Chronicle

The PASCAL ChronicleThe Golden RationLast summer when my parents and I visitedthe Louvre museum in Paris we saw thepainting of Mona Lisas’ face. I was so impressedby the beautiful natural look ofMona Lisa, that it made me wonder why thishappened. And I found the answer in theGolden Numbers…The Golden Number is a product not of mathematical imagination,but of a natural principle related to the laws of equilibrium.What do the pyramids in Egypt, Leonardo do Vinci's portrait ofthe Mona Lisa, sunflowers, the snail, the pine cone and yourfingers all have in common? The answer to this question lieshidden in a sequence of numbers discovered by the Italianmathematician Fibonacci. The characteristic of these numbers,known as the Fibonacci numbers, is that each one consists ofthe sum of the two numbers before it.Fibonacci numbers0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,1597, 2584, …Fibonacci numbers have an interesting property. When youdivide one number in the sequence by the number before it,you obtain numbers very close to one another. In fact, thisGOLDEN RATIO = 1.618number is fixed after the 13th in the series. This number isknown as the "golden ratio."233 / 144 = 1.618377 / 233 = 1.618610 / 377 = 1.618 987 / 610 = 1.618Based on the above Mona Lisa's face is a perfect golden rectangle,according to the ratio of the width of her foreheadcompared to the length from the top of her head to her chin.It has been scientifically proven that her face actually appearsin a golden rectangle, which also makes her face appearmore beautiful to human eyes.Panayiotis Parapanos 1D60Magic SquaresA magic square is a square of numbers consisting of thedifferent positive integers 1, 2, 3, 4,… arranged in such away that the sum of the numbers in any horizontal, vertical,or main diagonal line is always the same number,known as the magic constant. For example a 3x3 magicsquare consists of 3 rows and 3 columns.8 1 63 5 74 9 2This unique 3x3 magic square was known to the ancientChinese, who called it the Lo Shu. The magic constant is15.Horizontally we have: 8+1+6 = 15, 3+2+7 = 15,4+9+2 = 15Vertically we have: 8+3+4 = 15, 1+5+9 = 15,6+7+2 = 151 st Diagonal we have: 8+5+2 = 152 nd Diagonal we have: 6+5+4 = 15A version of a magic square 4x4 is:1 15 14 412 6 7 98 10 11 513 3 2 16You can easily check and find out that the magic constantis 34.As you can see from the following version of a 4x4 magicsquare:1 8 14 1112 13 7 215 10 4 56 3 9 16It has the same magic constant: 34. So there is not aunique 4x4 magic square.Panagiotis Antoniou 3B