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Philosophy of Structure 301-526B -- the online compendium

Winter 2006

                         Link to my  "underground city files" and the files of Mr. Jacques Besner for the benefit of students in Prof. Drummond's Civic Design Class.
                                                                                 Also, look here: http://www.ovi.umontreal.ca/home.html

Professor Pieter Sijpkes
Hello everybody. I copy below an email I sent to Juliy

Hi again Juliy,

The link works fine. I think your computer has an old version of the page in the 'cache' of your computer. You have to clear the contents of the 'cache'. In Netscape go to 'Edit' then to 'Preferences' then to 'advanced'  where you will see an option: 'clear cache' two times.
After reloading the page it should be ok.

Most of your other questions can be answered by reading James Gleick "Chaos the making of a new science". I have mentioned from the beginning of the course that that book is required reading for the course.
Chaos and fractals are related. For instance the Lorenz attractor describes the weather which is a chaotic phenomenon; but within the chaos of the weather there is an underlying order described by the Lorenz attractor which is fractal in its self-similarity at any scale. The same is true for clouds. They are also formed by chaotic processes, but at the same time clouds are fractal in that they are the same at all scales.

Fractal dimension can be any number between 0 and 3. Yes, fractal dimension can be a whole number like 2. The Peano curve has fractal dimension 2 for instance.

The difference between mathematical fractals and natural fractals is that the mathematical variety has an endless number of 'levels'. This means that fractal division can theoretically go on forever. In natural fractals there are only a few of such levels. For instance a tree will only have a 'fractal' branching pattern at about five or six levels.. We talk about'fractal-like behaviour'

Feedback is an absolutely essential mechanism in dynamic systems (Nature is one gigantic dynamic system). An 'echo' of a sound for instance is a feedback example. You clap your hands in an empty stone church, and the sound travels to the stone walls and vaults, and comes back a few seconds later to the observer, interacting with the outgoing sound. It may do this several times..thus producing  multiple echos. The original sound is reproducing and modifying itself, sometimes weakening, somtimes getting stronger, sometimes staying the same..
The stockmarket, traffic jams, there are an endless number of dynamic processes that involve feedback. And fractals (like the Mandelbrot set or the Koch coastline)  are all based on feedback loops.  

I must say that the answers I am giving to your questions could have been obtained by a few Google searches. As I mentioned several times in class: fractals and Chaos research 'lives' in computers, and the Google search engine is the 'encyclopedia' of that world.

I hope this helps.


Pieter Sijpkes

Last Class before study break on Friday February 17

You each will be asked in turn to outline the topic you have chosen to study for your final project.
Give some references that you will use.
The project carries more weight this year than in the past, since the final exam is no longer held and thus the final mark is only made up of  the midterm(40%) and the final project

The  material covered in the midterm is the same as it was in 2005, and is listed below.

in the second hour we will visoit the Redpath Museum


Assignment for this week, Friday February 3:
Have a look at this description of the work of Louis Kahn: http://members.tripod.com/freshness/images.html
Read up on the idea of Fractal Dimension in general, but specifically about Fractal Box Dimension, and how it is used to establish the fractal dimension of Architectural facades and plans.
Koch Fractal analyzed by Box Counting Dimension method: http://classes.yale.edu/Fractals/FracAndDim/BoxDim/KochBoxDim/KochBoxDim.html

A good introduction to box counting dimension can be found here: http://classes.yale.edu/fractals/FracAndDim/BoxDim/BoxDim.html
Also: http://classes.yale.edu/fractals/Labs/CoastlineLab/CoastlineLab.html
An interesting explanantion of powerlaw relations: http://classes.yale.edu/fractals/FracAndDim/BoxDim/PowerLaw/CrumpledPaper.html
Video Feedback: http://www.sweetandfizzy.com/fractals/index.html Also: http://www.zhurnal.ru/gallery/shohdy/vloop.html
Fractals in Architecture: http://math.unipa.it/~grim/Jsalaworkshop.PDF
Also: http://homepages.uel.ac.uk/1953r/fracarch.htm

An overview of fractal applications:
Read the following  very interesting observations on Fractal geometry::http://www.artmatrix.com/show.cgi?xfl7.memo  
This interpretation of fractal dimension is based on the three stages of chaos in a double pendulum: Complete regularity, cyclic regularity (or periodic regularity like the four seasons) and complete chaos.

last week's reading:

