Posted on 07/26/2002 11:24:55 AM PDT by Dengar01
Timing Suspicious On Mysterious Suburban Crop Circles
Could Eerie Mystery Have To Do With Movie Release?
Is it a case of mysterious crop circles -- or an elaborate movie hoax?
A soybean farmer in Naperville said the broken, concentric rings that appeared in a field off Diehl Road left him scratching his head.
"Have you ever heard of something so crazy?" Steve Berning said. "Unbelievable."
Berning said the circles appeared last weekend and damaged more than 10 percent of his 8-acre field.
The circles do resemble similar ones seen in England, but in this case, the timing of their appearance in the western suburb is a bit suspicious.
Two weeks from now, "Signs" hits the big screen. The movie starring Mel Gibson involves -- you guessed it -- mysterious crop circles.
William Leone, an investigator with the Mutual UFO Network, said soil analysis could determine whether the circles have human or extraterrestrial origins.
But Illinois Farm Bureau spokesman Dennis Vercler scoffed at that idea.
"Since I don't believe in UFOs -- at least not soybean-destroying UFOs -- I have to assume whoever did this did it intentionally as a malicious prank," Vercler said.
Meanwhile, Berning doesn't seem overly upset about the circles.
"There's some damage, which upsets me," Berning said. "But I'm more curious than anything. I"ll always be asking questions."
But Math makes the World Go Round, amongst other things... :)
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Squares that are bigger | Squares that are smaller |
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22 is 4 | 1/2=0·5 and 0·52 is 0·25=1/4 |
32 is 9 | 1/5=0·2 and 0·22 is 0·04=1/25 |
102 is 100 | 1/10=0·1 and 0·12 is 0·01=1/100 |
Here is a mathematical derivation (or proof) of the two values. You can skip over this to the answers at the foot of this paragraph if you like.Note that Phi is just 1+phi. As a little practice at algebra, use the expressions above to show that phi times Phi is exactly 1. Here is a summary of what we have found already that we will find very useful in what follows:Multiplying both sides by Phi gives a quadratic equation:
Phi2 = Phi + 1 or
Phi2 Phi 1 = 0We can solve this quadratic equation to find two possible values for Phi as follows:
Use your calculator to see that the values of these two numbers are 1·6180339887... and 0·6180339887...
- First note that (Phi 1/2)2 = Phi2 Phi + 1/4
- Using this we can write Phi2 Phi 1 as (Phi 1/2)2 5/4
and since Phi2 Phi 1 = 0 then (Phi 1/2)2 must equal 5/4- Taking square-roots gives (Phi 1/2) = +(5/4) or (5/4).
- so Phi = 1/2 + (5/4) or 1/2 (5/4).
- We can simplify this by noting that (5/4) = 5/4 = 5/2
- The two values of Phi are therefore:
- 1/2 + 5/2 and 1/2 5/2
Did you notice that their decimal parts are identical?
We will name the first value Phi and the second phi using the first letter to tell us if we want the bigger value (Phi) 1·618... or the smaller one (phi) 0·618... .
Phi phi = 1, Phi - phi = 1, Phi + phi = 5 | |
Phi = 1.6180339.. | phi = 0.6180339.. |
Phi = 1 + phi | phi = Phi 1 |
Phi = 1/phi | phi = 1/Phi |
Phi2 = Phi + 1 | (-phi)2 = -phi + 1 or phi2 = 1 phi |
Phi = (5 + 1)/2 | phi = (5 1)/2 |
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It starts from basic definitions called axioms or "postulates" (self-evident starting points). An example is the fifth axiom that
In Book 6, Proposition 30, Euclid shows how to divide a line in mean and extreme ratio which we would call "finding the golden section G point on the line".
