WEBVTT 00:00.000 --> 00:07.240 So now let's finally just end this video off with just a brief mainstream history of Fibonacci and more very interesting stuff 00:07.240 --> 00:12.940 Yeah, it's just it's been a habit that I've been going over in my earlier videos whenever I cover a concept in mathematics 00:12.940 --> 00:18.680 I would like to go discuss the mainstream history behind it and behind people behind it very interesting stuff 00:18.680 --> 00:23.940 So I'll continue with that theme. So let's dive into here. So on Wikipedia for Fibonacci 00:23.940 --> 00:28.260 Let's read up on him. So Fibonacci born in 00:28.260 --> 00:31.580 1175 common era or AD and 00:32.820 --> 00:34.660 He was and then he died around 00:35.180 --> 00:39.340 1250 common era was an Italian mathematician from the Republic of Pisa 00:39.340 --> 00:44.700 Considered to be the most talented Western mathematician of the Middle Ages. The name he is commonly called 00:45.340 --> 00:48.840 Fibonacci or Italian Fibonacci was made up in 00:49.380 --> 00:51.340 1838 by the Franco Italian historian 00:52.200 --> 00:57.000 Julian Libyir and or Libri and is short for 00:57.000 --> 01:02.020 Philius Banachi son of the Banachi and he also is known as 01:02.020 --> 01:05.260 Leonardo Banachi Leonardo of Pisa Leonardo 01:06.060 --> 01:11.320 Pizzano Bigelow or Leonardo Fibonacci. So yeah, so the name is apparently made up 01:12.020 --> 01:13.680 Very very interesting 01:13.680 --> 01:17.460 But anyways going for this Fibonacci popularized the Hindu Arabic 01:18.060 --> 01:24.840 Numeral system in the Western world primarily through his composition in 1202 of Libre Abasi book of calculation 01:24.840 --> 01:30.620 He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Libre Abasi 01:30.620 --> 01:34.040 So here's a image. Here's a statue of him zoom in 01:34.040 --> 01:42.900 They're very interesting statue of Fibonacci. It's got a pretty nice robe. It's going over. So statue Fibonacci 1863 by Giovanni 01:42.900 --> 01:46.000 Paganasi in the composite 01:46.940 --> 01:52.860 Composalato de Pisa, so you're born 1175 common era in Pisa in Italy 01:52.860 --> 01:54.760 And he died most likely 01:55.540 --> 01:58.680 1240 to 50 in most like in Pisa 01:59.700 --> 02:04.640 Occupation mathematician and he was known for Libre Abasi 02:05.340 --> 02:10.640 popularizing the Hindu Arabic numeral system in Europe Fibonacci numbers and his parents was 02:12.980 --> 02:18.020 Guglielimo, Giuliumo, that's what it pronounced. So history Fibonacci was born around 1175 to 02:19.460 --> 02:24.800 Guglielimo or Giuliumo a wealthy Italian merchant and by some accounts a consul for Pisa 02:25.520 --> 02:31.780 In console for Pisa. Guglielimo directed a trading post in Bugia a port in the 02:32.680 --> 02:33.960 Almoa had dynasties 02:34.820 --> 02:41.460 Sol Sol Anet Nesola net in North Africa Fibonacci traveled with him as a young boy 02:41.460 --> 02:44.360 And it was in buggy buggy oh or big yeah now 02:45.020 --> 02:48.140 Algeria that he learned about the Hindu Arabic numeral system 02:48.640 --> 02:55.200 So yeah, Fibonacci traveled extensively around the Mediterranean coast meeting with many merchants and learning about their systems of doing arithmetic 02:55.800 --> 02:59.820 He soon realized the many advantages of the Hindu Arabic system in 1202 02:59.820 --> 03:05.300 He completed the Libre Abasi book of Abascus or book of calculation which popularized the 03:05.300 --> 03:13.340 Hindu Arabic numerals in Europe Fibonacci became a guest of Emperor Frederick II who enjoyed mathematics and science in 03:13.340 --> 03:18.040 1240 the Republic of Pisa honored Fibonacci referred to as Leonardo Bigolo 03:18.780 --> 03:25.480 By granting him a salary in a decree that recognized him for the services that he had given to the city as an advisor on matters of 03:25.480 --> 03:26.920 Accounting and instruction to citizens 03:26.920 --> 03:34.200 The date of Fibonacci's death is not known, but it has been estimated to be between 1240 and 1250 in most likely in Pisa 03:34.200 --> 03:39.080 So Libre Abasi 12 or 2 you remain article if you want to go read that on Wikipedia 03:39.080 --> 03:44.880 So in the Libre Abasi 12 or 2 Fibonacci introduced the so-called modus 03:45.500 --> 03:51.720 Modus inter or interum method of in the Indians today known as a Hindu Arabic numeral system 12 or 2 03:51.720 --> 03:56.680 So his birthday he was born 1175 so believe that's about 27 years old 03:57.260 --> 04:01.600 Interesting so the book advocated numeration with the digits 0 to 9 and place value 04:01.600 --> 04:07.660 The book showed the practical use and value of the new Hindu Arabic numeral system by applying the numerals to commercial bookkeeping 04:07.660 --> 04:12.660 Converting weights and measures