Let's take a look at modern day money as it is used here in the United States, and see if it is proportionally up to the standard set by Mother Earth of ONLY exhibiting the Golden Ratio of 1.618 within the currency's dimensions of internal and external design elements.
Showing posts with label euclidean algorithm. Show all posts
Showing posts with label euclidean algorithm. Show all posts
RSA Encryption has been with us since 1987 securing our commercial transactions so that no one steals our private information and thus protects our bank accounts, our ICBM missiles tipped with nuclear warheads pointed at each other (be sure and see the movie: War Games - with Matthew Broderick, from the 1980s), and a whole bunch of other secrets we find so precious to keep in today's world.
Needless to say, I'm not a big fan of this scheme. I'd rather be poor (I am already, having lost my son to extreme honesty), then to have several nuclear warheads pointed at me and elsewhere. Without secrets, where would we be?
The first step in defrocking these secrets is to familiarize our self with how some of the more simpler encryption schemes operate. To this end, I present to you this description I gleaned from watching the following video on YouTube...
To aid us in studying his presentation, I have collected a few slides acting as reminders of Jordan's main points. Here is the first slide to help us encode and decode a secret message, "Hi"...
To help us understand the mechanics of how this encryption/decryption can actually happen, I've prepared a demonstration in JavaScript which is capable of performing all of the calculations on a scale small enough for it to handle, but not large enough to actually be secure...
The Euclidean Algorithm for finding the Greatest Common Denominator has always been a linear phenomenon repetitively subtracting the smaller - of a pair of integers - from the larger until a non-negative remainder smaller than the original subtrahend results. With each repetition, the remainder replaces the former subtrahend. This replacement scheme is what makes this style of computation a linear style incapable of breaking out of its limited view of the Golden Number series upon which it is built. This is not surprising, since the protocol for incrementing the Fibonacci series (of: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc upon which it is built) is also a linear phenomenon of adding the last number in the series with the integer immediately preceding it. So, the next number after 89 is found by adding it to 55 resulting in 144. To break out of this narrow definition of both Golden Numbers and the Euclidean Algorithm - with no offense intended toward Master Euclid, we must think in terms of a tablature of integers and take an integrated approach towards their computation. Let's begin with the Fibonacci series....
Instead of defining the Fibonacci series in terms of a series of integers, let's assume two series which increment in parallel with each other. It just so happens that these two parallel series are copies of each other but offset by one position. This is probably why we ignore their uniqueness and assume, for all intents and purposes, that there is only one series of numbers.
So, series 'a' got a head start over series 'b'. Now, how do we approximate the Golden Ratios of 1.618 and 0.618? Do we use the traditional method of dividing the smaller integer, among two neighboring integers, with its larger neighbor to its right? Or, do we integrate these two parallel series by dividing 'b' into 'a' for one ratio and 'a' into 'b' for another? Let's inspect the following regular pentagon of equal sides and equal angles (each angle is 108°)...
The answer is, yes we do integrate these two parallel series of integers, but it's not entirely obvious that there isn't only one series. For that premise to my hypothesis, we'll have to wait a little bit.
This Fibonacci series produces two roots, 1.618 and −0.618, which satisfy the following polynomial...
X2 − X1 − 1 = 0
Now, we'll move on to the next order of Phi, whose ratios
satisfy a third-order polynomial in one unknown, also known as a cubic
polynomial.
Now, we can approximate the three Golden Ratios of this 3rd order Fibonacci series by taking the following proportionate approximations among the diagonals of 'a', 'b', and 'c' of this regular septagon, or seven-sided polygon....
This progressive convergence upon a more accurate approximation results in the following three proportions...
To be able to make a 4th order polynomial in one unknown out of these four numbers will require that two of them are given a negative sign value:
−1 and
−0.532088886238
We can form this polynomial by multiplying these four values together.
But first, we have to turn them into linear expressions in one
unknown...
X1 = 2.87938524157
Subtract 2.87938524157 from both sides of the equal sign...
Simplifying further, yields a 4th order polynomial in one unknown...
X4− 2X3− 3X2 − X1 + 1 = 0
I hope these few examples have convinced you of a progressive tendency towards repeating a basic pattern to a higher and higher degree of Golden Ratios?
