Wednesday, 14 December 2011
Even educated people can be surprisingly reluctant to accept simultaneous events as coincidence. Say lightning strikes the church steeple just as old Aunt Mildred dies. Many will see a mystical connection between the events rather than view them as a weird but statistically plausible chance occurrence.
Coincidences are fun and fascinating, but they make scientists nervous. Case in point: the Andromeda galaxy. This nearest of all spiral galaxies could have had any size at all. In fact, Andromeda is the biggest and brightest within 50 million light-years of Earth. Astronomers, suspicious of what later turned out to be mere coincidence, initially doubted their distance measurements to the galaxy. We want our neighborhood to represent the universe, not be populated by oddballs.
Space, empty as it is, is crammed with coincidences. We gaze into a sky that displays only two disks, sun and moon, that just happen to appear the same size. The sun’s rotation and the moon’s revolution have nearly the same period of just under a month--coincidentally, of course. In the nineteenth century astronomers were thrown into a tizzy when they found that the planets’ distances from the sun (in astronomical units) seemed to follow a numerical sequence generated by taking the progression 0, 3, 6, 12, 24, 48, 96, 192, and 384, adding 4 to each, and then dividing by 10. The law, however, turned out to have no physical basis; it’s just a numerological quirk.
More instructive are the apparent coincidences, where chance is not involved. Novice astronomers are often amazed that the moon spins on its axis in the same period in which it revolves around Earth, keeping its far side forever hidden from our view. A closer look, however, reveals the noncoincidental explanation: our planet’s greater pull on the near side of the moon brakes the moon’s rotation and holds it in place like an invisible finger.
Similarly, planetary observers often saw particular dusky markings on Mercury when that planet completed its orbit; it seemed certain that Mercury had an 88-day rotation, in sync with its 88-day revolution around the sun. But radar imaging in the mid-1960s showed a rotation of 59 days, two-thirds of Mercury’s 88-day year. How, then, to explain the seemingly annually recurring features?
Again, gravitational tugs--only this time from the sun--are to blame. Mercury, like Earth, is not quite round. Instead, it has a slight egg shape, with bulges at either end of the planet. The bulges have a bit more mass than the rest of the planet, and so the sun tugs on them extra hard--harder still when Mercury is at perihelion, the point in its orbit closest to the sun. The tugs slow Mercury’s rotation so that one or the other bulge always faces the sun at perihelion. The result is not a synchronous orbit but a stable variation: in every revolution, Mercury rotates one and a half times. To observers, then, markings on the planet reappear every two revolutions--enough to falsely convince them of the features’ permanence and of a synchronous rotation that does not exist.
Had enough? Did you know that the number of astronomical units in a light-year (63,240) is virtually the same as the number of inches in a mile (63,360)? Or that the number of seconds in a day (86,400), times 10, is the same as the approximate diameter of the sun (864,000 miles)?
In times of crisis and heartbreak, many people's need for explanations of any sort seems to make them more open to the appeal of prophecies and coincidences. The Kennedy assassination, for example, led to a long list of seemingly eerie historical and numerical links between Kennedy and Lincoln.
|President Lincoln & President Kennedy|
Accepting such sham explanations can be more comforting than facing the awful acts directly, puzzling out their causes, and framing our responses. This may be part of the reason for the outpouring of superstition that sprang up on the Internet after the attack on the World Trade Center.
From 9-11 to ELSs
First there were the 11 numerologists whose e-mails began by pointing out that Sept. 11 is written 9-11, the telephone code for emergencies. Moreover, the sum of the digits in 9-11 (9 +1+1) is 11, Sept. 11 is the 254th day of the year, the sum of 2,5, and 4 is 11, and after Sept. 11, there remain 111 days in the year. Stretching things even more, the e-mails noted that the twin towers of the WTC look like the number 11, that the flight number of the first plane to hit the towers was 11, and that various significant phrases, including "New York City," "Afghanistan," and "The Pentagon" have 11 letters.
(Side note: The e-mails neglected to mention that 911 has a twinning property in the following rather strained sense: Take any three digit number, multiply it by 91 and then by 11, and, lo and behold, the digits will always repeat themselves. Thus 767 x 91 x 11 equals 767,767. Why? See below for the answer.)