calculations calculations of interest money changing and other applications 04:12.660 --> 04:15.500 The book was well received throughout in the 04:15.500 --> 04:22.040 Throughout educated Europe and had a profound impact on European thought no copies of the 12 or 2 edition are known to exist 04:22.040 --> 04:27.740 The 1228 edition however first section introduces a Hindu Arabic numeral system and 04:27.740 --> 04:32.260 Comparise the system with other systems such as Roman numerals and the methods to convert the other 04:32.760 --> 04:35.660 numeral systems into Hindu Arabic numerals 04:36.340 --> 04:41.000 Replacing the Roman numeral system is ancient Egyptian multiplication method and using an 04:41.800 --> 04:47.580 Abacus for calculations with a Hindu Arabic numeral system was an advance in making business calculations easier and faster 04:47.580 --> 04:50.180 Which led to the growth of banking and accounting in Europe 04:50.180 --> 04:56.340 The second section explains the uses of a Hindu Arabic numerals in business for example converting 04:56.860 --> 05:01.840 Different currencies and calculating profit and interest which were important to the growing economy 05:01.840 --> 05:07.060 Growing banking industry the book also discusses the rational numbers and prime numbers 05:07.060 --> 05:12.120 So here's a page from his 1228 edition of the book so page of Fibonacci's liberal 05:12.740 --> 05:18.540 Abassi from the Bibliotech National the frenzy showing in box on the right the Fibonacci sequence 05:18.540 --> 05:23.900 So the position of the sequence labeled in Roman numerals and a value in Hindu Arabic numerals and now yeah 05:23.900 --> 05:29.320 And I believe is it 1228 edition now, but it could be a different edition, but anyways, so we have 05:29.320 --> 05:35.740 Yeah, scan copy here. And yeah, so I believe these ones would be the Roman numerals 05:35.740 --> 05:39.240 I'm not sure which ones was probably some of these ones different lettering here 05:39.240 --> 05:43.340 And then I believe these are the other the Indian numbers similar to ours one two three four to nine 05:43.800 --> 05:46.840 Believe that's a 144. I'm not sure. I believe they're just different 05:48.320 --> 05:54.000 Numbers for theirs, but it's very very interesting stuff here. Let's just continue further 05:54.540 --> 05:57.220 Into this so now let's look at the Fibonacci sequence 05:57.220 --> 05:59.860 And there you can read more main article Fibonacci number 05:59.860 --> 06:05.140 I'll cover that actually in a bit so labor about see posed and solve the problem involving the growth of a population of 06:05.140 --> 06:11.940 Rabbits based on idealized assumptions the solution generation by generation was a sequence of numbers later known as Fibonacci numbers 06:11.940 --> 06:17.760 Although Fibonacci's labor about see can contains the earliest known description of the sequence outside of India 06:17.760 --> 06:22.180 The sequence has been noted by Indian mathematicians as early as the sixth century 06:22.700 --> 06:27.220 Fascinating in the Fibonacci sequence of numbers each number is the sum of the previous two numbers 06:27.220 --> 06:34.740 Fibonacci began the sequence not with zero one one two as modern mathematicians do but with one one two, etc 06:35.500 --> 06:38.160 He carried the calculation up to the 13th place 06:38.160 --> 06:43.460 And then 14th in modern computing because there's a zero get a zero of the next for a layer there 06:43.460 --> 06:45.940 Yes, yes, so that is two three three 06:45.940 --> 06:53.360 Though another manuscript carries it to the next place one one two three five eight thirteen twenty one thirty four fifty five eighty nine 06:53.960 --> 07:01.380 144 233 377 and Fibonacci did not speak about the golden ratio as a limit of the 07:01.380 --> 07:05.200 Ratio of consecutive numbers in the sequence very interesting 07:05.200 --> 07:09.500 Yes, and I'll explain the golden ratio as soon as I illustrated earlier above as well 07:09.500 --> 07:14.100 So like I say the 19th century of statue Fibonacci was constructed and raised in Pisa today 07:14.100 --> 07:18.000 It is located in the western gallery of the camp Ozana 07:18.000 --> 07:22.660 So our historical cemetery on the Piazza di 07:24.100 --> 07:27.360 Miracoli interesting stuff here guys. So here's a just image from him 07:27.360 --> 07:31.860 I found on our mainstream news site express.co UK's is drawing her 07:31.860 --> 07:34.120 a drawing or a photograph of him 07:34.120 --> 07:39.960 So this is Fibonacci believe you there's 27 when you wrote that books probably younger image of him 07:39.960 --> 07:43.640 So now let's quickly go over some of the mainstream history and 07:43.640 --> 07:49.060 Overview of the Fibonacci sequence and his close relationship