Creative Exploration of Structures
2010 Spring course atCreative Technology BA at University of Twente
Senior research fellow at Moholy-Nagy University of Art and Design
Associate Professor at UT - EWI
e-mail: zsofiATcs.utwente.nl
Time: Wednesday and Friday 9.45-12.30:interactive lectures
First occasion: 19 May
Location: Smart XP Lab
Course home: http://create.mome.hu/ruttkay/CES1
Assesment:
- active participation at lectures (20%),
- homeworks (30%) and
- written essay (25%) - also to be presented
- own project (25%) - also to be presented in order to get the credit, you have to perform at least 60%
Students:
Martijn Bruinenberg
Reinout Epke
Jan de Geus
Tom Heijnen
Joost Klitsie
Jan Kolkmeier
Thomas Kromkamp
Douwe Bart Mulder
Pieter Pelt
André Richter
Martina Schulze
Herjan Treurniet
Ineke Visser
Siewart van Wingerden
Essay topics proposed:
- Fibinacci numbers in nature -- Pieter Pelt -- 9 June
- Fibinacci numbers in art and architecture
- The history of Fibonacci numbers and their role in contemporary mathematics
- The myth and reality of the application of Golden section in antient and modern architecture -- Reinout Epke -- 11. June
- Survey critically and make a short presentation of the myth of the Golden section in human perception ("people find the golden proportion the most pleasing") -- Douwe Bart Mulder -- 11. June
- Survey critically and make a short presentation of the usage of the pentagram, as symbol -- Thomas Kromkamp -- 4.June
- Make an essay on comparing some (famous) spirals.
- The QR Code - the maths behind it, some range of usage -- André Richter -- 11 June
- Self-similar structures in nature
- Anamorphosis -- Herjan Treurniet -- 9. June
- Particle systems in (artistic) visualization -- Jan Kolkmeier -- 9.June
- Encryption -- Martijn Bruinenberg -- 11 June
- Fractals -- Siewart van Wingerden -- 9 June
Lectures
19.05: Infinity and beyond |
Topics:
- About the course
- who we are?
- what do you think of mathematics?
- maths as: way of thinking, attitude, inspiration and fun - also for artists - The (He)art of Mathematics
- schedule and way of working be active , ask questions - open your mind - Aleph, the Planet of Infinity
Hotel Omega - always full but always having a room
- for a single visitor
- for a tourist group in a bus with seats 1,2, 3, ...
Hotel OmegaSquare
- with floors 1,2, 3, ...; each with rooms 1, 2, 3, ...
- FIRE, FIRE - and the hotel is full! - Can Omega offer a room for each guests?
Clubs in Omega
- all guests, in all possible ways form clubs - the resulting clubs are quite many
- can the porter assign a room for each club?
- the answer is proven in an indirect way, with the 'triangular method of Cantor' - Other numbers than integers
Rational numbers
- definition: are the ration of two integers: n/m - we may also write them as (finite or infinite decimals)
- theorem: they are as many as the natural numbers - they are countable
Beyond rational numbers
- an old puzzle: double a square-shaped garden pond of 1x1 m in such a way, that the shape is square, and the 4 trees standing at the 4 vertices (corners) remain untouched
- theorem: square root 2 is not rational - a puzzling finding for the Pythagorans!
- proof: in an indirect way, assuming the opposite and concluding to a contradiction
- further theorems: square root 5, 7, ... is not rational - you can prove them yourself
Resources:- Cardinality and the continuum hypothesis
- Hilbert's paradox of the GrandHotel
- Aleph numbers in mathematics
- Hilbert hotel in Dinosaur comics
- The Eden project and
- Hilbert 23 problems presented in 1900
- A short summary of the Hilbert hotel problems
- irrational numbres - proof and history
- Send a mail with your blog/web page
- Invent two puzzles yourself, related to the Omega or OmegaSquare hotel,
one for which the answer is a countable set, and one for which is not.
Write them up (without solution)!
