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

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

Recommended Texts |

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 |

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 |

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 |

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.

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 |

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.

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

- Peter R. Cromwell: Polyhedra.
- Jay Kappraff, Connections, The Geometric Bridge between Art and Science.
- L. Christine Kinsey and Teresa E. Moore, Symmetry, Shape and Space.
- Dan Pedoe: Geometry and the Visual Arts.
- Dorothy K. Washbourn and Donald W. Crowe, Symmetries of Culture.
- Hermann Weyl, Symmetry.

I have compiled a list of additional references.

Please also consult my rough lecture notes.

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

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

- The Golden Ratio & Squaring the Circle in the Great Pyramid by Paul Calter.
- Polygons, Tilings, & Sacred Geometry by Paul Calter.
- The Golden section ratio: Phi by Ron Knott, University of Surrey.

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

Platonic solids from Kepler's Mysterium Cosmographicum, 1596 |

You can read more on my page about polyhedra.

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.

- Undergraduate research project on The Length of Vermeer's Studio by Madeleine CHEW Mei Ru and Christina LEE Yiwei, 2009.
- Master's thesis on Reconstruction of Vermeer's “The Music Lesson” by Aditya LIVIANDI, 2008.
- Master's thesis on The Inverse Problem in Perspective by CHIA Wan Ting, 2006.
- Undergraduate research project on Vermeer's Camera by YEH Ka Kei.
- Undergraduate research project on Perspective in Mathematics and Art by my student Kevin Heng. The project is a web page, but I also have a printer friendly PDF version.

- Drawing with Awareness by Marc Frantz of Indiana University gives an excellent introduction. For more details, please read his Lessons in Mathematics and Arts. My favorite is Lesson 3: Vanishing Points and Looking at Art.
- PerspectiveGeometry, a Web Site on the Theory, History and Practice of Perspective directed by Tomás García-Salgado.
- The Maths Year 2000 site has a nice page on anamorphic art. It includes a very good discussion of perspective.
- Alberti's Perspective Construction by Tony Phillips at Stony Brook is great! His JavaSketchpad applet is very useful when trying to understand Alberti's method.
- Perspective Modeler by Erik Vestergaard (partially in Danish).
- Handprint.
- Brunelleschi's Peepshow & The Origins Of Perspective by Paul Calter.
- Perspective Study The Flagellation by Marilyn Aronberg Lavin of Princeton University. You need to download a VRML plug-in/client like Cortona VRML Client. You can get the Java Runtime Environment (JRE) at java.com. Part of The Piero Project/ECIT - Electronic Compendium of Images and Text.
- Adventures in CyberSound: Drawing Aids to Perspective.
- The Geometry of Piero della Francesca by Mark A. Peterson of Mount Holyoke College.

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

- Vermeer's Camera, Uncovering the Truth Behind the Masterpieces. A Book by Philip Steadman.
- Vermeer's Camera from Grand Illusions.
- Lions Gate: Girl with a Pearl Earring.
- Essential Vermeer has many very interesting pages including Vermeer and The Camera Obscura and Geographical Distribution of Vermeer's Paintings.
- The Magic Mirror of Life: An Appreciation of the Camera Obscura.
- Vermeer - The Complete Works.
- Camera lucida.
- Camera Lucida: An Optical Illusion for Artists.

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

- David Hockney — Secret Knowledge.
- Art & Optics.
- Frequently Asked Questions (FAQ) by Charles M. Falco.
- Frequently asked questions (FAQs) about claims by David Hockney and Charles Falco on the purported use of optical devices by early Renaissance painters by David G. Stork.

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.

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

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

- Leonardo by Paul Calter.
- Leonardo da Vinci's Polyhedra from Virtual Polyhedra by George W. Hart.
- Dürer's Polyhedra from Virtual Polyhedra by George W. Hart.

Mandala by Timothy Hamons |

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. |

We will also study Penrose tiles.