with the golden ratio your hashtag 07:49.660 --> 07:53.940 Fascinating stuff so going further. Yeah, so he's from Wikipedia Fibonacci numbers 07:53.940 --> 07:59.940 So in mathematics the Fibonacci numbers are the numbers in the following integer sequence called the Fibonacci sequence and 07:59.940 --> 08:04.200 Characterized by the fact that every number after the first two is the sum of the preceding 08:04.200 --> 08:08.320 One so one one so one one then we have two then again 08:08.320 --> 08:15.080 We have two plus one is three two plus three is five three plus five is eight and so on so we have it all 08:15.640 --> 08:16.480 like that 08:17.280 --> 08:19.460 All right, so now often especially in modern usage 08:19.460 --> 08:23.800 The sequence is extended by one more initial digit and that's adding a zero there 08:23.800 --> 08:28.580 So again, this one one is the sum of zero and one and so on so by definition 08:28.580 --> 08:32.480 The first two numbers in the Fibonacci sequence are either one and one or zero and one 08:32.940 --> 08:37.880 Depending on the chosen starting point of the sequence and each subsequent number and the sum of the previous two 08:38.980 --> 08:45.400 No, is the sum of the previous two so the sequence f and Fibonacci numbers is defined by the recursion relation as I went over 08:45.400 --> 08:52.900 And that is basically f n equals f n minus one plus f n minus two or if zero equals one 08:53.540 --> 08:54.700 Actually, I just 08:55.740 --> 08:57.440 Thoughtlessly read that one on so yeah 08:57.440 --> 09:04.700 So it's this sequence with seed values or the initial values f one equals one and f two equals one or for modern usage 09:04.700 --> 09:09.080 Or modern computing usage usually it's f zero equals zero and f one equals one 09:09.080 --> 09:14.660 And now here is a tiling with squares whose side lengths are successive Fibonacci numbers 09:14.660 --> 09:18.980 So we have one one and two then go to side three then you go to five 09:18.980 --> 09:22.260 Then you go to eight like this and you go to 13 09:22.980 --> 09:28.120 It's very interesting. These are the side length. Yes, I believe these are the yeah, these are square 09:28.120 --> 09:31.520 So each one is 13 so this can be 13 like that and 09:32.600 --> 09:35.860 So on that is very interesting how they all line up perfectly like that 09:35.860 --> 09:39.020 Then it goes all the way to 21 and 21 lines up with that as well 09:39.020 --> 09:42.680 That's actually very very amazing. Yes. Yeah, I just want to add a quick note there 09:42.680 --> 09:48.820 So very interesting how the side lengths match up with the previous squares as they spiral around very interesting 09:48.820 --> 09:55.020 That's actually quite fascinating stuff and I hear here's further is a Fibonacci spiral 09:55.020 --> 09:58.480 Which is an expo an approximation to the golden spiral 09:59.020 --> 10:03.940 Created by drawing circular arcs connected at the opposite corners of 10:03.940 --> 10:09.800 Yeah, of squares in the Fibonacci tiling so these are the opposite corners like that goes to here 10:09.800 --> 10:12.080 This goes to here and you start off like that 10:12.700 --> 10:14.900 Very very fascinating stuff indeed 10:14.900 --> 10:19.800 And this one uses squares of size 1 1 2 3 5 8 13 21 10:19.800 --> 10:21.480 Very fascinating stuff 10:21.480 --> 10:25.560 So Fibonacci numbers appear to have first arisen and perhaps 200 BC and work by 10:25.560 --> 10:32.700 Ping ping gala on enumerating possible patterns of poetry formed from syllables of two lengths the Fibonacci 10:32.700 --> 10:38.520 Yes sequence is named after a tally mathematician Leonardo of Pisa known as Fibonacci 10:38.520 --> 10:41.420 Yeah, his 12 o 2 book 10:41.420 --> 10:45.580 Yeah, liver of us he introduced a sequence to Western European mathematics 10:45.580 --> 10:48.680 Although the sequence had been described earlier in Indian math 10:49.420 --> 10:56.240 Mathematics the sequence described in liver of us he began with f1 equals 1 Fibonacci numbers were later later 10:56.240 --> 10:59.260 independently discussed by Jonas Kepler in 10:59.260 --> 11:00.980 1611 in connection with 11:00.980 --> 11:06.360 Proximations to the Pentagon there. So yeah, it's very fascinating to actually go look into that 11:06.360 --> 11:10.380 Yeah, so their recurrence relation appears to have been understood from the early 11:11.200 --> 11:13.520 1600s, but it has been only in the past few 11:14.140 --> 11:18.320 Past very few decades that they have in general become widely discussed 11:18.840 --> 11:24.960 Interesting, so they are intimately connected with the golden ratio for example the closest rational approximations to the ratio are 11:24.960 --> 11:31.120 Into the golden ratio or two or one two over one three