If you are out of ideas, here is a question by me, send the answer (+ 1 own puzzle):
In the Omega hotel, having seen that all possible clubs are too many, now the manager is moree modest. He proposes to form all possible clubs with finite numbers of members. That is, all possible clubs with 1, 2, 3, 4, ... members are to be formed. Can he give a separate room in his Omega hotel for each club?
- Extra:
In the Omega hotel they have recently installed a series of rolling pavements (like in Schipol), to help people to get to their rooms fast. They are each faster and faster: It takes 10 seconds to get to room 2 from toom 1, 5 seonds to get to room 3 from room 2, 2.5 seconds to get to room 4 from room 3, ...
Can we get to any room within a certain time?
21.05: Fibonacci numbers
Fibonacci numbers- Three puzzles
Solve at least two of them!
Op-art towers
An op-art living district is to be built of 10 floor houses. Each house is to be painted according to the following rules:
a) each floor is painted black or white;
b) no two floors above each other may be black.
Maximum how many houses can be built, if all houses must be painted differently?
Slot machine
A slot-machine accpets 1 and 2 euros. How many ways can one spend 10 euros?
The order of the coins put into the machine matters.
Stairs
When running up stairs, you may take 1 or 2 stairs at a step.
How many ways can you go up 10 stairs?
- The Fibonacci numbers
- Add up the first Fibonacci numbers - what do you find?
Formulate a theorem, write up as a formula.
Now we must prove it. A handy method is full induction , with - in its simplest form - the following steps:
a) Check that the statement (the formula) is true for an initial number, usually n=0 or n=1.
b) Assume that the formula is true for some number n.
c) Show that then the formula is also true for the number (n+1). - The Fibonacci squares and the Fibionacci spiral
- Fibonacci numbers in nature - phyllotaxis, interactive tool in Mathematica Player
- How fast do they grow? The ratio of the neighbouring Fibonacci numbers approximates the Golden ratio
Resources:- Fibonacci mobiles
- Fibonacci design for knitting and for weaving
- Akio Hizume architect's towers
- painting by Colin Joye
- The Eden project and
- Peter Randall's sculpture
- Fibonacci architectural pathway by Marks Barfield
- We observed at class that the ration of neighbouring Fibonacci numbers seems to converge to a number around 1.6.
We showed that this limit - let's denote is by φ - must satisfy the quadratic equation
φ = 1 + 1/ φ
Solve this eaquation, and find the value of φ.
Is φ rational or irrational number? -
Make designs, experiment by using the Fibinacci squares, the spiral.
Make own variations of the spiral, use more spirals, build, draw, program, ...
To be sent: image(s) of your designs - Observe what happens if you add up every second Fibonacci number, starting with the first one.
Formulate a theorem, and prove it by induction. Send your proof, with well structured parts:
- some certain number n for which your statement holds
- show that IF the theorem (the formula) holds for n, THEN in will also fold for n+1.
26.05: The golden section
Topics:- the Golden Section: definition, with some some maths to deduce the value and the construction of the golden proportion
- golden rectange and golden triangles
- History of the maths of the Golden section
- the logarithmic spiral
- Eadem mutata resurgo ("Changed and yet the same, I rise again") Jacob Bernouli's grave stone - the engraving is NOT a logarithmic spiral! What then?
- the pentagram and star-like constructions
- myths of the golden section in the arts and sciences: be critical!
- Spirals and Java applets to play with them, or another one to see how the two directions emerge in seed arrangements
- Representing sea-shell surfaces and a Siggraph 92 paper - 3d models based on spirals
- whirling polygons and
- more on the golden proportion : construction, triangles, pentagram
- Criticism of the myth of the golden proportion in the arts by A. ven der Schoot
- Symbolism of the pentagram
- Construct a golden rectangle.
- Take an interval, and divide it according to the golden section.
- Construct the two kinds of golden traingles.
- Construct a pentagon, draw a pentagram in it.
- Check where you find in the pentagram the golden proportion.
- Make some design by using the golden proportion, golden rectangle and/or golden traingle and/or golden spiral
28.05: Pythagoras trees and other self-similar structures |
Topics:
- Proves for the Theorem of Pythagoras - 84 different ones!
