# STD1 – Visual Mathematics Science and Technology for Designers

## Visual Mathematics

### Science and Technology for Designers – STD1

2009 Fall course at **MOME**

**Lecturer: Zsófia Ruttkay** PhD, Senior research fellow, e-mail: ruttkayATmome.hu

** Time: Monday** 9.30-12.50 :interactive lectures followed by parctical work

** First occasion: ** 21 Sept, EXCEPTIONALLY 11.30-16.30 (with lunch break)

** Location: Room B102 ** – the room woth PCs, to the right from the staircase

** Course home: ** http://create.mome.hu/Ruttkay/STD1/

** Description:**

In the course students get acquainted with mathematical structures and principles behind designs. Starting off from existing designs – in a variety of medium and function like architectural ornaments, embroidery, basket weaving, cultic sand and cave drawings – students will analyse their structure and device the mathematical models behind. Topics to be discussed include: transformations and similarity, symmetries, tessellations, fractals, mazes, curves and regular and half-regular solids. After understanding the principle of creating such patterns, they will use them for own, artistic designs, by using traditional techniques (e.g. drawing, printing) or computer tools (like Photoshop, Inkscape) they are familiar with. This course will not provide support for using computers or teach to use special sw tools. Those who are familiar with programming in some appropriate language, may use the computer in a more advanced way to create own works.

** Prerequisites: **

Nothing in maths or programming, but curiosity for exploration and interest in analytical thinking. Familiarity with some computer tools for drawing, image processing is handy.

** Equipment: **

Pair of compasses, straight and triangular ruler, notebook, drawing-sketching gears. Own laptop if you have one.

For making own designs, students need to bring/find materials and equipments they wish to use (e.g. paper, paint, textile, …). Digital works are encouradged, but not a must.

** Assesment:**

- active participation at lectures and lab works (20%),
- studies, sketches on at least 3 topics (3×15%) and
- at least 1 fully finished, worked-out design (25%)
- with documentation and end-term presentation (10%).

** in order to get the credit, one has to perform at least 60% **

** Resources: **

- Visual Mathematics by Slavik Jablan Tesselations
- Modular Games with article
- Fractal Art by Janet Parke
- Symmetry by Pilar Moreno – Photography: reflection in water, architecture, cars, nature, …
- Quilts collection
- Spirolaterals by architect Robert Krawczyk
- Geometric Sculptures by George W. Hart
- Star patterns by Craig Kaplan with applet
- interacive synthetic images by Conal Elliott and a sw tool to experiment with
- Spirals, curves, circles and traingles by artist Chilla Benigna
- Spiral sculptures by Brent Collins
- Synetic structures by Frederick G. Flowerday
- Point of observation by Alexander Parkin

**Selection from the Visual Mathematics on-line exhibition**

- Tamas F. Farkas
- Rinus Roelofs

**Further links**

**Students:**

Yoav Amir |
Viktor Boritás |
Carolina Buzio |
Judit Kájel |
Anna Peto |
Tamás Pál |

Krisztina Sárközi | Anja Grunert | Paul Voggendelter | Yakim Voral Karoyan | Gert Pellens |

## Lectures

21.09: Fibonacci numbers |

**Topics:**

- introduction
- schedule and way of working
*starting on time, be active, switch off mobile, shut laptop*media and sw to be used - mathematics – programming – design – engineering
- a panorama of visual mathematics
- a puzzle – coloring houses with black and white, no 2 block floors on top of each other
- the Fibonacci numbers
- the Fibonacci squares, the Fibionacci spiral and the Golden Section spiral
- Fibonacci numbers in nature – phyllotaxis, interactive tool in Mathematica Player
- how fast do they grow? The Golden ratio

**Resouces: **

**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

**Assignments: **

- Fill in the form and send it back
- Send a recent digital photo
- Make designs, experiment by using the Fibinacci squares, the spiral. Make own variations of the spiral, use more spirals, build, draw, …

28.09: 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
- 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!

**Resouces: **

- whirling polygons and
- more on the golden proportion : construction, triangles, pentagram
- Symbolism of the pentagram

**Exercises for the lab: **

- 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.

