Golden Ratio
Rule of thirds
Project for Elos Interscience and Interart
Introduction
Golden Section is a mathematical law regarding a certain way of sectioning a distance such that the ratio between the whole distance and the bigger part of it (major part a) is equal to the ratio between the bigger part and the smaller part of it (minor part b).
( a + b ) / a = a / b
Conventionally this ratio is equal to (1 + q5) / 2 .1,618 , is noted with s (from section) or with j (Phidias is the Greek sculptor that tried for the first time to define it ).
The golden section was known by the ancient Egyptians and also by the
ancient Greeks (it is mentioned by Pythagoras ,
The first one that calls it Golden Section is Leonardo da Vinci . In our days we use that terminology or Golden Proportion coined by Martin OHM (1792- 1842).
One of the most important properties of the golden section is logarithmic spiral which can be found in nature at an amazing frequency: the disposal of sun flower seeds, in the morphology of some shells, comet tail, etc. Furthermore, most of the plant and animal species reveal the existence of this ratio between their components. Even the human body submits to the same law.
Another important mathematical law is the Fibonacci sequence:
 a0 = 0 , a1
= 1 , ,,,,,, ,an
=
an-1 + an-2
which has the limit for n
∞ is (1
+ q5) / 2. The biological law according to which any
form of life contains and repeats preceding stages can be found here.
Considering all the aspects presented above it must be very interesting to study the following issues :
Math : golden proportion, Fibonacci 's sequence, graphical representations of the logarithmic spiral;
History : Ancient
Biology : Biological laws: considerations about Fibonacci's sequence and Golden Section
Art
Recommendable Literature
De Gulden Snede (Ir. C.J. Snijders); De driehoek, Amsterdam, ISBN 90 6030 518 3
Pythagoras oktober 2000: De rij van Fibonacci
Pythagoras oktober 2001: Geheimen van de vijfhoek; De Gulden Snede I
Pythagoras december 2001: Moorse Kunst; De Gulden Snede II
Pythagoras februari 2002: Penrose tegels; De Gulden Snede III
Pythagoras april 2002: De Icosaëder; De Gulden Snede IV
Pythagoras juni 2002: Dodecader & Icosaëder; De gulden Snede V
Pythagoras augustus 2002: Gulden ruitenveelvlakken; De gulden Snede VI
Pythagoras oktober 2002: Veelvlakken prijsvraag
Besides you will find on the internet all kinds of information about this interesting subject.
The Project
The project consists of three parts:
1. A general introduction into the basic theory of the Golden Section and Fibonacci's Sequence with the help of the printed copy of the book 'De Gulden Snede'.
You may work on it whether quite individually or in small groups.
2. An investigation into a subject related to the Golden Section. You may choose a subject for yourself. It's allowed to cooperate in groups of at the most two students.
3. A practical investigation and a presentation during a studytrip.
For example: as has been shown in the article 'De Gulden Snede en de Moorse Kunst' you will discover in which way the Moorish architects in the late Middle Ages used the Golden Section in their beautiful mosaics (see: the Alhambra in Granada, Spain). Also the Dutch graphic artist Escher was very inspired by this Moorish Art.
4. It's also the objective of the project to exchange the results during the project or to apply for information to partner students at the other schools.
At the end of the project:
You have to write a report in English about the (basic theory of the) Golden Section and your own findings (to be handed in at the end of the school year).
This report will be counted as:
Practical Investigation (Praktische Opdracht) for maths
A Practical Investigation for English (handelingsdeel)
40 SLU
Jan de Jager
RSG Tromp Meesters Steenwijk