- self-similarity, exact or approximate
- design - recursive images: Dorste effect, also the Hungarian Macko sajt,
- logarithmic spiral
- trees of Pythagoras interactive applet and artistic designs , and with random variations for creation of realistic trees
- Construct the two kinds of golden traingles.
- Construct a pentagon, draw a pentagram in it.
- Check where you find in the pentagram the golden proportion.
Prove the theorem in the other direction, showing that IFin a triangle the longest side squared is equal to the sum of the squares of the other two sides, THEN the angle opposite the longest side is 90 degree.
Try to device a visual proof, and using what we have proven already. But other proves are also welcome
2.06: Fractals |
Topics:
- Creating self-similar structures by geometric replacement rules or iterative functions
- Koch curve and snowflake
- Sierpinski triangle and carpet
- dragon curve and paper folding of it
- Hilbert's and Peano's plane-filling curve
- Creating self-similar structures by escape values of functions mapping the plane to itself
Julia sets are images based on functions as f(z) = z2 +c or others, see tool to explore
z and c indicate complex numbers, that is points on the plane - thus f(z) is a mapping of the plane to itself
choosing z in black region the iterated sequence zn+1 = zn2 +c remains within a finite region
colored regions indicate how fast the sequence z (distance from the origo) grows to infinity - The Mandelbrot set is a "map" Julia sets of the same function with different c values
- The Mandelbrot set is self-similar, examine zooming in here
black is the region where the Julia set is connected
colors indicate distance from the black region - examples from nature: trees, broccoli, costal lines, mountaines, clouds, lightings, rivers, blood vessels,...
- making landscapes with random midpoint replacement algorithm
Resources:
- Mandelbrot and Julia set explorer
- Three dimensional fractals in SecondLife
- Fractal Geometry from Yale University - recommended readings, all aspects with a lot of examples
- interactive tools and examples from Boston University
- Fractal Foundation with on-line courses and gallery of pictures
- fractal generatying sw recommended: ChaosPro, UltraFractal
- Jock Cooper's fractal art
- World of Fractals
- Chaos game
- midpoint displacement with Processing
- Terragen photorealistic terrain generation sw
Assignments:
- Explore fractal art with some sw, save ones you like, try to make some natural phenomenon - plant, lighting, ...
- Make a landscape with midpoint displacement algorithm
4.06: Graphs |
Topics:
- We explored some sequences of problems, all which can be translated to statements about graphs
- Definitions of graphs and features
- graphs, edges, vertices, degree of a vertex
- directed / non-directed
- connected / not connected
- path, circle
- tree
- Theorems about graphs
- Sum of degree of vertices is even
- Necessary and sufficient condition for the existence of Euler path and Euler circle in directed and non-directed graphs
- Number of edges in a complete graph with n vertices
- Number of edges in a tree with n vertices
Resources:
Assignments:
- Finish the problems you did not do during class. For sure: Party 4-5 and Dominoes
- Complete the sentences, statements on page 3 - a summary of our conclusions about some charactersitics of graphs
9.06: Graphs - continued |
Topics:
- Particle systems in art applications - by Jan K.
- Anamorphoses - by Herjan
- Fibonacci numbers in nature - by Peter
- The golden section in nature - by Siebert
- The Euler formula for connected planar graphs
- The Euler formula for polyhedra
Assignments:
- Wrap up your essay, presentation and final design.
- We have seen for some planar graphs that the following formula held for the Edges, Vertices and Regions:
R + V -1 = E
Show by a proof that his holds for any planar, connected graph.
Hints: you may try induction. Another possibility is to go for a direct proof: see what happens to E and R if you remove edges so that the remaining graph is still connected, but finally does not contain any circle. - Finally, a last puzzle:
We cut our two opposite corner fields of a chess board. Can you cover the remaining 62 squares with 31 dominoes, if one dominoe covers 2 squares?
Please send your essay, talk, design and problem solutions by mail Thursday 21 pm, latest.
The last occasion is Friday, when we will wrap up all what we explored, have the remaining presentations and evaluate.