Golden Maths & Myths

chris cambré, Oct 31, 2024

Webpages with golden rectangles and spiral on temples, paintings and shells are numerous. It's clear: the golden section is popular. For Chris Impens, retired professor in Mathematics, [math]\varphi[/math] is just a number. He's blogging for years on the issue an recently he put it all thogether in the text [url=https://www.blogger.com/profile/08053866714451812324]Golden Maths & Myths[/url] you can freely download. WHen I discovered this text I immediately contacted Chris with a plan to tell the whole story in a GeoGebrabook. his text is truly a treasure. As a professor he teaches on spirals, as a surgeon he's filleting false claims. Where does the name 'golden section' come from? What's the exact position of the navel in the famous drawing by Leonardo da Vinci and how golden are the Parthenon and the Nautilus pompilius shell? Starting from the tekst by Chris Impens, additional info from his blogs and other tekst, I’ll try to take you with me on a fascinating discovery. If you’re still hungry you can always read the original full tekst version. It was a joy to learn many new things and I hope that you also will discover a story in whitch 'Maths' are at least as fascinating as the 'Myths'.

Table of Contents

  1. misconceptions and disclaimers
    1. the esthetic satisfaction of a golden rectangle
    2. The golden ratio is known for centuries
    3. core concept throughout history of architecture
    4. phi is omnipresent in math
    5. phi in nature
  2. the number phi
    1. definition
    2. construction of roots
    3. construction of phi
    4. construction of PHI
  3. phi and geometry
    1. a bit of trigonometry
    2. PHI in in a regular pentagon
    3. phi als the side of a regular decagon
    4. phi in a icosahedron
    5. icosahedron and dodecahedron
    6. distances in a dodecahedron
    7. trigonomic calculation
  4. Phi and the sequence of Fibonacci
    1. Fibonacci - rabbits
    2. exact and approximate Fibonacci formula
    3. alternative view on the sequence of Fibonacci
    4. golden rectangle and fibonacci rectangle
  5. the Vitruvian Man
    1. the Vitruvianse man - dimensions
    2. construction golden ratio
    3. circle with ratio phi
    4. Where's the navel? - construction
    5. Where's the navel? - calculation
    6. Know your classics
    7. other nonsense
    8. even more da Vinci nonsense
  6. Pacioli and the 'divina proportione'
    1. 'divine proportion'
    2. Trattato dell'architettura
    3. Diuina Proportio
  7. spirals and sunflowers
    1. parabolic spirals
    2. DIPAS
    3. efficiency
    4. 2 series of spirales
    5. the growth of sunflowers
  8. spirals and shells
    1. Fibonacci spiral
    2. spiral in a golden rectangle
    3. golden rectangle and golden spiral
    4. Nautilus pompilius
  9. Parthenon and Renaissance
    1. the Parthenon in pictures
    2. the Parthenon in numbers
    3. Renaissance
  10. the 'golden section' as an esthetic ideal
    1. how a construction became an esthetic ideal
    2. Vitruvius versus Zeising
    3. Zeising - Ghyka - Le Corbusier
    4. Le Corbusier
  11. what about Mondrian?
    1. What about Mondrian?

1.

misconceptions and disclaimers

In this first chapter we'll start with some and disclaimers. In the next chapters we'll focus on some separate aspects. Nice lecture is the last part of the text 'Golden Math & Myths' with a summary of statements from 'silly', 'very silly', 'absurd' to 'very absurd'.

  • 1. the esthetic satisfaction of a golden rectangle
  • 2. The golden ratio is known for centuries
  • 3. core concept throughout history of architecture
  • 4. phi is omnipresent in math
  • 5. phi in nature

1.1.

–

1.2.

–

1.3.

–

1.4.

–

1.5.

–

2.

the number phi

Although mostly [math]\varphi[/math] is named as a ratio in a segment or rectangle, in fact it's just an irrational number. And as other irrational numbers it can be constructed graphically using Pythagoras' theorem.

