Golden Section (φ² = φ + 1)

What this episode is about

This discussion explores the golden ratio, represented by the Greek letter phi (φ), highlighting its unique properties as an irrational number and its connections to the Fibonacci sequence. The talk delves into the historical context and significance of the golden ratio in art and nature, while also addressing misconceptions about its use in ancient Greek architecture.

Key takeaways

• The golden ratio, once a divine secret, is now an irrational party crasher in geometry and art!
• Not all beauty is Greek: the Parthenon's charm may not be rooted in the golden ratio after all.
• Fibonacci’s rabbits weren’t the only things multiplying; nature too finds harmony in the golden spiral.
• Penrose tiling: a non-repeating pattern that keeps mathematicians guessing—forever a puzzle in perfection!
• The golden ratio: more than a number; it's a universal thread woven through art, nature, and math.

What is discussed, and when

1. 0:00 – 3:00: golden ratio, Luca Pacioli
2. 3:00 – 6:00: golden ratio
3. 6:00 – 9:00: golden ratio, Leonardo of Pisa, penrose tiling
4. 9:00 – 12:00: golden ratio

Notable quotes from "Golden Section (φ² = φ + 1)"

• This talk is about one of the most fascinating numbers in mathematics, the golden ratio, often denoted by the Greek letter fi.

  about golden ratio · at 0:07

• In 1509, the Italian Luca Pacioli ascribed religious significance to it, calling it the divine proportion. And a century later, the mathematician and astronomer Johannes Kepler agreed,.

  about Luca Pacioli · at 1:14

• Using his kite and darts patterns he produced what is now known as a penrose tiling as shown in the center.

  about penrose tiling · at 6:53

• Fibonacci's book contained many arithmetical problems including his famous one about rabbits.

  about Leonardo of Pisa · at 8:43

Who and what is mentioned in this episode?

People

• Leonardo of Pisa — Person · Referenced historical figure · 1 mention · first at 7:55
  Introduced as Fibonacci, connected to the Fibonacci sequence and his influential arithmetic text that helped spread Hindu–Arabic numerals in Western Europe.
• Luca Pacioli — Person · Referenced historical figure · 1 mention · first at 1:14
  Mentioned for naming the golden ratio the 'divine proportion' and attributing religious significance to it in 1509.

Concepts & topics

• golden ratio — Concept · Core topic · 7 mentions · first at 0:21
  Central mathematical concept discussed in the episode, with applications in geometry, art, and nature.
• penrose tiling — Concept · Framework · 1 mention · first at 6:44
  A type of non-repeating tiling related to the golden ratio, with applications in science.

Full transcript with timestamps

0:17 This talk is about one of the most fascinating numbers in mathematics, the golden ratio, often denoted by the Greek letter fi. But what is the golden ratio and what properties does it have? We'll also meet the Fibonacci numbers. But what are these and how are they related to the golden ratio? And how are these topics related to art and nature? The number five is the positive solution of the quadratic equation x^2 = x + 1 [snorts] and is 1/2 of 1 + the<unk> of 5 or 1.618033 etc. An irrational number which goes on forever.

1:04 Although some of its geometrical properties can be traced back to ancient Greece, its description as golden did not emerge until the 19th century. In 1509, the Italian Luca Pacioli ascribed religious significance to it, calling it the divine proportion. And a century later, the mathematician and astronomer Johannes Kepler agreed, describing it as a precious jewel and claiming it to be elemental to God's creation of the universe. Because 5^ 2= 5 + 1, we obtain its square by adding one. So 5^ 2 is 2.618, etc. If we then divide the quadratic equation by phi and rearrange it, we get 1 / 5 = 5 - 1. And so we obtain its reciprocal 1 over 5 by subtracting 1. So 1 over 5 is 0.618 etc.

2:02 Also turning this last equation upside down, we find that phi = 1 / 5 - 1. That is 1.618 618 = 1 / 0.618. And we'll need this last result shortly. As we'll see, much of the interest in the golden ratio arises from geometry, but assertions about its geometrical origins are often illfounded. For example, claims have been made that the proportions of the Pathonon in Athens and other Greek buildings were based on it. Aesthetically, some buildings may seem too short and fat or too long and thin as shown on the right, while a rectangular building with proportions in golden ratio may be considered to have the perfect shape.

2:53 But although the Greeks knew about the golden ratio, there's no evidence that they designed their buildings to accord with it. A golden rectangle is a rectangle whose side lengths are in the ratio of 5:1. Above on the left is such a rectangle with sides five and one. And if we now remove the left hand square with side one, we're left with a vertical rectangle whose sides are one and five minus one. Now this smaller rectangle has the same shape as the original one. And to see why, we recall from our earlier result shown here that 5 over 1, the ratio of the side lengths of the original rectangle is the same as 1 / 5 - 1, the ratio for the second rectangle.

3:45 If we now remove a square from the second rectangle and then a square from the third rectangle and continue in this way, we produce the pattern of rectangles above on the right. Here the resulting rectangles while getting smaller and smaller all have the same shape. On this picture we can then draw a succession of circular arcs that approximate a spiral pattern known as the golden spiral. As shown below, this spiral converges to a point. In fact, it's the point where diagonals of the first two rectangles cross and is 618 of the way between the top and bottom of the rectangle and 618 of the way between the two sides.

4:34 And it's also a point where some artists consciously or unconsciously have placed the main focal point of their paintings to which the eye is naturally directed. Such a spiral is called a logarithmic spiral. and it arises in nature in the form of a nautilus shell as shown on the right. It also appears in whirlpools in the head of a sunflower as we'll see later and in other places also. So the golden ratio features in both art and nature. Indeed, the golden ratio arises throughout geometry. For example, in a regular pentagon, the length of any of its diagonals is five times the length of any of its five sides.