over two five over three eight over five and so on 11:31.120 --> 11:37.160 That's fascinating. So that you see the Fibonacci numbers there two three then five then eight and so on 11:37.160 --> 11:40.700 That's actually very amazing. So Fibonacci numbers appear unexpectedly 11:41.520 --> 11:47.160 Often in mathematics so much so that there is an entire journal dedicated to their study the Fibonacci quarterly 11:47.760 --> 11:54.540 applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data 11:54.540 --> 11:56.380 structure in 11:56.380 --> 12:00.100 Grounds called Fibonacci cubes used for interconnecting parallel and distributed systems 12:00.100 --> 12:05.100 Very very fascinating. They also appear in a biological setting such as branching leaves of 12:05.100 --> 12:12.300 Philo taxes the arrangement of leaves on a stem the fruit sprouts of a pineapple the flowering of an artichoke and 12:12.300 --> 12:15.640 uncurling fern and the arrangement of a pine cones 12:16.460 --> 12:24.980 Bracks, so yes, you'll see this everywhere and the Fibonacci numbers are the sums of the shallow diagonal shown in red of Pascal's 12:25.560 --> 12:29.500 Triangle so notice how this goes up to one. So this one here if you have one 12:29.500 --> 12:35.220 This is gonna be one be some of zero and one and this one here some of one and zero 12:35.220 --> 12:39.920 And then if you have over here two, it's gonna be some of one plus one is two 12:40.640 --> 12:45.900 And then the three is gonna be one plus two is three the six as you can see is this some of the top two 12:45.900 --> 12:47.540 So three and three six 12:48.120 --> 12:52.920 So you go to here 35 20 plus 15 is 35 and here if you sum all these 12:52.920 --> 12:54.240 Shallow angles 12:54.840 --> 12:58.600 It's actually pretty amazing shall diagonally get one then you get one again 12:58.600 --> 13:05.460 Then you get your two and you get this was made to one just three and five eight thirteen twenty one thirty four 13:05.460 --> 13:12.360 So that's absolutely fast. Yeah. Yeah. So now let's look at the golden ratio 13:12.360 --> 13:19.080 And go over here and let's read up on this one mathematics to quantities on the goal or in the golden ratio 13:19.080 --> 13:25.840 If the ratio is the same as the ratio of their sum to the larger of the two quantities and the figure on the right 13:25.840 --> 13:28.860 Oh, it's actually below illustrates a geometric relationship 13:28.860 --> 13:34.180 Express algebraically for quantities a and b with a greater than b greater than zero 13:34.180 --> 13:39.840 We have over here a plus b over a if it equals to a over b 13:39.840 --> 13:45.820 Then this is by definition phi. This is phi or the golden ratio where the Greek letter phi 13:46.340 --> 13:49.980 Written like this lowercase phi or phi represents the golden ratio 13:49.980 --> 13:55.960 It is an irrational number with a value of and this is exactly the one I calculated exercise five 13:55.960 --> 14:02.140 So one plus square root five divided by two equals one point six one eight zero three three nine eight eight seven 14:02.140 --> 14:07.940 And I mean so this is the limit of the ratio of successive numbers of the Fibonacci sequence 14:07.940 --> 14:09.540 Yeah in exercise 14:10.520 --> 14:15.860 Four and now remember the exercise four I just want to scroll up quickly because I think I may have mentioned 14:15.860 --> 14:21.640 Earlier that the limit of the sequence is golden ratio. That's actually not the case. It's 14:22.220 --> 14:24.160 It's the ratio. So here number here 14:24.160 --> 14:28.280 So we let a a n equals to fn plus one divided by fn like that 14:28.280 --> 14:31.240 So that's the ratio of the six successive terms of the Fibonacci sequence 14:31.240 --> 14:35.800 But the Fibonacci sequence limit is well just keeps getting larger and larger. So it's infinity. So yes 14:35.800 --> 14:40.460 So it's the ratio and I showed here then that when you do the ratio and becomes like this 14:40.460 --> 14:46.960 One point six one eight. It's just fascinating stuff indeed. Let's just go back over here 14:47.760 --> 14:49.400 The golden ratio 14:49.400 --> 14:53.220 Yeah, so the golden ratio is called the golden mean or golden section and Latin 14:53.940 --> 14:54.500 sectio 14:55.480 --> 14:58.660 Aruria other names include extreme and mean ratio and 14:58.660 --> 15:05.680 Medial section divine proportion divine section or Latin sectio divina golden proportion golden cut 15:05.680 --> 15:11.800 And golden number mathematicians since Euclid have studied the properties of the golden ratio including its appearance in the dimensions of a regular 15:11.800 --> 15:16.220 Pentagon so you have to dig into that is very fascinating and in a golden rectangle 15:16.220 --> 15:21.280 Which may be cut into