Penrose tiles |

- Introduction to Tilings (Science U) includes a page on Wallpaper Groups with links to pages about the individual groups that includes animations!
- Symmetries of Culture by Donald W. Crowe includes the flowchart from the book by Crowe and Washburn.
- Symmetry and Pattern The Art of Oriental Carpets by The Textile Museum gives a nice overview of the 17 wall paper groups at Symmetry of Rugs.
- Mathematics Museum (Japan) has several interesting pages, including Seventeen Kinds of Wallpaper Patterns. It shows that the Japanese used 16 of the 17 groups and it also includes a little examination!
- Wallpaper Groups by David E. Joyce of Clark University. David has a lot of other exciting stuff, too. Don't forget to check out his Java version of Euclid.
- 17 Wallpaper Groups by Hop.
- The Discontinuous Groups of Rotation and Translation in the Plane by Xah Lee.
- There is a lot of information at Tiling
Plane & Fancy by Steven R. Edwards at Southern Polytechnic State University, part
of the University System of Georgia.
- Patterns and tilings arranged by their symmetry groups gives a nice overview of the 17 wall paper groups.
- Identifying the 17 Plane Symmetry Groups.
- Tilings from Historical Sources includes Chinese Patterns.

- 17 Wall Paper Symmetry Groups to Create a Regular Division of the Plane by Hans Kuiper.
- Dror Bar-Natan's Image Gallery Symmetry Tilings contains wonderful examples of all the 17 tilings. He also has a flowchart based on Brian Sanderson's Pattern Recognition Algorithm.
- Symmetric Patterns at the Alhambra by Susan Addington and David Marshall.
- Crystallography Now by George Baloglou.
- M.C. Escher and 17 Wallpaper Patterns by Mark Yates.
- crompton tessellations by Andrew Crompton.
- Tilings and Geometric Ornament from The University of Washington, Department of Computer Science & Engineering. This includes Taprats, Computer-Generated Islamic star patterns.
- Gallery by Mike Field at University of Houston.
- Symmetry and the Shape of Space by Chaim Goodman-Strauss.
- Geometry and the Imagination by John Conway, Peter Doyle, Jane Gilman and Bill Thurston.
- The 14 Different Types of Convex Pentagons that Tile the Plane.
- One of the biggest problems when classifying patterns is to distinguish between p3m1 and p31m. Frank Farris has an excellent page on p3m1 versus p31m. It's part of his paper on Vibrating Wallpaper.

- My student Ruth Poh Kim Muay did an undergraduate research project on Frieze Patterns in Ming Ceramics.
- Frieze Patterns in Cast Iron by Heather McLeay.

- Tyler Applet.
- Symmetries, Patterns & Tessellations Constructed With The Geometer's Sketchpad by Allan Bergmann Jensen.
- Kali at Science U. Instructions for using Java Kali.
- Escher Web Sketch by Wes Hardaker and Gervais Chapuis.
- KaleidoMania! Interactive Symmetry from Key Curriculum Press.
- Tess from Pedagoguery Software Inc.

- Jan Abas' Islamic Art Site by Jan Abas at University of Wales, Bangor.

- Tilings from The Geometry Junkyard by David Eppstein.
- Symmetry and Tessellations by Jill Britton.

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 |

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. |

- Jo Edkins' Maze Page.
- EncycloZine Mazes and Labyrinths.
- Caerdroia - the journal of mazes and labyrinths.
- Mid-Atlantic Geomancy: Labyrinths Section.
- Labyrinths and Mazes from Crystalinks.
- mazemaker.com by Adrian Fisher Maze Design.
- Through Mazes to Mathematics by Tony Phillips at SUNY, Stony Brook.
- World Mathematical Year 2000 poster in the London Underground from the Isaac Newton Institute for Mathematical Sciences.

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.

- The World of Escher.
- Mathematical Art of M.C. Escher -- Platonic Realms.
- The Polyhedra of M.C. Escher from Virtual Polyhedra by George W. Hart.