**Assignments: **

- Make some design by using the golden proportion, golden rectangle and/or golden traingle and/or golden spiral
- 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") or in fine arts
- Survey critically and make a short presentation of the usage of the pentagram, as symbol

28.09: Regular polygons and polyhedra |

**Topics:**

- regular polygons and its symmetries, the notion of the dihedral symmetry group, Dn
- starred polygons: place n equidistant points pn a circle, and connect every 2nd, 3rd, … observe what you get
- studying polyhedra: in the footsteps of two giants: Leonardo da Vinci and Donald Coxeter
- the regular or Platonic solids – names and VR models
- why only 5?
- duality of: cube-octaeder, dodecaeder – icozaeder, tetraeder – tetraeder
- their characterisation and symmetries
- their planar layout for constructing paper models
- other relationships

- the half-regular or Archimedian polyhedra – definition, names, notation and VR models
- why only 13?
- their symmetries
- constructing their paper models

- regular and half-regular polyhedra in the arts

**Resources: **

- The King of Infinite Space site about the book
- resource of nets to make paper models
- Virtual Polyhedra site by George Hart

**Exercises for the lab: **

- make a paper model of an Archimedian solid
- design a solid with 12 faces of identical 5-sided but non-regular polygons, make its net

**Assignments: **

- make a stellated polyhedron model of paper, with own design of the faces (color, material,…)
- make (a series of) models in some interesting medium: big size of bamboe edges, or like jewellery pieces, …

05.10: Symmetries and friezes |

**Topics:**

- The notion of symmetry, symmetry transformations on the plane:
**translation, rotation, reflection, glide reflection** - Frieze: a one-dimensional repetitive pattern
- Friezes in the history of art – see Hungarian folk art embroidery examples
- Symmetries of friezes – there are only 7 types of friezes, according to the 7 symmetry groups – see the table with names and examples.
- Coloured friezes – see exercise 6.
- The Möbius strip

**Resources: **

- Katona, Júlia: Samplebook of Creative Ornamental Art, Typotex Publisher – a great book with a lot of examples of friezes from different cultures, and grids to draw them
- Java applet to explore the structure of the 7 frieze patterns

**Exercises: **

- Draw a simple form on an A4 sheet, and construct where the following transformations will move it on the plane:
**translation**of length 2 cm, in the direction parallel to the shorter side of the paper,**rotation**by 30 degree around the middle point of the paper as rotation center, clockwise;**reflection**on one of the diagonals of the paper**glide reflection**on the other diagonal of the paper, of distance 3 cm

- What is the effect of
**repeating a symmetry transformation**: A translation? A rotation? A glide reflection? - What is the result of
**performing two reflections**one after the other? Think of the possible relative location of the two mirror lines! - Draw a simple motive, and using this,
**sketch different friezes**(on ‘sqare paper’). - Analyze your designs: identify their symmetries, put down the according name. If you have not done all possible types, draw the missing ones.
- Analyze the
**symmetries of friezes from different cultures**– here is the exercise sheet. - Analyze the
**coloured friezes**: does the colouring change the symmetries, and the size of the unit tile to create the frieze? Here are some examples from Maori designs and some others from the Alhambra, sketched by M. C. Escher. - It is easy to design a frize for a column or a vase, which are like cylinders. But how to do it for a Möbius strip? Make a Möbius strip from paper and print or draw on it a continous frieze. Experiment with colouring to make your design emphasize the single-sidedness.

**Assignments: **

Make your work or collected photos electronically available at your own site!

- Make a family of friezes in the medium you like, with at least one example of each of the 7 types. Indicate the type.
- Go out for a hunt for friezes in Budapest. Take photos, and make a collection with note of the exact location, and the type of frieze.

Collect at least one example of all the 7 types. A few tips for good locations: Párizsi udvar, Szervita square – and the staircases of the houses around. The iron fences of the corridors around the internal couryard and stuccos in some of the houses on Nagykörút and Király utca are also interesting. Do not miss the building of the Music Academy on Liszt Ferenc Square, and the building of the Photo Museum nearby.