There are many more of these after-the-fact manipulations, but the problem should be clear. With a little effort, we could do something similar with almost any date or any set of words and names.
The situation is analogous to the Bible codes, which I have discussed in a previous column. People search the Bible for equidistant letter sequences (ELSs) that spell out words that are relevant to an event and that can be said to have "predicted" it. (ELSs are letters in a text, each separated from the next by a fixed number of other letters.) Consider the word "generalization" for an easy example. It contains an equidistant letter sequence for "Nazi" as can be seen by capitalizing the letters in question: geNerAliZatIon.
There were e-mails and Web sites claiming the Bible contains many ELSs for "Saddam Hussein," "bin Laden," and also much longer ones describing the heinous acts at the World Trade Center. Unlike the original Bible codes, whose faults were rather subtle, these longer ELSs are purely bogus.
Nostradamus, Rorschach, and the 'Devil'
The most widely circulated of the recent e-mail hoaxes involves the alleged prophecies of the 16th century mystic and astrologer Nostradamus. Many verses were cited, most complete fabrications. Others were variations on existing verses whose flowery, vague language, like verbal Rorschach inkblots, allows for countless interpretations.
One of the most popular was "The big war will begin when the big city is burning on the 11th day of the 9th month that two metal birds would crash into two tall statues in the city and the world will end soon after." Seemingly prescient, this verse was simply made up, supermarket tabloid style.
Credit to abcNews.
- π=3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 ...
- A sphere has two sides. However, there are one-sided surfaces.
- There are shapes of constant width other than the circle. One can even drill square holes.
- There are just five regular polyhedra
- In a group of 23 people, at least two have the same birthday with the probability greater than 1/2
- Everything you can do with a ruler and a compass you can do with the compass alone
- Among all shapes with the same perimeter a circle has the largest area.
- There are curves that fill a plane without holes
- Much as with people, there are irrational, perfect, complex numbers
- As in philosophy, there are transcendental numbers
- As in the art, there are imaginary and surreal numbers
- A straight line has dimension 1, a plane - 2. Fractals have mostly fractional dimension
- You are wrong if you think Mathematics is not fun
- Mathematics studies neighborhoods, groups and free groups, rings, ideals, holes, poles andremovable poles, trees, growth ...
- Mathematics also studies models, shapes, curves, cardinals, similarity, consistency,completeness, space ...
- Among objects of mathematical study are heredity, continuity, jumps, infinity, infinitesimals,paradoxes...
- Some numbers are square, yet others are triangular
- The next sentence is true but you must not believe it
- The previous sentence was false
- 12+3-4+5+67+8+9=100 and there exists at least one other representation of 100 with 9 digits in the right order and math operations in between
- One can cut a pie into 8 pieces with three movements
- A clock never showing right time might be preferable to the one showing right time twice a day
- Among all shapes with the same area circle has the shortest perimeter
It is a remarkable coincidence that
27/12 is very close to 3/2.
Harmony occurs in music when two pitches vibrate at frequencies in small integer ratios.
For instance, the notes of middle C and high C sound good together (concordant) because the latter has TWICE the frequency of the former. Middle C and the G above it sound good together because the frequencies of G and C are in a 3:2 ratio.
In the 16th century the popular method for tuning a piano was to a just-toned scale. What this means is that harmonies with the fundamental note (tonic) of the scale were pure; i.e., the frequency ratios were pure integer ratios. But because of this, shifting the melody to other keys would make the music sound different (and bad) because the harmonies in other keys were impure!
So, the equal-tempered scale (in common use today), popularized by Bach, sets out to "even out" the badness by making the frequency ratios the same between all 12 notes of the chromatic scale (the white and the black keys on a piano). Thus, harmonies shifted to other keys would sound exactly the same, although a really good ear might be able to tell that the harmonies in the equal-tempered scale are not quite pure.
So to divide the ratio 2:1 from high C to middle C into 12 equal parts, we need to make the ratios between successive note frequencies 21/12:1. The startling fact that 27/12 is very close to 3/2 ensures that the interval between C and G, which are 7 notes apart in the chromatic scale, sounds "almost" pure! Most people cannot tell the difference!
What a harmonious coincidence!!!!