<-------- 1 ---------> A G B g 1gEuclid used this phrase to mean the ratio of the smaller part of this line, GB to the larger part AG (ie the ratio GB/AG) is the SAME as the ratio of the larger part, AG, to the whole line AB (ie is the same as the ratio AG/AB). If we let the line AB have unit length and AG have length g (so that GB is then just 1g) then the definition means that
GB = AG or 1g = g so that 1g=g2 AG AB g 1Notice that earlier we defined Phi2 as Phi+1 and here we have g2 = 1g or g2+g=1.
g = | 1 +5 | or g = | 1 5 | |
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It seems that this ratio had been of interest to earlier Greek mathematicians, especially Pythagoras (580BC - 500BC) and his "school".
A G B
x 1
so that AB is now has length 1+x. If Euclid's "division of AB into mean and extreme ratio" still applies to point G, what quadratic equation do you now get for x? What is the value of x?
The ratio of the length of a face of the Great Pyramid (from centre of the bottom of a face to the apex of the pyramid) to the distance from the same point to the exact centre of the pyramid's base square is about 1·6. It is a matter of debate whether this was "intended" to be the golden section number or not.
According to Elmer Robinson (see the reference below), using the average of eight sets of data, says that "the theory that the perimeter of the pyramid divided by twice its vertical height is the value of pi" fits the data much better than the theory above about Phi.
The following references will explain circumstantial evidence for and against:
There are no extant records of the Greek architects' plans for their most famous temples and buildings (such as the Parthenon). So we do not know if they deliberately used the golden section in their architectural plans. The American mathematician Mark Barr used the Greek letter phi () to represent the golden ratio, using the initial letter of the Greek Phidias who used the golden ratio in his sculptures.
Luca Pacioli (also written as Paccioli) wrote a book called De Divina Proportione (The Divine Proportion) in 1509. It contains drawings made by Leonardo da Vinci of the 5 Platonic solids. It was probably Leonardo (da Vinci) who first called it the sectio aurea (Latin for the golden section).
Today, mathematicians also use the Greek letter tau (), the initial letter of tome which is the Greek work for "cut" as well as phi.
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Phi has the value | 5 + 1 | and phi is | 5 1 | . |
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Here is the decimal value of Phi to 2000 places grouped in blocks of 5 decimal digits. The value of phi is the same but begins with 0·6.. instead of 1·6.. .
Read this as ordinary text, in lines across, so Phi is 1·61803398874...)
Dps: 1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 50 28621 35448 62270 52604 62818 90244 97072 07204 18939 11374 100 84754 08807 53868 91752 12663 38622 23536 93179 31800 60766 72635 44333 89086 59593 95829 05638 32266 13199 28290 26788 200 06752 08766 89250 17116 96207 03222 10432 16269 54862 62963 13614 43814 97587 01220 34080 58879 54454 74924 61856 95364 300 86444 92410 44320 77134 49470 49565 84678 85098 74339 44221 25448 77066 47809 15884 60749 98871 24007 65217 05751 79788 400 34166 25624 94075 89069 70400 02812 10427 62177 11177 78053 15317 14101 17046 66599 14669 79873 17613 56006 70874 80710 500 13179 52368 94275 21948 43530 56783 00228 78569 97829 77834 78458 78228 91109 76250 03026 96156 17002 50464 33824 37764 86102 83831 26833 03724 29267 52631 16533 92473 16711 12115 88186 38513 31620 38400 52221 65791 28667 52946 54906 81131 71599 34323 59734 94985 09040 94762 13222 98101 72610 70596 11645 62990 98162 90555 20852 47903 52406 02017 27997 47175 34277 75927 78625 61943 20827 50513 12181 56285 51222 48093 94712 34145 17022 37358 05772 78616 00868 83829 52304 59264 78780 17889 92199 02707 76903 89532 19681 98615 14378 03149 97411 06926 08867 42962 