a square and a small rectangle with the same aspect ratio 15:21.280 --> 15:28.380 The golden ratio has also also been used. He has been used to analyze portions of natural objects such as 15:29.040 --> 15:34.420 Yes, as well as man-made systems such as financial markets in some cases based on debius fits the data 15:34.420 --> 15:41.640 So yes, interesting. So the golden ratio appears in some patterns in nature including this spiral regiments of leaves and other plant parts 15:41.640 --> 15:45.880 some 20th century artists and architects including Le Corbusier and 15:45.880 --> 15:46.440 Solva 15:47.420 --> 15:51.300 Salvador Dali have proportioned their works to approximate the golden ratio 15:51.300 --> 15:56.780 Especially in the form of the golden rectangle in which the ratio of the longer side to the shorter side is 15:56.780 --> 16:01.320 The golden ratio and I believe in this portion be aesthetically pleasing 16:01.320 --> 16:06.740 So here is a line segments in the golden ratios. You have a and b where the summation a plus b 16:06.740 --> 16:11.000 Divided by a is yes, so a plus b is to a as a is to b 16:11.000 --> 16:17.160 So it's the same ratio and here's the golden rectangle with longer side a and shorter side b 16:17.160 --> 16:22.460 One place adjacent to a square with side lengths a and b will decide lengths a 16:22.460 --> 16:29.500 Will produce a similar or golden rectangle with longer side a plus b and shorter side a 16:29.500 --> 16:31.660 So yes, we have a golden 16:32.260 --> 16:37.980 Rectangle over here. So a plus b over a that's that's the golden rectangle as well as a 16:38.820 --> 16:41.680 Over b so a over b golden rectangle golden 16:42.300 --> 16:44.960 Rectangle and then the overall one is gonna be golden 16:45.820 --> 16:51.920 Rectangle put both them together and also what you could do here now what I'm gonna do is well, I'm gonna divide this out 16:52.840 --> 16:53.360 by 16:54.040 --> 16:55.040 divided out by 16:55.040 --> 16:59.220 B so if you divide it out by b I'll just divide every single side 16:59.220 --> 17:05.000 We're gonna get a smaller rectangle obviously just to illustrate it. So it'll have as b over b is one 17:05.580 --> 17:07.260 a over b is 17:07.260 --> 17:13.520 Phi and then over here is gonna be well phi a over b and this is well 17:14.100 --> 17:17.000 a over b is equal to phi 17:17.800 --> 17:19.700 So Phi so we have a 17:21.440 --> 17:22.120 A 17:22.120 --> 17:28.840 Square phi like that and this is fine like that. So yes, there's one that's why it's a fascinating stuff 17:28.840 --> 17:30.060 So we're going further 17:30.660 --> 17:36.800 Now relationship the Fibonacci sequence the mathematics of the golden ratio in the Fibonacci sequence are intimately interconnected 17:36.800 --> 17:40.780 The Fibonacci sequence is 1 1 2 3 5 8 17:40.780 --> 17:46.780 13 21 34 55 89 144 233 37 7 6 10 17:46.780 --> 17:47.700 87 17:48.220 --> 17:52.220 And a cool closed form expression for the Fibonacci sequence involves a golden ratio 17:52.220 --> 17:57.480 And you get ready like this fn equals to any simplified Phi n minus 17:58.180 --> 18:01.540 negative Phi to power of negative n divided by a 18:01.540 --> 18:07.000 Squared Phi so the golden ratio is a limit of the ratios of successive terms of the Fibonacci 18:07.000 --> 18:11.500 Sequence or any Fibonacci like sequence as originally known shown by Kepler 18:11.500 --> 18:18.500 So limit as and reports infinity of fn plus 1 divided by fn equals to Phi as I show it 18:18.500 --> 18:25.040 This is exactly exercise 4 in other words of the Fibonacci numbers divided by its immediate predecessor in the sequence 18:25.040 --> 18:30.580 The quotient approximates Phi for example 9 8 7 over 6 10 18:30.580 --> 18:36.960 It is going to be roughly equal to Phi like that and so on so for example 2 over 1 is going to be well 18:37.480 --> 18:40.840 It's going to be 2 so then then 3 over 2 will be 18:40.840 --> 18:45.470 1.5 so it gets closer and closer to that number and 18:46.580 --> 18:48.940 Then 5 over what's 5 over 3? 18:49.680 --> 18:53.100 That is let's see what 5 or 3 is so I'm going to type it in quickly 18:53.940 --> 18:55.840 5 over 3 18:56.360 --> 19:02.660 equals 1.667 and so on so that's faster than 8 over 5 19:03.760 --> 19:10.720 Equals 1.6 so yes, you'll eventually get closer and closer and there is numbers in the Phi I mean in the 19:10.720 --> 19:16.680 Later parts in the Fibonacci sequence so it's fascinating something so these approximations are all 19:16.680 --> 19:22.300 Alternately lower and higher than Phi and converge to Phi as the Fibonacci sequence increase 19:22.300 --> 19:27.100 So yes, they all turn out pretty amazing. Let me just type that in over here actually and note here 19:27.100 --> 19:30.300 I just want to