I'm going to move on, now, to explain how my expanded hypothesis of the Euclidean Algorithm is deduced from these prior examples...
Let's begin with the first diagram...
If we were to take the last two numbers in column 11, namely: 144 and 89, we could take the Greatest Common Divisor of these two integers, couldn't we? But how would the operation look the traditional way?
144 ÷ 89 = 1, with 55 remaining.
89 ÷ 55 = 1, with 34 remaining.
55 ÷ 34 = 1, with 21 remaining.
34 ÷ 21 = 1, with 13 remaining.
21 ÷ 13 = 1, with 8 remaining.
13 ÷ 8 = 1, with 5 remaining.
8 ÷ 5 = 1, with 3 remaining.
5 ÷ 3 = 1, with 2 remaining.
3 ÷ 2 = 1, with 1 remaining.
2 ÷ 1 = 2, with 0 remaining.
1 is the Greatest Common Divisor of 144 and 89.
Nobody would want to complicate this simple approach. It's so elegant as it is. But we'll have to in order to prepare ourselves for its subsequent expansion to higher ordered Golden Series of Numbers. Actually, we should have used the modulo operator, rather than division, since we're ignoring the quotient and only paying attention to the remainder...
144 % 89 = 1, with 55 remaining.
89 % 55 = 1, with 34 remaining.
55 % 34 = 1, with 21 remaining.
34 % 21 = 1, with 13 remaining.
21 % 13 = 1, with 8 remaining.
13 % 8 = 1, with 5 remaining.
8 % 5 = 1, with 3 remaining.
5 % 3 = 1, with 2 remaining.
3 % 2 = 1, with 1 remaining.
2 % 1 = 2, with 0 remaining.
1 is the Greatest Common Divisor of 144 and 89.
Here's a demonstration of Euclid's Algorithm from YouTube done in the usual way...
Now it's time for my method...
The Greatest Common Divisor of 144 and 89
two integers
144
89
sort from left to right
89
144
shift to the right
*
89
replace * with a zero
0
89
stack the two rows of integers,
the 'sort' on top of 'replace',
and take their modulo
89
%0
89
144
%89
55
sort from left to right
55
89
shift to the right
*
55
replace * with a zero
0
55
stack the two rows of integers,
the 'sort' on top of 'replace',
and take their modulo
55
%0
55
89
%55
34
sort from left to right
34
55
shift to the right
*
34
replace * with a zero
0
34
stack the two rows of integers,
the 'sort' on top of 'replace',
and take their modulo
34
%0
34
55
%34
21
sort from left to right
21
34
shift to the right
*
21
replace * with a zero
0
21
stack the two rows of integers,
the 'sort' on top of 'replace',
and take their modulo
21
%0
21
34
%21
13
sort from left to right
13
21
shift to the right
*
13
replace * with a zero
0
13
stack the two rows of integers,
the 'sort' on top of 'replace',
and take their modulo
13
%0
13
21
%13
8
sort from left to right
8
13
shift to the right
*
8
replace * with a zero
0
8
stack the two rows of integers,
the 'sort' on top of 'replace',
and take their modulo
8
%0
8
13
%8
5
sort from left to right
5
8
shift to the right
*
5
replace * with a zero
0
5
stack the two rows of integers,
the 'sort' on top of 'replace',
and take their modulo
5
%0
5
8
%5
3
sort from left to right
3
5
shift to the right
*
3
replace * with a zero
0
3
stack the two rows of integers,
the 'sort' on top of 'replace',
and take their modulo
3
%0
3
5
%3
2
sort from left to right
2
3
shift to the right
*
2
replace * with a zero
0
2
stack the two rows of integers,
the 'sort' on top of 'replace',
and take their modulo
2
%0
2
3
%2
1
sort from left to right
1
2
shift to the right
*
1
replace * with a zero
0
1
stack the two rows of integers,
the 'sort' on top of 'replace',
and take their modulo
1
%0
1
2
%1
0
1 is the Greatest Common Divisor of 144 and 89.
I call this process unzipping, because I'm reducing 144 and 89 in the same manner that created them in the first place...
