Mathematics in Art and Architecture (GEK1518)

Melencolia I by Albrecht Dürer, 1514 The Ambassadors (1533), by Hans Holbein the Younger (1497/8 - 1543)
Melencolia I by Albrecht Dürer, 1514 The Ambassadors (1533), by Hans Holbein the Younger (1497/8 - 1543)

Welcome to Mathematics in Art and Architecture!

Objectives of the Module Topics to be Covered Practical Information and Assessment IVLE Course Page with Discussion Forum
Recommended Texts      

Course Content

Pyramids The Golden Ratio The Platonic Solids and Polyhedra Perspective
The Ambassadors by Holbein Symmetry, Patterns and Tilings Mazes and Labyrinths The Art of Escher
Kaleidoscopes Music    

Additional Information

Lecture Notes Tutorials and Homework Assignments Old Exams Past Homework
Past Projects Project Topics References Web links
Art Figures: Mathematics in Art. An Exhibition at the Singapore Art Museum      
Luca Pacioli by Jacopo de Barbari, 1495
Luca Pacioli by Jacopo de Barbari, 1495

Objectives of the Module

The goal of the course is to study connections between mathematics and art and architecture. You will see how mathematics is not just about formulas and logic, but about patterns, symmetry, structure, shape and beauty. We will study topics like tilings, polyhedra and perspective.

After taking this course you will look at the world with new eyes and notice mathematical structures around you.

Raphael's School of Athens, 1510-11
Raphael's School of Athens, 1510-11

Topics to be Covered

We start by studying tilings. They occur in many settings, and have a rich mathematical structure. The Platonic solids and polyhedra have inspired people throughout the ages. The golden ratio has fascinated many people, but we will take a critical look at whether it was really used in art and architecture. Symmetry and patterns are important in ornamental art in all cultures. Among the most famous are the Islamic patterns at Alhambra. Perspective originated in the Renaissance and changed the way we look at the world. Many artworks are rich in mathematical structure. We will look at the works of Escher and Holbein. Some of the applications of mathematics in architecture that we will look at are the Parthenon and military engineering. Other beautiful applications of geometry are kaleidoscopes, mazes and labyrinths, the fourth dimension and optical illusions. We finish by looking at applications of mathematics in music.

Picture from Crystalinks

Practical Information and Assessment

This course is one of the new General Education Modules at the NUS.

I will be away on conference leave from 6/1/04 to 12/1/04, so there will be no lectures on Tuesday 6/1 and Friday 9/1.

There will be three hours of lectures and one hour of large group tutorial each week. The time slots in 2003/2004 Semester 2 are Tuesday and Friday 10-12 in LT22. I will go from 10.00 to 10.50, take a 10 min break and go from 11.00 to 11.50. There will be two tutorial groups. The last session on Friday will be a large-group tutorial, and there will also be another tutorial group. You only need to attend one of these.

I use a cordless microphone and walk around in class and ask questions. But don't worry, I only ask easy questions! I also like to create physical demonstrations to illustrate the concepts, and I often need “volunteers” for this. I am not afraid of looking silly, and I hope you are not either!

If you send me e-mail, please use the module code GEK1518 in the subject. Otherwise you may end up in my spam folder. This is especially important if you use a non-NUS e-mail address.

The final exam counts 40% of your grade. You have to do a project that counts 30%. The projects are done in groups of four to six students. There will be also be two homework that count 15% each.

Please do the homework in the same group as you do the project. If you are planning to do a very special topic, and you're having a hard time finding somebody interested in it, I MAY also approve individual projects or groups of two or three. The chances of me approving such requests are best if you approach me early.

The first homework and the project proposal are due Tuesday 10/2/2004, the second homework is due Tuesday 2/3/2004, and the project is due Tuesday 9/3/2004. The exam will be Thursday 15/04/2004 pm.

Many topics will only be touched upon in lectures, and you may explore them further on your own in the projects. I have a list of possible topics, but I also encourage you to propose your own topics and send them to me for approval. I hope that you will be able to find something that you are enthusiastic about. However, the project must have some scientific angle. I cannot be a pure arts or cultural project.

The project can be a normal paper project, a web page, a physical model or a combination of all these. I don't have any set rules about length or scope of the project, but I have some guidelines.

The project proposal should include the title, the names of the members of the group, a brief outline, and a list of the main references. One or two pages is enough.