Read everything about Federation Square in Melbourne designed by Lab Architecture Studio: http://www.labarchitecture.com/.  (Explorer only)
also read:



Fractint Download site: http://spanky.triumf.ca/www/fractint/fractint.html

A view on mathematics before the discovery of the 'Butterfly Effect":

"Implicit in western cultural heritage is the notion that the nature is governed by mathematical laws. The doctrine of Determinism states that if we would but know nature's (mathematical) laws, we only need to plug in the relevant measurements in order to predict the future. Cause and predictable effect. Also, natural shapes imitate the perfections of Euclidian figures such as triangles and circles. In the words of Galileo Galileo, "Mathematics is the language with which God wrote the Universe" Therefore an understanding of mathematics is a pre-requisite to understanding and appreciation of nature... indeed all scientific study"

A view of mathematics after the discovery of the "Butterfly Effect":

God may have written the universe in mathematics, but it's getting pretty evident that he didn't use the equations and formulas studied in school mathematics when he did it. These all imply that natural phenomena obey some mathematical differentiable function or another. Applied mathematicians have known for some time that mathematics does not dictate nature; but rather imperfectly attempts to describe it. Recent discoveries have buried classic determinism very deeply. 

Two opinions supporting the notion that nature is the master and mathmatics is the servant:

"So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality."

Albert Einstein
Geometry and Experience

"What we observe is not nature itself, but nature exposed to our method of questioning."

Werner Heisenberg
Physics and Philosophy

Link to midterm exam
A list of topics to study for the 2005 Mid Term exam:
The information discussed in the course this year so far can be grouped under four main headings:
1. Fractal geometry, (know these five names, and how they relate to chaos theory)
2. Chaos Theory, (the limits of predictability, the butterfly effect)
3. The use of computers in Architecture, and
4. Different structural systems used by architects  and engineers 
throughout history:  http://nisee.berkeley.edu/godden/index.html
The Fractal geometry and Chaos Theory section has the following four parts:


what is a fractal  (definition)

What is the difference between a natural (physical) fractal and a mathematical fractal

Draw the generation of a fractal

What are some of the basic attributes of  fractals (self similarity, iteration, dimension) Describe.

Give some examples of fractals , natural and mathematical

It is thanks to the computer that we are now looking at patterns that could only be glimpsed by pioneers like Poincare, and Koch
Why exactly is that the case?

Euclidian dimension is either  0 (a point), 1 (a line) 2 (a plane) 3 (a volume)
Fractal dimension can be any number (fractions included) between 0 and 3
Dimension depends on the observer: look at a ball of string .. it looks from far away like a point (O dimension) closer up like a sphere (dimension 3) and closer again like a thread (dimension 1) and closer again (inside the string) like a fractal volume ( dimension between 2 and 3, depending how coarse the thread is.)
Dimension of a physical fractal can be determined by the box counting method

Dynamic Systems

We have looked at statical systems (structures) like architecture, and looked at dynamical systems
( like the weather, the stockmarket, erosion of a landscape, growth of a city...they all have a time variable in them. )
Dynamical systems rely on feedback

The Lorenz attractor  

Descibe the Lorenz attractor.


The butterfly effect.

 Describe the buterfly effect. What was the general thinking in science before Conrad Lorenz discovered it.
What are some of the aspects that have come to be understood in recent years about it/ (extreme sensitivity to initial conditions..
"The river is never the  same" Lao Tse.

Discussion question: Do you see any application of these new insights in architecture?

Look at some work by interesting offices like the following ones:








Exam preparation:
Be familiar with the contents of James Gleick book on Chaos

Be familiar with terms such as Fractals, fractal geometry and fractal dimension.Be able to describe the term strange attractor and two examples thereof.
Know the difference between statical and dynamical systems.
Know how to explain the concept of feedback with some example.
Study the interactive diagram we have used. Here is the latest version

Look at the variety of physical structures on this website as we did in class.


"Architecture is the art of creating physical structures to suit social, emotional, religious political structures"

"Structures, structures, structures everywhere....."
                                                                                                             Matthew Lella, former student in the course
"I think the next century will be the century of complexity"
                                                                                                                                                                                          Stephen Hawking

How big is this grain..? (see at bottom of Course Outline)

Students 2002

Course Progress 2002


An investigation into the nature of various forms of structure that will hopefully lead to a better understanding
of Order, Disorder, Chaos, Stability, Nature, Human Constructs, Animal Constructs, History and the Future.