By Samuel Gugliotta
For Variety

LIVING in a world that seems so our of whack with our ideals of hope, justice, love and peace, it might be a helpful counterpoint to our trials and tribulations to glance, if only for a moment, at some aspect of our universe in which beauty and balance still hold sway over our lives. The aspect I have chosen to look at is the notion of symmetry.
Symmetry is a “big” idea. Ideas, of course, do not have physical characteristics. They are not, strictly speaking, big, small, heavy, light, blue, yellow, or furious. Ideas are not in the world like tables, chairs, houses, cars, and bill collectors. Still, symmetry is a “big” idea because its “instances” or “exemplifications” are so pervasive and various. It applies to almost every area of human endeavor. For example, in one of its senses, as applied to art, it is defined as “beauty of form arising from balanced proportions.” Living on the beautiful island of Saipan, you need only to look at a flower or a leaf to see many aspects of symmetry. In fact, when teaching the subject, letting students explore the environment for examples of symmetry is an excellent way to show the connection between mathematical pattern and the physical world.
Forms of symmetry are exemplified in the evolution of the universe, sub-atomic physics, the construction of our bodies, the senses of beauty, reason, justice, and the unity of opposites, like good and evil, sadness and joy, yin and yang. It is precisely because symmetry is so common, and always present, that we take it for granted and it is so difficult to notice. As Whitehead remarked, “We habitually observe by the method of difference. Sometimes we see an elephant and sometimes we do not.”
In mathematics, a geometrical figure is said to be symmetrical when certain operations, such as rotations and reflections, are performed on them and they remain unchanged. These form the congruence motions of the figure, and are associated with algebraic structures known as “groups.”
For example, a square has rotational symmetry every time it is rotated by 90 degrees. In every such “rigid motion” the square will appear to be exactly the same. It also has line symmetry with respect to its diagonals and lines drawn to the centers of its sides. A cube, believe it or not, has 48 such symmetries, remaining unchanged after 48 different turnings, reflections and flippings.
The letters of the English alphabet exhibit various types of symmetry. The letters A H I M O T U V W X Y (as well as the human body) all have bilateral symmetry. They remain unchanged when the vertical line drawn through the center of the shapes is considered as the line of reflection. When such shapes are reflected in a mirror, they remain unchanged. Another way to see this is that if you place a mirror vertically down the center of the letters, you will still see the same letter.
The letters H I O X B C D E K all have reflective symmetry around the horizontal line drawn through the center of the shapes. Of these, H I O X have both vertical and horizontal reflective symmetry.
The letters H I O X also have a third type of symmetry known as rotational symmetry. Rotating these letters by 180 degrees (or looking at them upside down) gives the same letter. This property is also shared by N S and Z.
The letter richest in symmetry is O, the ancient shape of perfection. It has reflective symmetry around an infinity of diameters, and rotational symmetry through an infinity of rotations. Thus O is infinitely symmetrical, and thus an apt symbol for the immense secrets lurking behind zero, as well as the mandala (magic circle) of self, consciousness, and world.
The symmetry, and near symmetry of letters and numbers are involved in palindromic recreations and designs. A palindrome is a word or number which reads the same way backwards or forwards.
The Zoological Society of San Diego publishes a magazine called ZOONOOZ. This word reads the same backwards, forwards, and also upside down.
The phrase NOW NO SWIMS ON MON does not read the same backwards but it does upside down.
The phrase YOU CAN CAGE A SWALLOW, CAN’T YOU, BUT YOU CAN’T SWALLOW A CAGE, CAN YOU? has reversibility with respect to words, and not letters. If you turn the word CHOICE upside down and view it in a mirror you will see the same word.
Words like WOW LEVEL REDDER DEIFIED read the same backwards or forwards and so are palindromes, properly speaking. So are whole sentences such as A MAN, A PLAN, A CANAL — PANAMA! or WAS IT A BAT I SAW? or RISE TO VOTE, SIR or NAME NOW ONE MAN or LIVE NOT ON EVIL.
Numerals have many symmetry and broken symmetry properties that allow for quaint and entertaining puzzles. Rotating digits by 180 degrees (that is, turning them upside down) often result in shapes which may be interpreted as letters. These are good games to play on calculators Consider, for example, the number 7734. If you think of the “4” in the “old fashioned” way, that is, as the lines open on top, and not the closed lines of the typewritten 4, you will see that inverted it becomes an “h,” while “3” becomes and “E,” and the two sevens inverted are two “L’s.” Thus turning 7734 upside down gives you that sinister location directly opposite to heaven.
Another old puzzle is to write the number 11030 down and ask someone to add two lines to transform the number into a person. The answer is HOBO.
Another old favorite, to be done with a calculator, is to tell the story of the 337 Arabs and 337 Israelis who battled over a square property 8424 meters on a side. If you sum the squares of 337 and 8424, you find out who won by turning the result, 71077345, upside down. (Did you try it? Shall I spoil your fun and tell you the inverted numerals spell SHELLOIL?)
Lastly, here is a mirror puzzle. Consider the apparently outlandish addition statement:
3414 + 340 + 74813 = 43374813
In order to show that this sum is “correct” hold up the statement to a mirror and you will be enlightened. Now this time I won’t spoil your fun. It’s good to look in the mirror now and then, not for your vanity, but to notice many properties of that big idea, symmetry, in the reflection of your mind. And last of all, when symmetry is thought of in the artistic sense of the balance and unity of opposites, know that you sadness is balanced with joy, hidden perhaps in a subconscious which compensates distortions of conscious perception.
Puzzles
1. Suppose there are 500 voters in San Vicente. All of them voted on the two issues of allowing casinos on Saipan, and funding a new gym for the high school. Three hundred seventy-five were in favor of casinos and 275 in favor of the new gym. If there were 40 votes cast against both issues, how many votes were cast in favor of both issues?
2. Can you find two numbers such that their product, quotient, and difference are the same?
3. What is the smallest odd integer which has the property that if its cube is added to twice its square, the result is a perfect square?
Answers To Last
Week’s Puzzles
7203^2 + 343^3 = 98^4
2000 meters long
20, 15, 12, is one solution, the least.