of scenes that transform when rotated: Topsys and Turvys and Topsys and Turvys Number 2. And there’s !OHO!, by Rex Whistler.


(Image from Amazon UK, used for review purposes only.)

Probably one of the more impressive displays of this kind of art is Gustav Verbeek’s The Upside-Downs of Little Lady Lovekins and Old Man Muffaloo, a weekly strip that ran from 1903 to 1905, for a total of 64 strips. You read the first 6 panels normally, then turn the page over and read the same panels again to finish the story. GoComics carries the reprints now.


(From GoComics)

Another example of pictures changing when you rotate them 90 degrees is the rabbit duck.


(Image from the wiki article on ambiguous images.)

Plus two landscapes by German painters from the Renaissance.

Salvador Dali illustrated Maurice Sandoz’s The Maze, which supposedly has several plates where turning the image causes it to look like something else, but I can’t find any example artwork on the net showing this.

Finally, a couple puzzles.
Oliver Lee, age 44, lives at 312 Main Street. He asked the city to give him the license plate 337-31770 for his car. Why?

A basket contains more than 6 eggs, which are either white or brown. If x is the number of white eggs and y is the number of brown eggs, then what values are needed such that the sum of x and y turned upside-down is the product of x and y? That is, how many eggs are there?

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