5:25 Then, as shown on the right, an isosles triangle formed from two sides of the pentagon and one diagonal has an angle of 108° and two angles of 36°. While a triangle formed from one side of the pentagon and two diagonals has one angle of 36° and two angles of 72°. Moreover, the number five turns out to be twice the cosine of 36°. By combining these triangles in two different ways, the Oxford professor and Nobel Prize winner Sir Roger Penrose constructed the two shapes below known as the kite and the dart to which we'll turn in a moment.

6:14 Suppose that you wish to tile the floor of your bathroom with tiles that are all identical regular polygons. This can be done, but only in three ways. One is a familiar tiling made from squares. But there are also tilings with equilateral triangles and with a honeycomb pattern of regular hexagons. All three of these tilings can be extended as far as we like and look the same everywhere. In the 1970s, Roger Penrose constructed a different type of tiling. And on the right you can see him standing on a version of it.

6:53 Using his kite and darts patterns he produced what is now known as a penrose tiling as shown in the center. These tilings are interesting in that unlike the regular floor patterns shown above. They never repeat anywhere however far out we go. They're also important scientifically in that they turn up in the form of quasi crystals. solids used in the construction of many practical items such as LED lights and non-stick frying pans. Closely related to the golden ratio is the Fibonacci sequence of numbers. This is a sequence beginning with 1 1 2 3 in which each successive number is the sum of the previous two. For example, 13 is 8 + 5 and further on 144 is 89 + 55.

7:55 Born around the year 1170, Leonardo of Pisa has been known as Fibonacci since the 19th century. While traveling in Africa, he learns about our Hindu Arabic numerals which he then publicized in Western Europe in his influential Liba Abaki or book of calculation. Here in this early manuscript, you can see the Fibonacci numbers in a vertical column on the right. But in fact these numbers predate Fibonacci by a long way having arisen earlier in the context of Indian poetry where mathematicians would count the rhythms with a given number of beats short or long obtaining these numbers as their results.

8:43 Fibonacci's book contained many arithmetical problems including his famous one about rabbits. A farmer has a pair of baby rabbits. Rabbits take two months to achieve maturity and then give birth to another pair each month. How many pairs of rabbits are there after a year? For the first two months, there's just the original pair. But in month three, a new pair arrives, giving two in total. In month four, the original pair produces another pair, but the new pair hasn't yet produced. So, there are three pairs and so on.

9:24 And as you can see, the numbers of pairs after successive months are the Fibonacci numbers. In fact, it's as easy as 1 2 3. Interestingly, Fibonacci seems to have had no particular interest in this problem, and it didn't become popular until the 19th century. But what has all this to do with the golden ratio? In the early 17th century, Kepler considered the ratios of successive Fibonacci numbers. And these ratios appear here in two columns 1 over 1, 2 over 1, 3 over2, 503, and so on. And as he discovered and as you can see the ratios in the left hand column gradually increased towards the golden number while those on the right decrease towards it with their common limiting value being five.

10:24 Here we see how the Fibonacci numbers can be arranged in a spiral pattern that resembles our earlier golden spiral. Here, starting from the middle are squares with sides 1 1 2 3 5 8 and 13 and further ones can then be added at will. Also, like the golden ratio, Fibonacci numbers appear throughout nature. For example, the numbers of seeds in the spiral patterns of a sunflower head are often Fibonacci numbers such as 34, 55, or 89. An amusing paradox popular in the 19th century was described by Lewis Carol, mathematics teacher and author of Alice's Adventures in Wonderland. Here an 8 by8 grid of 64 squares is cut into four pieces which are then rearranged to give a 5x13 rectangular grid of 65 squares.

11:27 So where does the extra square come from? The paradox arises because when we form the 5 by13 rectangle on the right from the four pieces of the 8 by8 square there's a very narrow parallelogram in the center whose area is that of one small square. And in a similar way, we can cut and reassemble a 13x13 grid of 169 squares to give an 8x 21 grid of 168 squares. But in this case, a square is destroyed rather than created. Before we had 5 * 13 - 8 2 is + 1 and now we have 8 * 21 - 13^ 2 is minus 1. But we notice that all these numbers 5, 8, 13, and 21 are Fibonacci numbers. And for all similar examples, a single square is either added or destroyed.

12:36 Below you can see Lewis Carol's own drawings for generating the general case together with a table which shows the Fibonacci numbers that arise. I'd like to conclude with an unexpected feature of the Fibonacci number 89. The decimal form of its reciprocal 1 over 89 begins with 0.01 1 2 35 clearly exhibiting the early Fibonacci numbers. Indeed, 1 over 89 is exactly equal to the infinite sum of numbers shown in the middle where the Fibonacci numbers appear after lots of zeros or below in fractional form as 1 / 10^ 2 + 1 / 10 cubed + 2 / 10 4th + 3 over 10 5th and so on.

13:37 And with that truly remarkable result, I'd like to thank you for listening.

Questions about this episode

What is "Golden Section (φ² = φ + 1)" about?

This discussion explores the golden ratio, represented by the Greek letter phi (φ), highlighting its unique properties as an irrational number and its connections to the Fibonacci sequence. The talk delves into the historical context and significance of the golden ratio in art and nature, while also addressing misconceptions about its use in ancient Greek architecture.

What is mentioned in this Oxford Mathematics episode?

4 people, works and ideas are mentioned, including golden ratio, penrose tiling, Leonardo of Pisa, Luca Pacioli. Each one on this page links to its own page with every episode that mentions it.

How long is "Golden Section (φ² = φ + 1)"?

The episode runs 14 min; the transcript is about 1,728 words in 16 timestamped sections.

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