illustrate this further so an MES note 19:30.300 --> 19:33.600 Here's the Fibonacci sequence and here is Phi 19:34.140 --> 19:36.920 So let's just remove this dot 19:36.920 --> 19:42.620 Yes, so put Phi over there and let's see what happens when you do that the term so 1 over 1 that is obviously 19:43.820 --> 19:45.520 So yeah, now we have over here 19:45.520 --> 19:52.300 So 1 over 1 is 1 2 over 1 is equal to 2 and then going further here. We have 3 19:53.020 --> 19:56.780 If so then the next one is is yet 3 over 2 19:56.780 --> 20:03.600 Let's see how it looks like and the next one is we have it's 5 over 3 20:04.420 --> 20:10.260 The next one is we have let's remove the spaces. So now we have next one is 8 over 20:10.860 --> 20:12.740 3 and 8 over 5 20:12.740 --> 20:18.540 Equals 1.6 and then we have next one is 13 over 20:20.080 --> 20:21.380 Over 8 20:22.580 --> 20:25.200 And the next one we have is 20:26.160 --> 20:27.040 21 over 20:27.920 --> 20:28.360 13 20:29.500 --> 20:36.820 So you can see it's a yes, it's going above and below the 1.6 when it's initially it's below above below above 20:37.740 --> 20:42.340 Below and then this one's going to be above and then this is below and then the next one 20:42.340 --> 20:47.600 Yes, very very fascinating stuff. The next one is 34 divided by 21 20:48.240 --> 20:54.420 Now we have this is above above 1.6 18 and then we have 55 over 20:55.020 --> 21:01.640 89 equals I mean the other way around we have 55 over 34 21:04.120 --> 21:05.480 Over 34 21:06.500 --> 21:12.260 And this equals that yet below and then we have next one 89 21:12.900 --> 21:14.860 Yeah, 89 over 21:15.900 --> 21:19.380 55 equals 1.6 18 that's a bit higher 21:19.380 --> 21:23.440 So there's 1.6 182 and then keep going on. This is pretty fun 21:24.340 --> 21:28.040 114 over 89 equals one point 21:28.860 --> 21:30.540 Yeah, that's off. So 144 21:31.880 --> 21:33.060 Like that 21:35.780 --> 21:39.920 So 1.6 18 and then 233 over 21:41.070 --> 21:41.460 144 21:42.420 --> 21:48.300 That's getting pretty cool. And then 230 at 377 divided by 21:49.180 --> 21:56.280 233 equals 1.6 18 and then see I think I think my calculator is rounding up 610 over 21:56.280 --> 21:57.120 3 21:58.380 --> 21:59.300 7 7 22:00.080 --> 22:03.520 Equals. Yeah, it's just rounding up and so on so on. Yeah 22:03.520 --> 22:09.240 So it actually I just rounds up and so on but what we can do here is now I'll put a space here 22:09.240 --> 22:15.100 So instead of I could put just type in pie and I'll see just just put equals and equals there 22:15.100 --> 22:19.340 This is a capital letter. I just made it capitals and there is there's a fine 22:20.020 --> 22:21.380 fascinating stuff indeed 22:22.880 --> 22:27.440 Now I'll put a race the last digit so because it sums up and just put a dot dot dot because yeah, otherwise 22:27.440 --> 22:34.100 I would round up. So yeah, because eight nine and there's the five actually actually that's not the exact five 22:34.100 --> 22:38.560 Let's see what the exact five is and exact five is yeah, eight eight seven 22:39.820 --> 22:42.380 We have eight eight seven. Yeah, that's all you round it up. So anyways 22:42.920 --> 22:47.480 Those are just the sidetracks want to get in those. I didn't realize it alternates like that 22:47.480 --> 22:50.300 It was very fascinating stuff. So now let's look at the golden angle 22:50.300 --> 22:54.940 so in geometry the golden angle is a smaller of the two angles created by 22:55.780 --> 23:02.920 sectioning the circumference of a circle according to the golden ratio that is into two arcs such that the ratio of the 23:02.920 --> 23:08.860 Length of the larger arc to the length of the smaller arc is the same as the ratio of the full 23:08.860 --> 23:14.360 Circumference to the length of the larger arc very very fascinating. So yeah, basically 23:14.360 --> 23:16.360 We're gonna do the same thing with 23:16.360 --> 23:22.420 Arc arc lengths. So the total to the larger and it's the larger to the smaller 23:22.420 --> 23:27.400 So that's all this algebraically let a plus b be the circumference of a circle 23:27.400 --> 23:29.080 divided into a 23:29.080 --> 23:36.360 Longer arc of length a and a smaller arc of length b such that you have two arcs that form one full circumference 23:36.360 --> 23:41.600 And then we have the golden ratio over here equals to five and the golden 23:41.600 --> 23:44.420 Oh, yeah golden angle here the golden angle is is the 23:44.820 --> 23:47.540 Golden I could be like that or I mean this is fine 23:47.540 --> 23:48.680 This ratio is fine 23:48.680 --> 23:55.580 But then the golden angle is gonna be the angle subtended by the smaller arc of length b 23:55.580 --> 23:56.740 It measures approximately 23:56.740 --> 23:59.100 137.5 077 23:59.760 --> 24:04.100 Etc. Or