26757 56052 31727 77520 35361 39362 1000 10767 38937 64556 06060 59216 58946 67595 51900 40055 59089 50229 53094 23124 82355 21221 24154 44006 47034 05657 34797 66397 23949 49946 58457 88730 39623 09037 50339 93856 21024 23690 25138 68041 45779 95698 12244 57471 78034 17312 64532 20416 39723 21340 44449 48730 23154 17676 89375 21030 68737 88034 41700 93954 40962 79558 98678 72320 95124 26893 55730 97045 09595 68440 17555 19881 92180 20640 52905 51893 49475 92600 73485 22821 01088 19464 45442 22318 89131 92946 89622 00230 14437 70269 92300 78030 85261 18075 45192 88770 50210 96842 49362 71359 25187 60777 88466 58361 50238 91349 33331 22310 53392 32136 24319 26372 89106 70503 39928 22652 63556 20902 97986 42472 75977 25655 08615 48754 35748 26471 81414 51270 00602 38901 62077 73224 49943 53088 99909 50168 03281 12194 32048 19643 87675 86331 47985 71911 39781 53978 07476 15077 22117 50826 94586 39320 45652 09896 98555 67814 10696 83728 84058 74610 33781 05444 39094 36835 83581 38113 11689 93855 57697 54841 49144 53415 09129 54070 05019 47754 86163 07542 26417 29394 68036 73198 05861 83391 83285 99130 39607 20144 55950 44977 92120 76124 78564 59161 60837 05949 87860 06970 18940 98864 00764 43617 09334 17270 91914 33650 13715 2000
Phi's value in binary to 500 places is:
Phi to 10,000,000 places!
Simon Plouffe of Simon Fraser University notes that Greg J Fee programmed a method of his to compute the golden ratio (Phi) to ten million places in December 1996. He used Maple and it took about 30 minutes on a 194MHz computer. Have a look at the first part with 15,000 decimal places. The rest are organised in several files which you can investigate using this index.
1·10011 11000 11011 10111 10011 01110 01011 11111 01001 01001 11110 00001 01011 11100 11100 11100 11000 00001 10000 00101 100 11001 11011 01110 01000 00110 10000 01000 01000 00100 01001 11011 01011 11110 01110 10001 00111 00100 10100 01111 11000 200 01101 10001 10101 00001 00011 10100 00110 00001 10001 11010 01010 10010 01110 11001 11111 10000 10110 00101 01001 11101 300 00100 11110 11011 11111 00000 01101 00011 10000 01000 10110 11010 11011 11110 00110 00001 00111 11110 00000 01100 01000 400 01101 11100 00100 10010 10000 10000 00001 10000 00000 01011 00000 11101 01100 10010 11101 00100 00001 11100 11001 10101 500Neither the decimal form of Phi, nor the binary one nor any other base have any ultimate repeating pattern in their digits.
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Here is the connection the other way round, where we can discover the Fibonacci numbers arising from the number Phi.
The graph on the right shows a line whose gradient is Phi, that is the line
Let's call these the Fibonacci points. Notice that the ratio y/x for each Fibonacci point (x,y) gets closer and closer to Phi=1·618... but the interesting point that we see on this graph is that
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The basic Fibonacci relationship is
F(i+2) = F(i+1) + F(i) The Fibonacci relationship |
The graph shows that the ratio F(i+1)/F(i) seems to get closer and closer to a particular value, which for now we will call X.
If we take three neighbouring Fibonacci numbers, F(i), F(i+1) and F(i+2) then, for very large values of i, the ratio of F(i) and F(i+1) will be almost the same as F(i+1) and F(i+2), so let's see what happens if both of these are the same value: X.
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X = | F(i+1) | = 1 + | F(i) |
F(i) |
F(i+1) |
X = | F(i+1) | = 1 + | F(i) | = 1 + | 1 |
F(i) |
F(i+1) |
X |
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X2 = X + 1 |
Remember, this supposed that the ratio of two pairs of neighbours in the Fibonacci series was the same value. This only happens "in the limit" as mathematicians say. So what happens is that, as the series progresses, the ratios get closer and closer to this limiting value, or, in other words, the ratios get closer and closer to Phi the further down the series that we go.
