By Samuel Gugliotta
For Variety

THE four-letter word I am thinking of is the “m” word — “math.” Over the years I have entertained myself with various definitions of the subject. For example, “Math is a form of poetry,” or “Math is the science of symbolic diversity,” or “Math is the melody of your neurons.” I could continue with definitions which hint at the deep and pervasive nature of the subject, but to what avail? The sad paradox is that many students see math as a boring, irrelevant, and difficult subject. It seems that educators are adept at subtracting all beauty and pattern from the subject, making it as boring and irrelevant as possible. Degree requirements and “exit” tests replace the old-fashioned hickory stick.
One way to counter the negative attitudes our misguided teaching has instilled in students is to hearken back to ancient mathematics; to the time when religion, mathematics, science and art were all One, and numbers formed the principles of All Things. This could create the motivation for students to see math in a different way, and each example discussed could be generalized into modern algebraic principles we are anxious for students to learn.
For example, it seems that all ancient cultures gave special significance to the numbers 3, 4 and 5, especially in their role as the sides of a right triangle, and illustrating what became known as Pythagoras’s theorem- that is, 3-squared plus 4-squared equals 5-squared (not to mention the interesting pattern that 12 = (3)(4)).
According to Plutarch, the Egyptians interpreted this triangle as indicative of the principle of fertility. The horizontal base, of length 4 was associated with Isis, the Great Mother. The perpendicular side, of length 3 was the erect father, Osiris. The hypotenuse, or longest side 5 was the son, Horus. Now this is similar to the Greek tradition. Three is the first male (odd) number, 4 a female number, and 5 the number of marriage.
On another, related interpretation, 3 represents Spirit, 4 represents Matter, and 5 is Soul, partaking of both Spirit and Matter.
From such a beginning a teacher could develop the beautiful formulas which generate all Pythagorean Triples.
We could continue: the sum of 3, 4, and 5 is 12, the number of divisions of the zodiac. The cube of 12 (12 x 12 x 12) is 1728 and this number is thought by many scholars to be the “fatal” or “geometric” number mentioned in book 8 of Plato’s Republic (546A ff). It is generally agreed that the passage in question is the most obscure of all of Plato’s writings. I urge all of you experience the unique amazement of reading the most perplexing and mysterious paragraph in all of Western literature. According to Plato, all mortal things have a natural cycle of becoming and perishing, and 1728 years is the natural cycle for the perfect state.
This number, 1728, has some interesting properties. Permutations of the digits of 1728 produce two “vampire numbers”: 21 x 87 = 1827, and 27 x 81 = 2187.
To continue, 17 is approximately the diagonal of a square of side 12; and 28, the second “perfect number” is also the number of days in the lunar cycle. The cube of the product of the first five integers is 1728000. Moreover, the cube of two multiplied by the sum of the cubes of 3, 4 and 5 is 1728.
Lastly, 1728 is divisible by the number formed of its first and last digits (18), and also the number formed by its middle digits (72). I know of no other four-digit number, all digits different, which has this property.
Thus from such interesting ancient math, the teacher could introduce such topics as divisibility, powers, etc. (Coincidentally, Swift devotes the last paragraph of chapter three of Gulliver’s Travels to the number 1728. Gulliver is to receive the amount of rations each day equivalent to the rations sufficient for 1728 Lilliputians.)
If you multiply 1728 by the square of 5, the result is 43200, and this number (multiplied by various powers of 10), according to Joseph Campbell in, “The Inner Reaches of Outer Space,” is ubiquitous in many ancient cosmologies. In the Hindu sacred epics, the number of years in the present cycle of time, the Kali Yuga, is 432,000. At the same time, in the Icelandic Eddas, there are 540 doors in the warrior’s hall, and on the Day of the Wolf, 800 warriors will pass through each door. 540 x 800 = 432,000. The Babylonian empire flourished for 432,000 years. In addition, 1728 x 3 x 5 = 25,920, which is the Platonic Great Year, the time required for one complete cycle of the twelve zodiacal signs. Campbell notes that if you divide 25,920 by 60, the basic sexagesimal unit of astronomical measurement, the result is 432. In general, for the ancients, the human world was thought of as a reflection of a celestial city, ordered by the creator according to mathematical pattern.
Ancient number theory is also pervasive in the Bible. In (John:21:11) we have, “Simon and Peter went up, and drew a net of land full of great fishes, an hundred and fifty and three; and for all there were so many, yet was not the net broken.” Now 153 is the sum of the cube of its digits; that the cube of 1 plus the cube of 5 plus the cube of 3 equals 153. Coincidence? I don’t think so.
In (Genesis 32:14) we have “Jacob gives Esau 220 goats as a gesture of friendship.” Now Pythagoras investigated “friendly numbers” long ago. And sure enough 220 is a “friendly number” since the sum of its proper divisors is equal to 284 and the sum of the proper divisors of 284 is 220.
Puzzles
1. If you add 1 to 48, you get the square number 49. If you add 1 to half of 48, you get the square number 25. Can you find another number with this same property?
2. Can you arrange the nine digits, 1 through 9, into two groups, such that if you divide the number formed by one group by the other, the result is 2? Can you do this also to produce the results of 3, 4, 5, 6, 7, 8, and 9?
3. There are six ways in which you can arrange the nine digits, 1 through 9, into three groups of two, three, and four, so that the first two numbers multiplied equals the third. For example, 12 x 483 = 5796. Can you find any of the other five ways?
Answers to last week’s puzzles
1. Believe it or not, the cliff-hanging person will have a probability of 1/2 of falling over the cliff.
2. Let T = 6, A = 0, C = 1, H = 2. Then 600600600 = 5(120120120). That is TAA-TAA-TAA = (T - 1)(CHA-CHA-CHA).
3. It is mathematically impossible to connect the lines in such a way that no lines will cross.