It is not so easy to get into houses. You could combine your hunt with visits to open ateliers during the Design Week Design Week.

12.10: Tesselations with ploygons – Symmetries of wallpaper patterns |

**Topics:**

*tesselation of the plane*by using congruent examples from a set of polygon types, such as:- By regular polygons

*regular tesselation*with a single type of regular polygons: square, tirangle, hexagon grid

*semiregular or Archimedean tesselation*with 2 or more types of regular polygons, verteces are identical, non-uniform

*non-uniform tesselation*with 2 or more types of regular polygons, but the verteces are not identical, uniform - non-regular polygons, e.g. the Cairo tesselation

- By regular polygons
- dual of a tesselation
- periodic versus aperiodic tesselations
- The 17 wallpaper groups: symmetries, visualization, naming
- Recognizing wallpaper patterns: flowchart, examples

**Resources: **

- Symmetry Groups Summary
- Patterns from the Alhambra
- Java applet to explore symmetries and make own tesselations, you can save your designs!
- The know 14 types of tesselations by convex pentagons – still an open problem if more exist
- Regular Division of the Plane Drawings by M. C. Escher
- Print grid paper : square, triangular, hexagonal
- japanese wallpaper patterns
- Math and the Art of M. C. Escher
- Wikipedia Wallpaper Groups

**Exercises: **

- Exercise sheet to identify structure of wallpapers
- On-line exercise to identify rotational symmetry
- On-line exercise to identify the type of wallpaper symmetry
- Explore patterns from The Grammar of Ornament , originally published in 1856 by Owen Jones (1808-74), and some otherresources.

Look at an individual patterns and try to identify its type. By enlarging the image, you can check the answer.

**Assignments: **

- Design own wallpaper patterns. Work with the on-line sw above, or your own drawing program, or by making print stamps (e.g. cut from potatoe), or hand drawing on grids …
- Collect tesselation examples from the MOME, from Budapest. Analyse them.

19.10: Creating tesselations |

**Topics:**

- Tesselation with rotated and mirrored versions of a single square tile, see Sebastien Truchet tilings , and some more possible tilings
**Coding of tesselations,**patterned and random tesselations and the processing code for single and mirrored tiles to generate them.- Tesselation with
**identical non-convex tiles of different colours,**see examples by Slavik Jablan. - Tesselation with
**2×2 tiles:**a basic one, its mirrored version and their colour complements – KrabceK by Paul Clark - Making Escher-like tiling displaying figures.
- by translation
- by mirroring
- by rotation around edge midpoints
- by rotation around vertices
- using two tiles of different shapes

- Penrose tilings as non-periodic tesselations

**Resources: **

- Op-tiles by Slavik Jablan.
- Coloured tilings by Slavik Jablan.
- Tesselation database
- Makoto Nakamura – check out the animations, my favorites are: bird-dog, and circle , and this one.
- Escherization with sw by Craig Kaplan
- Metamorphoses in Escher’s art by Craig Kaplan
- Non-periodic tilings, also in architecture
- BRIDGES 2010 in Pecs where you can show your things
- Offf 2010 in Paris

**Exercises: **

- Draw ornaments for generating op-art and colour tesselations. Use curves, patches – but bare in mind that the tiles should be connected by the decoration.
- Analyse how Escher created his prints.
- Design some Escher-like tesselations by using different techniquesand grids – see the form here.

**Assignments: **

- Design a set of tiles: op-art, or colour. Create tesselations with them.

You may use the processing code above, a drawing sw or printing technique. - Make some Escher-like tilings of your own.
- Design tesselation for one of the regular solids, and make the paper model.
- Design some interactive tesselation (game) for some public space, and program it in processing.

26.10: NO LECTURE |

2.11: Knot designs |

9.11: |

16.11: Fractals and Discussion of individual projects |

23.11: Optical illusions |

30.11: Project presentations |

page updated by Zsófia Ruttkay — 5.10.2009

style based on www.style.org