By analyzing the manner in which these golden numbers zip themselves up and unzip themselves back down, we see that they have a very unique identity in that their collective quotients are always 1 and their remainders yield the prior set of Fibonacci numbers. Try and do that with non-Golden numbers!
We'll move on to the cubic order of Golden Numbers and see this same pattern repeat itself. And since the standardized, conventional method of performing the Euclidean Algorithm won't apply to this scenario, I must dispense with it and use my own...
Summary,
It can be said of Golden Numbers, of any degree Golden Polynomial for which their Golden Ratios are the solutions, that their unique property is that they close themselves. Only a Golden Number can be added to another Golden Number, from the same set, to build up a Golden Series. And only a Golden Number from the same set can be subtracted from another Golden Number to produce another smaller Golden Number from the same set.
It must also be emphasized that division is not occuring here, only repetitive subtraction, since we're not interested in the quotient – we're only interested in the remainder. Since the modulo operator is equivalent to repetitive subtraction, we use it to reduce integers during performance of the Euclidean Algorithm.
But we're engaging a special case, here, of using only Golden Numbers – not any integer – in the performance of the Euclidean Algorithm to find the Greatest Common Divisor among them. So, we need not utilize the services of the modulo operator. We may replace it with subtraction since we'll only be performing repetitive subtraction once yielding a quotient of one.
The Greatest Common
Divisor of 636, 793 and 353
three integers
636
793
353
sort from left to right
353
636
793
shift to the right
*
353
636
replace * with a zero
0
353
636
stack the two rows of integers,
the 'sort' on top of 'replace',
and perform subtraction
353
− 0
353
636
−353
283
793
−636
157
sort from left to right
157
283
353
shift to the right
*
157
283
replace * with a zero
0
157
283
stack the two rows of integers,
the 'sort' on top of 'replace',
and perform subtraction
157
− 0
157
283
−157
126
353
−283
70
sort from left to right
70
126
157
shift to the right
*
70
126
replace * with a zero
0
70
126
stack the two rows of integers,
the 'sort' on top of 'replace',
and perform subtraction
70
− 0
70
126
− 70
56
157
−126
31
sort from left to right
31
56
70
shift to the right
*
31
56
replace * with a zero
0
31
56
stack the two rows of integers,
the 'sort' on top of 'replace',
and perform subtraction
31
− 0
31
56
− 31
25
70
− 56
14
sort from left to right
14
25
31
shift to the right
*
14
25
replace * with a zero
0
14
25
stack the two rows of integers,
the 'sort' on top of 'replace',
and perform subtraction
14
− 0
14
25
− 14
11
31
− 25
6
sort from left to right
6
11
14
shift to the right
*
6
11
replace * with a zero
0
6
11
stack the two rows of integers,
the 'sort' on top of 'replace',
and perform subtraction
6
− 0
6
11
− 6
5
14
− 11
3
sort from left to right
3
5
6
shift to the right
*
3
5
replace * with a zero
0
3
5
stack the two rows of integers,
the 'sort' on top of 'replace',
and perform subtraction
3
− 0
3
5
− 3
2
6
− 5
1
sort from left to right
1
2
3
shift to the right
*
1
2
replace * with a zero
0
1
2
stack the two rows of integers,
the 'sort' on top of 'replace',
and perform subtraction
1
− 0
1
2
− 1
1
3
− 2
1
sort from left to right
1
1
1
shift to the right
*
1
1
replace * with a zero
0
1
1
stack the two rows of integers,
the 'sort' on top of 'replace',
and perform subtraction
1
− 0
1
1
− 1
0
1
− 1
0
1 is the Greatest Common Divisor of 636, 793 and 353.
In conclusion,
The study of the Infinite Range of Golden Ratios, and the Golden Numbers which spawn them, yields a fuller appreciation for the Euclidean Algorithm and a more efficient method for discovering the Greatest Common Divisor among any quantity of integers.
Here are some JavaScripts to bring my method to life! This first one generates a few integers at random and multiplies them against another small integer to find their Greatest Common Divisor.
This next JavaScript will accept any quantity of non-zero integers so long as they are not equal each other...