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.

- The Invention of The Kaleidoscope at about.com.
- Kaleidoscope Collector - How Kaleidoscopes Work.
- Kaleidoscope Heaven.
- Kiera.com.
- Kaleidoscope applet by David E. Joyce of Clark University.
- Kaleidoscope Tessellations by Ephraim Fithian.

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

- Music and Computers by Dan Rockmore and Larry Polansky of Dartmouth College.
- The Well-Tempered Scale by Jim Loy.
- Pythagoras Music and Space by J. Boyd-Brent.
- Math and Music by Online Math Applications.
- Mathematics and Music by Dave Rusin.
- The Equal Tempered Scale and Some Peculiarities of Piano Tuning by Jim Campbell.
- A Short History of Tuning and Temperament excerpted from an article by Chas Stoddard.

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

- Symmetry in Art and Architecture
- The Beauty of Polyhedra
- Tilings and Polyhedra
- Polygons and Tilings
- The Golden Ratio
- Polyhedra
- Patterns
- Additional Patterns
- Mazes and Labyrinths
- The Mathematics of Painting

- Tutorial 1 Solutions
- Tutorial 2 Solutions
- Tutorial 3 Solutions
- Tutorial 4 Solutions
- Tutorial 5 Solutions
- Tutorial 6 Solutions
- Tutorial 7
- Tutorial 8 Solutions
- Tutorial 9
- Tutorial 10 Solutions

I have a separate page for past homework.

I have a separate page for past projects.

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.

- Tilings
- Tilings.
- Penrose tiles.

- Polyhedra
- Polyhedra.
- Stellations of polyhedra.

- Golden Ratio
- The golden ratio in art, architecture and nature.
- Fibonacci numbers.
- Growth and ratios in nature.
- Proportions in the human body.
- Proportions in architecture.
- The proportions of the Egyptian pyramids.

- Symmetry and Patterns
- The 17 wallpaper patterns at Alhambra.
- How many of the 17 wallpaper patterns did the Chinese know?
- Patterns in Islamic art and architecture.
- Ornamental symmetry in Singaporean architecture.
- Symmetry patterns in Asian art.
- Tangrams.
- Celtic knots.

- Perspective
- Perspective in paintings.
- Perspective in Chinese art.

- Art
- The Art of Escher.
- The Ambassadors by Holbein.
- Origami.

- Architecture
- Why are fortresses often pentagons?
- The geometry of war.
- Visual illusions in the Parthenon.
- Geodesic domes
- Domes.

- Geometry
- Mazes and labyrinths.
- The fourth dimension.
- Optical illusions.
- Kaleidoscopes.

- Music
- The mathematics of music.
- Chinese musical scales.

- Symmetry in Nature
- Soap bubbles.

I have a separate web page with references.

- Soon after I started planning this course, I discovered
the Mathematics Across the Curriculum project at Dartmouth College. They have several exciting courses.
- Geometry in Art & Architecture developed by Paul Calter.
- Pattern developed by Pippa Drew and Dorothy Wallace.

- Art, Architecture and Mathematics at Leeds University.
- Investigating Patterns Symmetry and Tessellations by Jill Britton. Please remember to check out her activity links.

- Surfaces Beyond the Third Dimension, Computer Generated Images and Video by Tom Banchoff and associates.
- Gallery by Mike Field at University of Houston.

- Math Awareness Month - April 2003: Mathematics and Art
- mathartfun.com - Where math, art & fun come together!
- ISAMA, The International Society for the Arts, Mathematics, and Architecture, maintained by John M. Sullivan.
- The Mathematics Archives has a section on Art & Music.
- The Math Forum at Swarthmore has a page on Architecture.
- Art - Mathematics and the Liberal Arts by Todd Hammond.
- The Geometry Junkyard by David Eppstein.
- Nexus Network Journal is a peer-reviewed online research resource for studies in architecture and mathematics.

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

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