Please submit the proposal, the project and the homework in both hard copy in class and soft copy in the IVLE workbin. I prefer to read the hard copy, so if you create a web page, please print out a hard copy, too. I realize that the print out may not do full justice to your web page, but it will give me time to read the text before I look at your page. If you have animations or other things that you can't print out, please include a note where you indicate which parts of the web site I should look more closely at.

If the project is a web page and you have a server to put it on, you can just submit a file with the URL. However, I would appreciate it if you could also give me the files on a CD, or zip the files into one file and upload.

If your project includes a physical model, please let me know if you want it back. Some of them I may ask to keep, but some of them are too bulky, and I must either throw them away or return them to you quickly.

I have a page with links to some past projects.

The first homework is to make paper models of the five Platonic solids and the thirteen Archimedean solids. The second is to collect five pictures (per person) of mathematically interesting objects around you. So if there are x people in your group, I expect 5x pictures. I have a page with some highlights from the second homework.

Julians Bower turf labyrinth - Alkborough, Humberside, England - medieval
Julians Bower turf labyrinth - Alkborough, Humberside, England - medieval

IVLE Course Page

I have a course page at IVLE, the Integrated Virtual Learning Environment at the NUS. It has a discussion forum that I encourage you to use.

Cover of Pedoe's book

Recommended Texts

Unfortunately, there's no text that is suitable. Some of the material will be taken from books like:

I have compiled a list of additional references.

Please also consult my rough lecture notes.

Pyramids

Pyramid Pyramid

We start by studying the geometry behind the Egyptian pyramids. A lot of this is controversial, see the paper by Markowsky.

Pyramids

The Golden Ratio

Golden ratio

The golden ratio and proportions were important in Greek culture. A lot of this is controversial, see the paper by Markowsky.

The Golden Ratio

The Platonic Solids and Polyhedra

Picture
Find out more about the polyhedra my student Kavitha and I have made with Zome System.

Platonic solids

Archimedean solids

Platonic solids from Kepler's Mysterium Cosmographicum, 1596
Platonic solids from Kepler's Mysterium Cosmographicum, 1596

You can read more on my page about polyhedra.

Perspective

Flagellation of Christ by Piero della Francesca, late 1450s; in the National Gallery of the Marches, Urbino, Italy.
Flagellation of Christ by Piero della Francesca, late 1450s; in the National Gallery of the Marches, Urbino, Italy.

Next comes the Renaissance and the origin of perspective and projective geometry.

Student Papers on Perspective

The Origin of Perspective

Vermeer and optics

Many scholars believe that the Dutch painter Johannes Vermeer (1632-1675) used a camera obscura.

David Hockney — Secret Knowledge.

The painter David Hockney believes that optical aids were used even earlier. This is a more controversial theory.

The Ambassadors by Holbein

The Ambassadors (1533), by Hans Holbein the Younger (1497/8 - 1543)
The Ambassadors (1533), by Hans Holbein the Younger (1497/8 - 1543)

The Ambassadors by Holbein is a famous example of Anamorphosis. Do you see the strange object on the floor? Close your left eye, put your face close to the computer screen near the right side of the picture. You will then see a skull! If you can't get it to work, you can cheat and look at a picture of it. Please check out my page about The Ambassadors by Holbein.

Dürer and da Vinci

We also look at the work of Dürer and da Vinci.

Vitruvius man in Leonardo's notebook Albrecht Dürer, 1525
Vitruvius man in Leonardo's notebook Albrecht Dürer, 1525

Dürer and da Vinci

Mandala by Timothy Hamons
Mandala by Timothy Hamons

Symmetry, Patterns and Tilings

Ornamental patterns are important in all cultures. Among the most famous are the Islamic patterns at Alhambra. This brings in symmetry groups and crystallography. We study symmetry in the plane and the wallpaper and frieze groups. Symmetry of Rugs gives a nice overview of the 17 wall paper groups.