in radiance two point three nine nine nine and so on 24:04.620 --> 24:10.580 It's a fascinating stuff and the name comes from the golden angles connection to the golden ratio the exact value of the golden angle is 24:10.580 --> 24:13.260 Then you type it here thirty three hundred sixty 24:14.260 --> 24:18.500 Divided by one minus one over five. Oh, yeah, let's put this five. That's fine 24:18.500 --> 24:26.880 And then and so on you get this degrees about 180 times three minus square five and so on or 360 divided by five 24:26.880 --> 24:32.600 Square or in radiance two pi over five squared. So yes fascinating stuff indeed 24:32.600 --> 24:35.840 So here it is if you were to draw this out 24:35.840 --> 24:39.280 So where the equivalents follow from well-known algebraic properties of golden ratio 24:39.280 --> 24:45.080 So here's the golden angle is the angle subtended by the smaller red arc. So there is our 24:46.000 --> 24:46.500 137 24:47.000 --> 24:48.420 dot dot dot degrees 24:49.500 --> 24:54.840 When two arcs that make up a circle are in the golden ratio, so there we have the golden ratio 24:55.360 --> 24:59.640 Across there. So again a plus b over a equals a over b 25:00.440 --> 25:04.900 Just fascinating stuff indeed and now here's an example of the angle between successive 25:05.500 --> 25:11.280 Floor florets and some flowers is the golden angle. So if you have one, it's gonna be 137 25:11.920 --> 25:17.520 Then the next one so there's one there's two there's three and so on then there's gonna go to the four here 25:17.520 --> 25:22.160 I'm gonna go four four is gonna go to the five fives gonna go to the six six is gonna 25:22.160 --> 25:26.560 They go to the seven eight nine ten fascinating fascinating stuff 25:26.560 --> 25:29.840 So as you see it everywhere so the angle between them are going there 25:29.840 --> 25:36.520 So now let's look at the golden spiral so in geometry a golden spiral is a logarithmic spiral whose growth factor is 25:36.520 --> 25:43.740 Phi the golden ratio that is a golden spiral gets wider or further from its origin by a factor of 25:43.740 --> 25:47.080 Phi for every quarter it turns and 25:47.800 --> 25:53.380 Yeah, this is in polar coordinates a golden spiral with it with initial radius one has the following polar equation r equals 25:53.380 --> 25:55.500 Phi the power of 25:55.500 --> 25:58.800 the angle to the angle of 25:58.800 --> 26:02.740 Theta times two over pi that's in radians 26:03.460 --> 26:10.380 So yes, you then what you end up having is this spiral across like that where again where definition here 26:10.380 --> 26:11.920 So it gets further 26:11.920 --> 26:17.260 From the origin by factor of Phi for every quarter turn it makes 26:17.260 --> 26:22.060 So yes, it goes here quarters and go across there quarters and go across there 26:22.060 --> 26:25.900 It goes further by factor of Phi. So yes, it looks like that Phi 26:26.660 --> 26:30.540 I mean it looks like the approximation to it the Fibonacci spiral 26:30.540 --> 26:36.460 So your golden spiral assuming a square has a length a side length one the next smaller square is 1 over Phi 26:37.100 --> 26:41.360 Wide the next width is 1 over Phi square and it's the next smaller one 26:41.360 --> 26:44.960 So the each one's width is going to be 1 over Phi squared and 1 over Phi cubed 26:44.960 --> 26:49.020 So it's very interesting stuff and here's Fibonacci spiral approximates a golden spiral using 26:49.020 --> 26:54.040 Quarter circles inscribed in squares of integer at Fibonacci numbers shown again 26:54.040 --> 27:01.040 This is a sort of one above 1 1 2 3 5 8 13 and 21 and so on so this looks like that spiral there 27:01.040 --> 27:05.500 Very fascinating stuff. I can find a good comparison between the two but it may be in later videos 27:05.500 --> 27:10.680 So anyways, I'll finally let's go over some golden and Fibonacci spirals in nature and 27:10.680 --> 27:18.200 Art design corporations even advertising and Fibonacci sequences and so on so here's galaxy goes spiral like that very fast 27:18.200 --> 27:25.180 So there's a hurricane and there is a flower and there is another leaf like that going in in there 27:25.180 --> 27:33.520 There's a shell some other leafs or some other plants like that and here is this is a model of the spiral and 27:34.320 --> 27:38.940 Here is another flower the inside and here's even water water spinning a vortex 27:38.940 --> 27:44.500 That's following that looks like a spiral Fibonacci slash golden ratio and here is this 27:44.500 --> 27:47.560 It's a snail here's some grass 27:48.280 --> 27:54.380 Hurricane from Bob. There's a seahorse and so on and these even the sequences of tree branches 27:54.380 --> 27:58.220 You got one one and you have two three five eight 27:58.220 --> 28:02.920 