Figure 3. We can do a similar thing with the Golden Rectangle. We can make a set of Whirling Rectangles that produces a similar logarithmic spiral. Again this spiral converges at the intersection of the two blue lines, and these ratio of the lengths of these two lines is in the Golden Ratio.
I will outline the proof that the ratio of the lengths of the two diagonals is indeed the Golden ratio. An interesting thing happens when we work with these whirling rectangle. Suppose we take a rectangle of side 1 unit and a rectangle of side 2 units and we put them side to side in the following fashion and draw the appropriate segments to form a rectangle.
If we continue to create rectangles in this fashion we will get a series of whirling rectangles like the one of above. The following picture shows several such rectangles, and the lengths of their sides. If we take ratios of the length we will see that the series of whirling rectangles will begin to estimate the Golden Ratio. Hence as we increase the number of squares we get a figure that begins to look more and more like the Golden Rectangle. It might also be noticed that there is something special about the sides of the squares.
If we list them we have, 1, 2, 3, 5, 8, 13, This of course is the famous Fibonacci sequence. As will be shown in the rest of the essay, the Fibonacci sequence and the golden ratio are intertwined with each other. Leonardo Fibonacci began study of this sequence by posing the following problem in his book, Liber Abaci How many pairs of rabbits will be produced in a year, beginning with a single pair , if in every month each pair bears a new pair which becomes productive from the second month on?
Of course, this problem gives rise to the sequence 1, 1, 2, 3, 5, 8, 13, This particular sequence has some interesting properties.
Instead of proving these, I will justify them by use of a spreadsheet. I began by finding the first 40 Fibonacci numbers, which will be used in most of the demonstrations. The values can be found in column A.
One of the first propeties that might be noticed is that for two positive integers a and b, if a b then F a F b. The converse also holds true If F a F b , then a b , and proof of these two properties can be found in Maxfield and Maxfield Another property of the Fibonnaci numbers is that no two consecutive numbers in the sequence have a common prime factor. As an example consider F 7 through F 10 All though we can look at any of the first 40 numbers , and see that no consectutive pair has a common factor.
The proof is as follows. Another property that is verifiable by looking at the first forty terms of the sequence, is that the Fibonnaci sequence is complete with respect to the positive integers.
This means that every positive integer can be written as the sum of finite terms from the sequence with no term being used more than once. He continued doing research right up until his sudden death while relaxing with a cup of tea.
For almost all of the last 17 years of his life he was totally blind. The breadth of Euler's knowledge may be as impressive as the depth of his mathematical work. He had a great facility with languages, and studied theology, medicine, astronomy and physics. His first appointment was in medicine at the recently established St.
Petersburg Academy. On the day that he arrived in Russia, the academy's patron, Catherine I, died, and the academy itself just managed to survive the transfer of power to the new regime.
In the process, Euler ended up in the chair of natural philosophy instead of medicine. Euler is best remembered for his contributions to analysis and number theory, especially for his use of infinite processes of various kinds infinite sums and products, continued fractions , and for establishing much of the modern notation of mathematics.
Euler's greatest contribution to mathematics was the development of techniques for dealing with infinite operations. In the process, he established what has ever since been called the field of analysis , which includes and extends the differential and integral calculus of Newton and Leibniz. Euler used infinite series to establish and exploit some remarkable connections between analysis and number theory.
Euler's uncritical application of ordinary algebra to infinite series occasionally led him into trouble, but his results were overwhelmingly correct, and were later justified by more careful techniques as the need for increased rigor in mathematical arguments became apparent. We'll see Euler's name more than once in the remainder of the chapter. Prove this. Collapse menu 1 Logic 1.
Logical Operations 2. Quantifiers 3. De Morgan's Laws 4. Mixed Quantifiers 5. This would include all infinities, also. The Super Set is inclusive of all number in every form. The difference is that the second set fits into the space between 0 and 1 on a number line while the first set extends from 1 infinitely into spac ealong the same number line … and then there are the negative integers: -1,-2,-3,-4 etc.