Did you notice that we have not used the two starting values in this proof? No matter what two values we start with, if we apply the the Fibonacci relationship to continue the series, the ratio of two terms will (in the limit) always be Phi!
i | ... | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ... |
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Fib(i) | ... | 55 | 34 | 21 | 13 | 8 | 5 | 3 | 2 | 1 | 1 | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | ... |
1 | = 1, | 1 | = 0.5, | 2 | = 0.666.., | 3 | = 0.6, | 5 | = 0.625, ... |
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Phi = 1 + 1 = 1 + 1/( 1 + 1/( 1 + 1/( 1 +.. ))) 1 + 1 1 + 1 1 + ..This unusual expression is called a continued fraction since we continue to form fractions underneath fractions underneath fractions.
This continued fraction has a big surprise in store for us....
Mathematicians call all these fractional (and whole) numbers rational numbers because they are the ratio of two whole numbers and it is these number fractions that we will mean by fraction in this section.
It may seem as if all number can be written as fractions - but this is, in fact, false. There are numbers which are not the ratio of any two whole numbers, eg 2=1.41421356... , =3.14159..., e=2.71828... . Such values are called ir-ratio-nal since they cannot be represented as a ratio of two whole numbers (ie a fraction). A simple consequence of this is that their decimal fraction expansions go on for ever and never repeat at any stage!
Can we write Phi as a fraction?
The answer is "No!" and there is a surprisingly simple proof of this. Here it is. [This proof was given in the Fibonacci Quarterly, volume 13, 1975, page 32, in A simple Proof that Phi is Irrational by J Shallit and later corrected by D Ross - see below.]
Suppose we could write Phi as A/B where A and B are two integers. If this was possible then we can choose the simplest form for Phi and write Phi=p/q (p and q are integers again) but now p and q will have no factors in common. What we now show is that this leads to a logical contradiction. The only assumption we have made is that Phi can be written as a fraction and, since this will lead to a logical impossibility, then this assumption must be wrong - i.e. Phi cannot be written as a fraction.
The definition of Phi (and also of phi) is that it satisfies the equation
Phi2 Phi = 1 (*)So, if we are assuming that Phi can be written as p/q, we substitute this in:
(p/q)2 p/q = 1Since q is not zero, we can multiply through by q2 to get:
p2 pq = q2 (**)but we can factorise the left hand side, so
p(p q) = q2Since the left hand side has a factor of p then so must the right hand side. In other words p is a factor of q2.
Note there is an error in the paper quoted above, which is corrected in the next paragraph and in A Letter to the Editor from David Ross in Fibonacci Quarterly vol 13 (1975) page 198.
Also, by re-arranging the equation marked (**) above, we have:
p2 = q2 + pq = q(q + p)so q, being a factor of the right-hand side must also be a factor of the left-hand side, which is p2. But again, since p and q have no common factor except 1 then q also must be 1 too!
Here is the contradiction if both p and 1 are 1, then p/q is 1 and this does not satisfy our original equation for Phi, the one marked (*).
So we have a logical impossibility if we assume Phi can be written as a proper fraction.
The only possibility that logical allows therefore is that Phi cannot be written as a proper fraction - Phi is irrational.
The answer lies in the continued fraction for Phi that we saw earlier on this page.
If we stop the continued fraction for Phi at various points, we get values which approximate to Phi:
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The proper mathematical term for these fractions which are formed from stopping a continued fraction for Phi at various points is the convergents to Phi. The series of convergents is
1 | , | 2 | , | 3 | , | 5 | , | 8 | , | 13 | , | 21 | , ... |
1 | 1 | 2 | 3 | 5 | 8 | 13 |
The convergents start with 1/1 which is F(1)/F(0)
where F(n) represents the n-th Fibonacci number.
To get from one fraction to the next, we saw that we just take the reciprocal of the fraction and add 1:
so the next one after F(1)/F(0) is
1 + 1 = 1 + F(0) = F(1)+F(0) F(1)/F(0) F(1) F(1)
But the Fibonacci numbers have the property that two successive numbers add to give the next, so F(1)+F(0)=F(2) and our next fraction can be written as
1 + 1 = 1 + F(0) = F(1)+F(0) = F(2) F(1)/F(0) F(1) F(1) F(1)
So starting with the ratio of the first two Fibonacci numbers the next convergent to Phi is the ratio of the next two Fibonacci numbers.