  • 1. definition
  • 2. construction of roots
  • 3. construction of phi
  • 4. construction of PHI

2.1.

–

2.2.

–

2.3.

–

2.4.

–

3.

phi and geometry

While making calculations in pentagons or decagons, you'll soon meet the number [math]\varphi[/math]. In this chapter you'll discover why just one equality is responsible for this evidency.

  • 1. a bit of trigonometry
  • 2. PHI in in a regular pentagon
  • 3. phi als the side of a regular decagon
  • 4. phi in a icosahedron
  • 5. icosahedron and dodecahedron
  • 6. distances in a dodecahedron
  • 7. trigonomic calculation

3.1.

–

3.2.

–

3.3.

–

3.4.

–

3.5.

–

3.6.

–

3.7.

–

4.

Phi and the sequence of Fibonacci

Often the sequence of Fibonacci is linked to the golden section. The further you go into the sequence the more the quotient of consecutive numbers approaches [math]\Phi[/math]= 1.618... . In this chapter you'll discover why.

  • 1. Fibonacci - rabbits
  • 2. exact and approximate Fibonacci formula
  • 3. alternative view on the sequence of Fibonacci
  • 4. golden rectangle and fibonacci rectangle

4.1.

–

4.2.

–

4.3.

–

4.4.

–

5.

the Vitruvian Man

Speaking or reading about the [i]golden section[/i], very soon is reffered to the [i]Vitruvian Man[/i], a drawing by Leonardo da Vinci, as a link between Art and Maths going back to the Renaissance and Classic Antiquity. Da Vinci himself wrote a number of ratio's on his drawing and this is the ideal start to examing the exact place of the famous navel. A question worthwhile to ask is what are we talking about? There are thin, fat, athletic, slim and athletic men, with short or long arms. So does 'nature' only fit with the ideal man? And what about the woman?

  • 1. the Vitruvianse man - dimensions
  • 2. construction golden ratio
  • 3. circle with ratio phi
  • 4. Where's the navel? - construction
  • 5. Where's the navel? - calculation
  • 6. Know your classics
  • 7. other nonsense
  • 8. even more da Vinci nonsense

5.1.

–

5.2.

–

5.3.

–

5.4.

–

5.5.

–

5.6.

–

5.7.

–

5.8.

–

6.

Pacioli and the 'divina proportione'

  • 1. 'divine proportion'
  • 2. Trattato dell'architettura
  • 3. Diuina Proportio

6.1.

–

6.2.

–

6.3.

–

7.

spirals and sunflowers

The position of sunflowerseeds is all but random. One clearly sees spirals. In this chapter you learn more about it.

  • 1. parabolic spirals
  • 2. DIPAS
  • 3. efficiency
  • 4. 2 series of spirales
  • 5. the growth of sunflowers

7.1.

–

7.2.

–

7.3.

–

7.4.

–

7.5.

–

8.

spirals and shells

  • 1. Fibonacci spiral
  • 2. spiral in a golden rectangle
  • 3. golden rectangle and golden spiral
  • 4. Nautilus pompilius

8.1.

–

8.2.

–

8.3.

–

8.4.

–

9.

Parthenon and Renaissance

Search on 'golden ratio', select 'Pictures' and you'll find Leonardo da Vinci, shells and the Parthenon. In fact, illustrating the golden ratio on a picture of the Parthenon is quite difficult because of perspectiver distortion. So let's look at all these pictures and the real dimensions. A doctoral study brings more light upon ratio's in the Parthenon.

  • 1. the Parthenon in pictures
  • 2. the Parthenon in numbers
  • 3. Renaissance

9.1.

–

9.2.

–

9.3.

–

10.

the 'golden section' as an esthetic ideal

  • 1. how a construction became an esthetic ideal
  • 2. Vitruvius versus Zeising
  • 3. Zeising - Ghyka - Le Corbusier
  • 4. Le Corbusier

10.1.

–

10.2.

–

10.3.

–

10.4.

–

11.

what about Mondrian?

  • 1. What about Mondrian?

11.1.

–

Information