Visual Mathematics

Science and Technology for Designers - STD1

2009 Fall course at MOME

Lecturer:   Zsófia Ruttkay PhD, Senior research fellow, e-mail:

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:

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.

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.

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.


  • active participation at lectures and lab works (20%),
  • studies, sketches on at least 3 topics (3x15%) 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%



Yoav Amir
Viktor Boritás
Carolina Buzio
Anna Peto
Tamás Pál
Anja Grunert

Paul Voggendelter

Gert Pellens


21.09:   Fibonacci numbers

  • 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: Assignments:
  1. Fill in the form and send it back
  2. Send a recent digital photo
  3. 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: Resouces: Exercises for the lab:
  1. Construct a golden rectangle.
  2. Take an interval, and divide it according to the golden section.
  3. Construct the two kinds of golden traingles.
  4. Construct a pentagon, draw a pentagram in it.
  5. 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
  • 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

  • Resources: Exercises for the lab:
    1. make a paper model of an Archimedian solid
    2. design a solid with 12 faces of identical 5-sided but non-regular polygons, make its net
    1. make a stellated polyhedron model of paper, with own design of the faces (color, material,...)
    2. make (a series of) models in some interesting medium: big size of bamboe edges, or like jewellery pieces, ...
    05.10:   Symmetries and friezes

    Topics: Resources: Exercises:
    1. Draw a simple form on an A4 sheet, and construct where the following transformations will move it on the plane:
      1. translation of length 2 cm, in the direction parallel to the shorter side of the paper,
      2. rotation by 30 degree around the middle point of the paper as rotation center, clockwise;
      3. reflection on one of the diagonals of the paper
      4. glide reflection on the other diagonal of the paper, of distance 3 cm
    2. What is the effect of repeating a symmetry transformation : A translation? A rotation? A glide reflection?
    3. What is the result of performing two reflections one after the other? Think of the possible relative location of the two mirror lines!
    4. Draw a simple motive, and using this, sketch different friezes (on 'sqare paper').
    5. Analyze your designs: identify their symmetries, put down the according name. If you have not done all possible types, draw the missing ones.
    6. Analyze the symmetries of friezes from different cultures - here is the exercise sheet.
    7. 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.
    8. 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.
    Make your work or collected photos electronically available at your own site!
    1. Make a family of friezes in the medium you like, with at least one example of each of the 7 types. Indicate the type.
    2. 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

    • tesselation of the plane by using congruent examples from a set of polygon types, such as:
      1. 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
      2. non-regular polygons, e.g. the Cairo tesselation
    • dual of a tesselation
    • periodic versus aperiodic tesselations
    • The 17 wallpaper groups: symmetries, visualization, naming
    • Recognizing wallpaper patterns: flowchart, examples
    Resources: Exercises:
    1. Exercise sheet to identify structure of wallpapers
    2. On-line exercise to identify rotational symmetry
    3. On-line exercise to identify the type of wallpaper symmetry
    4. 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.
    1. 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 ...
    2. Collect tesselation examples from the MOME, from Budapest. Analyse them.
    19.10:    Creating tesselations

    Topics: Resources: Exercises:
    1. 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.
    2. Analyse how Escher created his prints.
    3. Design some Escher-like tesselations by using different techniquesand grids - see the form here.
    1. 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.
    2. Make some Escher-like tilings of your own.
    3. Design tesselation for one of the regular solids, and make the paper model.
    4. Design some interactive tesselation (game) for some public space, and program it in processing.
      Clean up your works made so far, that we can put them on-line.
    • Save images as max 300 dpi .jpeg files, size large web format (640x480 pixels).
    • NAME the images as: topic_sequencenumber_anythingelse.jpg E.g. anythingelse may be the title, or symmetry type.
    • Collect your files in folders named after a topic, if you have many images for a theme.
    • Make a starting TableOfContent, in html or as text tile, with some information about the images.
    • Put everything in a folder with your family name as folder name, and have it copied to a stick for the next occasion.
    26.10:    NO LECTURE

     2.11:    Fractals

    Topics: Resources: Assignments:
    1. Make 2d or 3d Pythagoras trees of your own design
    2. Create fractals with own generation rules - draw a few iterations
    3. Make some fractal art images using a fractal sw like UltraFractal
    4. Make fractal tesselations
    9.11:   Mirror curves   

    Topics: Resources: Assignments:
    1. make own mirror curves
    2. make own lunda designs
    3. experiment with using more than 2 colors
    16.11:   Optical illusions and Discussion of individual projects
    Guest talk by Tamas Waliczky
    30.11:    Project presentations
    • Project presentations
    • Interactive wall
    • Offf trip
    • evaluation - see form here
    • picnic