One frequently asked question is whether all the 17 wall paper groups can be found in the Alhambra. In her 1944 thesis, Edith Müller found 11, and not 17 as has often been claimed. Two more were described in Branko Grünbaum, Zdenka Grünbaum and G.C. Shephard, Symmetry in Moorish and Other Ornaments, Comp. Math. Appl, 12B (1986), 641-653. R. Péres-Gómez, The Four Regular Mosaics Missing in the Alhambra, Comp. Math. Appl., 14 (1987), 133-137, claims to have found the last four. However, as far as I can tell, he does NOT include a picture of p3m1. You may also want to look at Coxeter's review in Math. Review. José María Montesinos includes pictures of all 17 in his book “Classical Tessellations and Three-Manifolds”, but I don't understand how he manages to see p3m1 in his pictures. If anybody can clear this up for me, I would be very grateful.

Fatehpur Sikri, Tomb of Salim Chishti, 1573-4. From Blair and Bloom, The Art and Architecture of Islam, p. 273.
Fatehpur Sikri, Tomb of Salim Chishti, 1573-4. From Blair and Bloom, The Art and Architecture of Islam, p. 273.

We will also study Penrose tiles.

Penrose tiles
Penrose tiles

The 17 Wallpaper Groups

The 7 Frieze Groups

Interactive

Jali, red sandstone, Rajasthan, late 17th/18th century, Asian Civilisations Museum's collection, Singapore Wooden panel, 18th - 19th centuries, North Africa (probably Morocco), Asian Civilisations Museum's collection, Singapore
Jali, red sandstone, Rajasthan, late 17th/18th century, Asian Civilisations Museum's collection, Singapore Wooden panel, 18th - 19th centuries, North Africa (probably Morocco), Asian Civilisations Museum's collection, Singapore

Islamic Art

Collections of Links about Symmetry, Patterns, Tilings and Tessellations

Mazes and Labyrinths

Hampton Court maze Hampton Court maze
Hampton Court maze Hampton Court maze

Many ornamental patterns are related to topology, for example mazes. Is there a difference between a maze and a labyrinth? Traditionally, the terms have been considered to be synonymous, but around 1990 people interested in the spiritual aspects of labyrinths devised a terminology where a labyrinth is unicursal and a maze multicursal. This means that a labyrinth has only one path with no branches and no dead ends, in other words, no choice, while a maze is a logical puzzle with branches and possibly dead ends.

Cretan labyrinth
Cretan labyrinth

Unfortunately, there are problems with this terminology. Theseus would not need Ariadne's thread in a labyrinth, and the turf mazes in Britain have been called turf mazes all along, although most of them are unicursal.

Labyrinth at Chartres Cathedral. Photo: Jeff Saward.
Labyrinth at Chartres Cathedral. Photo: Jeff Saward.

Mazes and Labyrinths

The Art of Escher

Circle Limit III by M.C. Escher, 1959
Circle Limit III by M.C. Escher, 1959

The work of Escher is rich in mathematical content. Much of it is related to hyperbolic geometry.

Escher

Kaleidoscopes

Kaleidoscope from Kiera.com/
Kaleidoscope from Kiera.com

The kaleidoscope is a beautiful application of geometry. It was invented by Sir David Brewster, a Scottish scientist, in 1816. He named his invention after the Greek words, kalos or beautiful, eidos or form, and scopos or watcher. So kaleidoscope means the beautiful form watcher. Brewster's kaleidoscope was a tube containing loose pieces of colored glass and other pretty objects, reflected by mirrors or glass lenses set at angles, that created patterns when viewed through the end of the tube.

Kaleidoscopes

Pythagoras

Music

Ever since Pythagoras, there has been a close relationship between mathematics and music.

Cover of Gödel, Escher, Bach

Music

Lecture Notes

Here are some rough lectures notes. They are under construction!

Tutorials and Homework Assignments

Homework

Tutorials

Old Exams

Past Homework

I have a separate page for past homework.

Past Projects

I have a separate page for past projects.

Project Topics

Most of the topics we talk about in lecture can be extended to projects. Here are some suggestions. I also encourage you to propose your own topics and send them to me for approval.

References

I have a separate web page with references.

Course Pages and Books

Computer Art

Other links

Art Figures: Mathematics in Art. An Exhibition at the Singapore Art Museum.

I was mathematical consultant for the exhibition “Art Figures: Mathematics in Art” at the Singapore Art Museum.


Helmer Aslaksen
Department of Mathematics
National University of Singapore
helmer.aslaksen@gmail.com

Web Server Statistics for Helmer Aslaksen, produced by Analog.

Valid XHTML 1.0!

I use the W3C MarkUp Validation Service and the W3C Link Checker.