This always happens:
if we have F(n)/F(n1) as a convergent to Phi, then the next convergent is F(n+1)/F(n).
We will get all the ratios of successive Fibonacci numbers as values which get closer and closer to Phi.
You can find out more about continued fractions and how they relate to splitting a rectangle into squares and also to Euclid's algorithm on the Introduction to Continued Fractions page at this site.
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In fact, you can start with many values but not all (for instance 0 or -1 will cause problems) and it will still converge to the same value: Phi.
Phi2 = Phi + 1or, taking the square root of both sides:
Phi=(Phi+1)Can you see why we must start with a number which is not smaller (i.e. is not more negative) than 1?
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Phi has the value | 5 + 1 |
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We want to find a formula for the numbers (x, say) "that have the same decimal part as their squares". So, if we subtract x from x2, the result will be a whole number because the decimal parts were identical. Let's call this difference N, remembering that it is a whole number.
So
x = | 1 ± (1 + 4N) |
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N: | 1 | 2 | 3 | 4 | 5 | ... |
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1+4N: | 5 | 9 | 13 | 17 | 21 | ... |
For example: if we choose N=5, then the number x (that increases by exactly 5 when squared) is
x = | 1 ± (1 + 45) | = | 1 ± 21 | = 2.791287847.. and x2 = 7.791287847... = 5 + x |
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Another example: take Phi, which is (1 + 5)/2 or (1 + (1+41))/2 so that N=1. Thus we can "predict" that Phi squared will be (N=)1 more than Phi itself and, indeed, Phi=1.618033.. and Phi2=2.618033.. .
We can do the same for other whole number values for N.
More generally: There is nothing in the maths of this section that prevents N from being any number, for instance 0·5 or . Suppose N is pi (=3.1415926535... ). We can find the number x that, when squared, increases by exactly ! It is
x = | 1 ± (1 + 4) | = | 1 ± 12.566370614... | = 2·3416277185... |
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We look at some other numbers similar to Phi but in a different way on the (optional) Continued Fractions page. This time we find numbers which are like the Golden Mean, Phi, in that their decimal fraction parts are the same when we take their reciprocals, ie find 1/x. They are called the Silver Means.
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The Mathematical Magic of the Fibonacci Numbers | Fibonacci Home Page This is the first page on this topic. Where to now??? The next page on this Topic is... |
The Golden String |
If you apologize immediately then maybe, just MAYBE, I won't post my colonoscopy pictures.
Leni
Anyway, as soon as the movie flops, we can all get back to real life.
The reasons I have for not retracting "anything" I wrote are as follows:
1) you have not provided a scintilla of evidence to refute anything I wrote.
2) you have provided NO evidence that the Math Depts. of the top 50 Universities teach technical courses in "Sacred Geometry," which should be trivially easy to do, if it is as Mathematically important as you keep making it out to be.
3) your bloviating threats to "prove you to be a complete, total, utterly ignorant fool" don't impress me in the least.
4) and last, but not least, I decline your "invitation" to retract my statements because, how shall I put this...........
I already have a degree in Mathematics from a top-25 University.
To recapitulate my point, "Sacred Geometry" isn't part of the curricula of serious Mathematics Departments, and serious Mathematicians, like Peano, Cantor, Hilbert, Gödel, Whitehead, and Russell, didn't, and don't, spend their time working on it or worrying about what the value of "phi" is to a bazzillionth decimal place.
Hence, the idea that Space Aliens would travel the Galaxy to stomp "messages" about "Sacred Geometry" in farmer's fields makes no sense at all, at least to someone formally trained in Mathematics.
So spare me your assinine threats to "prove me a fool" and so forth, and spare me your private FReepmail telling me your opinion of my intellect and how closed-minded you think I am. But you are entitled to your opinion.....
Hehe...I love it when you're subtle! :)
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