And we are taught this is all random the result of an explosion common man! Scientifically speaking: do we ever get order out of chaos? How many times do we have to blow up a junkyard of airplane parts before we have a jetliner or a single-engine prop-plain, The odds for success dwindle after each attempt? To improve our chances we blow up all the junkyards once. Will we get a Rolls Royce or even a pinto? The laws were set or created by prior knowledge, to deny it is only the result of faulty information or input.
It seems the more knowledge one learns the less wisdom they retain or can attain! I have been working on an Angelic translator version of a 10 place system. I can say that your poem is very much on target, and I can show you the test of your equation in order for you to move on in either a quadratic solution or possibly a quantum solution.
Reply back for more info. Oh no, no, no! What does the paragraph mean? Refer to this set when needed. How did we get this list? Begin with 0. The 1st whole number after 0 is 1. I'll explain what I mean by "approaches. See how our ratio went from 2, 1. And we started at 0, right? Nor I, but he may be referring to the Temple in Jerusalem.
The sanctuary was the ultimate design ratio of phi. You can also use phi to calculate the circumference of a circle to within. Which pretty much makes pi obsolete. Go the golden mean! Finally, someone said it. Way, way beyond mystifying. Love your point math is math! Do you see why? Also, is there anything I can make a scaled model of to show?
Please reply to this by 6th July. Same with from your shoulder to your fingertips divided by elbow to fingertips, and hip to floor divided by knee to floor. Try it. Try not to look for important truths about the natural world in fiction bestsellers. Check your data. The navel to height ratio is very close to phi for many people, including myself.
Even in his classic debunking article, Ph. George Markowsky measured his own family and found an average ratio of 1. I thought Phi is only one Mathematics constant. Thanks for your share. Great documentary and informative. Might take a college course in logic to get there though.
I have been very focused on sacred geometry this year. I have received much guidance regarding the creation of geometric forms and their uses. I love how numbers pull together so many different aspects of life. Like Phi and the Pythagorean musical scale.
The relationship between 3 and 2. That precise a measurement on an ancient crumbling pyramid is nonsensical. It would seem these numbers are especially doctored to fit the phi number. Buddhism for example. Not all Buddhists are non-theistic. The top-left-pointing swastika is the four-square tetragram which represents the current pole star Polaris and the four positions of the big dipper Ursula major. This is symbolic of the heavenly way of Life Eternal abiding in Jesus Christ.
The way is for those who see Him unveiled, as well as those who see Him veiled, but believe it to be a mediator, as He came both to give life, and life more abundantly, by all, three measures of grace.
Perhaps if this were better understood, things like the Chinese genocide of Tibetans would not be occurring. That is a number worth considering. A few of clarifications here. The four positions of the big dipper sp. Ursa Major is top-right. Before it was used for infamy, the top-right swastika was not only a Buddhist symbol, but it was the basis of the Basque cross, the lauburu.
It appears to be spinning counter-clockwise, which is the direction of the stars when one looks at Polaris. That symbol appears to be spinning clockwise. That is the same direction as the appearance of the yin-yang symbol, which represents the motion of the sun and moon when you face away from Polaris. Be thoughtful. If i have words consisting numbers of 30, 40, 1 then how will i calculate that according to this golden ratio? And what is the process of calculating this, im not getting this.
To find the golden ratios of any number, just multiply it by 1. Golden ratio relationships with 30 are thus found at You can round that to 49 and 19 if needed. This mind-blew me so good.
I love the orgasmic effect it instigates in my mind. Truly a large phenomena, and will always be on my mind. Phi is much more than amazing. Nobody in my year knows what it is either. I did my student teaching with a guy who is a Vietnam vet with 3 Purple Hearts, a three decade devotee of Parmahansa Yogananda and the Self-Realization Church, and my best friend. The city we live in, where he has spent practically his whole life, has a Central Ave. If you drop a location pin from a Google map app on a cell phone for N.
The place he has been learning to be enlightened for most of his adult life. He was married there, and even renewed his vows there. But I can guarantee that he was never taught that his International Church, started almost ears ago, has locations worldwide, with addresses, area codes, zip codes, phone number, etc. How do I tell him? Do I tell him? Mr Hedding i suggest you 1. Study history a bit better there are many deceptions and lies. Suggestion 1 is your best bet because if you understood Phi you would see it proves intelligent design.
Alan I suggest you 1.
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