Yeah, I believe these are yeah lengths. This one is one. This one is two. This one is three 28:02.920 --> 28:08.880 Then it has five then it has eight and so on and each one breaks up into it like that 28:08.880 --> 28:13.460 And so on very fascinating stuff not exactly sure but the details but yes very fast 28:13.460 --> 28:16.680 and here's some pineapples grow in a numerical sequence as well and 28:17.400 --> 28:19.920 Here's how hard to zoom in so eight parallel rows 28:19.920 --> 28:23.740 So that is eight parallel rows this way of scaling spiral 28:23.740 --> 28:31.160 Gradually and it has 13 parallel rows spiraling at a medium slope like this and it also has 28:31.160 --> 28:38.500 21 parallel rows of spot a scale spiraling steeply all the way across fascinating stuff and you have one two three five 28:40.060 --> 28:44.520 And consider the drawing of a pineapple here where we have numbered the hexagons 28:44.520 --> 28:49.900 The numbering has been done by the following rule the lowest hexagon was assigned to zero zero 28:50.520 --> 28:54.920 And then the hot next higher gets a one remember this continues on the backside not visible in the picture 28:54.920 --> 29:00.800 Then the next higher is assigned the number two and so on notice the hexagon 42 is slightly longer 29:00.800 --> 29:07.960 I mean slightly higher than hexagon 37. Yeah, whereas the 42 is 42 is slightly higher than 37 and 29:08.520 --> 29:15.160 Basically, yeah, right here. You should be able to identify three distinct spirals one will have the hexagon zero five ten 29:15.160 --> 29:18.660 So where is the zero five? This is zero five 29:19.360 --> 29:25.400 Yeah, ten I believe that ten is on this side there the spirals like that the second will have hexagon zero thirteen 29:25.400 --> 29:27.560 So there's a zero thirteen 29:28.160 --> 29:35.980 26 going upwards like that and the third will have will include zero eight sixteen so zero eight sixteen 29:35.980 --> 29:39.700 So and then basically a similar to one above we have a horizontal 29:39.700 --> 29:45.700 We have a medium slope and then we have a very steep slope slope up top there and there 29:45.700 --> 29:49.900 And now look at the common difference between the spirals you will find a difference between five eighteen 29:51.340 --> 29:54.900 Thirteen you have five eight and thirteen so there's five this way 29:54.900 --> 30:00.360 There is and this one is a bit smaller than the other ones and this one's going to be eight to be thirteen across 30:00.800 --> 30:02.860 To be thirteen across there. This should be 30:03.860 --> 30:11.660 So that this one follows a five eighteen thirteen over here, but then this pineapples a leaf has a bigger 30:12.260 --> 30:13.180 relative to its 30:14.360 --> 30:17.140 Hexagons and those go eight thirteen twenty one. So anyways 30:18.140 --> 30:19.480 That's a very interesting 30:20.840 --> 30:27.540 Yeah, you're very interesting pineapple us Fibonacci sequence. So yeah, there's this way is this way and then there's the vertical one 30:27.540 --> 30:34.060 Just count them up next time you eat a pineapple. Here's an egg also Fibonacci sequence across there 30:34.780 --> 30:39.820 Let's see. Here's even continents. You could I see Africa like that starting from there and goes all the way down 30:39.820 --> 30:44.940 Very fascinating stuff. There's a Mona Lisa very fascinating stuff indeed 30:45.720 --> 30:47.780 so going further this is 30:48.560 --> 30:53.400 Sonic the hedgehog and there's even websites can also have it visually set up like that 30:53.400 --> 30:57.880 So little tabs on top there and it scrolls over for the big 30:58.680 --> 31:04.320 Main stuff that you gotta look at the main article. So there's natural geographic. You can also see it has one there 31:04.320 --> 31:11.640 Here's another website here, and yeah, so there it is to set up in all these blocks and Fibonacci blocks and 31:13.200 --> 31:16.080 Even here. This is even a movie covers 31:16.080 --> 31:20.800 You see yeah, you can see the kind of spiral here across and there's a spiral across there and 31:20.800 --> 31:26.080 And yeah, very very fascinating stuff and here is a 31:27.080 --> 31:28.720 the dream piano 31:29.940 --> 31:35.100 Advertisement there it is the Fibonacci sequence as well. It just looks cool now, and yeah, I believe 31:35.760 --> 31:41.460 Apple logo probably has it too. It's very interesting stuff here. I haven't confirmed but it looks like it 31:41.460 --> 31:43.720 got a bunch of spirals here and 31:44.500 --> 31:48.100 Yeah, and even a ball thrown in there a wet tennis ball like that 31:48.660 --> 31:53.500 Spiraling across there's Fibonacci in the waves. There's Fibonacci hurricane galaxies 31:54.940 --> 31:59.820 And I even trumps here. So yes, yeah pretty cool stuff