Models of polyhedra made with Zome System | Workshop on building polyhedra with secondary school students from the Gifted Education Programme |

- Do you want me to give a lecture?
- Abstracts of
my favorite lectures
- The Mathematics of the Public Holidays of Singapore
- The Mathematics of the Chinese Calendar
- Heavenly Mathematics: Observing the Sun and the Moon from Different Parts of the World
- When Does the Sun Rise in Singapore and What is “Wrong” with the Singapore Flag?
- The Mathematics of Painting
- Symmetry in Art and Architecture
- Symmetry in Art: The Mathematics of Chinese Ceramics
- The Beauty of Polyhedra
- Tilings
- The Mathematics of Sudoku
- Proposing, Supervising and Assessing Student Research Projects in Mathematics

- Brief biography for publicity purposes
- Upcoming lectures
- Past lectures

I want everybody to appreciate the beauty of mathematics and astronomy, and to see its significance and relevance to the world around us. I believe that this is an important part of being a professor of mathematics and I'm more than happy to give public lectures. However, as you can see from my list of past lectures, I have become quite busy. I'm afraid that I therefore have to cut down on the number of lectures at individual schools, and I will prioritize schools that fit the university's outreach program. However, I'm more than happy to talk to groups of teachers.

I need a cordless clip-on microphone because I need my hands free to hold my models. I also need a computer projector for my laptop. If you don't have a projector, please let me know, and I will bring OHP transparencies.

These lectures can be given as one hour lectures or as longer workshops with more hands-on activities. If you're interested in the workshops, they can be arranged through the Singapore Mathematical Society.

Have you always wondered why Chinese New Year, the end of Ramadan (Eid ul-Fitr/Hari Raya Puasa), Easter Sunday and Deepavali fall on different days each year? Then this is the talk for you! I will give an overview of the Gregorian, Chinese, Islamic and Indian calendars. The Gregorian calendar is fairly simple, while the three other involves deep mathematical problems. However, there are simple rules of thumb that allow you to predict Chinese New Year, the end of Ramadan and Deepavali with an error of at most one day. (Sure way to impress your relatives!) I will also discuss various historical and cultural aspects of the calendars.

I hope that this talk will make you more conscious of the mathematics of the world around you, and give you knowledge that you will enjoy sharing with others for the rest of your life.

Have you always wondered why Chinese New Year falls between January 21 and February 21? Chinese New Year is the main holiday for more than a quarter of the world's population. Yet very few people know how to compute the date. The exact rules are very complex, but I will give some simple rules of thumb that allow you to predict the date with an error of at most one day. (Sure way to impress your relatives at reunion dinner!)

You will learn why it was necessary to have more than 100 reforms of the Chinese calendar, why the lunar calendar is not useful to farmers, why the modern Chinese calendar was designed by a German, why the Mid-Autumn Festival is celebrated on the 15th day of the 8th month, why year 2000 was a golden dragon year, why Lunar New Year should really be called Lunisolar New Year and other aspects of the science and history behind the Chinese calendar that I believe every educated person should know.

Optional hands-on: Use a computer to study the variation in the date of your Chinese birthday.

Most astronomy books are written from a “high-northern-latitude-centric” point of view. I will discuss the motion of the Sun and the Moon from a “hemispherically-correct” point of view, with special emphasis on the needs of “latitudinally-challenged” observers. Some of the questions we will address are: Why do clocks go clockwise? What does the orbit of the Moon around the Sun look like? Which day does the Sun rise earliest in San Francisco, Singapore or Sydney? How do you tell the difference between a waxing crescent Moon and a waning crescent Moon in San Francisco, Singapore or Sydney?

I hope this talk will make you more conscious of the mathematics of the world around you, and give you knowledge that you will enjoy sharing with others for the rest of your life.

Where does the Sun rise? Most of you will say that it rises in the East, but in fact it only rises due East on two days each year.

On the North Pole the Sun moves clockwise in the course of the day. On the South Pole it moves counterclockwise. So how does it move in Singapore?

In the northern hemisphere, the days are long in June and short in December. In the southern hemisphere it's the other way around, and near the equator the length of the day is about the same all year round. Most people therefore think that in Singapore the sun will rise at more or less the same time each day. This is not true!

In Singapore, the difference between the longest and the shortest day is only about 9 minutes, but the difference between the earliest and latest sunrise is almost 31 minutes! The longest day is on June 21 and the shortest day is on December 21, which should not come as a surprise to anybody who has ever heard of the June and December solstices. But why on earth does the earliest sunrise in Singapore fall on November 5 and the latest sunrise on February 13? In order to understand this, we will talk about something called the equation of time and the analemma.

The crescent on the Singapore flag is supposed to represent a young, ascending nation. On the front of the flag we see a left crescent, on the back a right crescent. So what exactly does a young Moon look like in Singapore? The answer is that it will look like the Moon on the Singapore Coat of Arms, namely a bottom Moon!

And by the way, do you know what the orbit of the Moon around the Sun looks like? Do you think it has loops or curves in and out? No it doesn't! It curves left all the time!

You will discover that you can learn a lot about astronomy by doing simple observations in your daily life. This will make you see how mathematics can help you understand the world around you.

In the early 1400's there was a revolution in European art. For the first time, painters were able to paint realistically. The secret was the new mathematical theory of perspective. The key idea is that a picture corresponds to a specific viewing point.

However, this does not fully explain the sharp jump in the quality of the paintings. Scholars and artists like David Hockney and Philip Steadman suggest that part of the explanation is the use of optical aids like the camera obscura. We will look at why Vermeer's paintings suggest the use of optical aids.

You will learn how you can sometime determine the best place to stand when looking at a picture, and we will discuss whether a “perfect” perspective drawing really reflects what we see.

The goal of this talk is to change the way you look at the world around you. You will learn how to use mathematics to analyze and describe patterns and symmetry in art and architecture, using a variety of examples, including Chinese ceramics, Islamic art and buildings in Singapore. We will focus on the symmetry of the patterns, and not the motif itself. Different cultures have a preference for different symmetry types, and this can be used to determine the origin and date of artifacts.

Optional hands-on: Learn how to determine the type of different patterns.

We will study the mathematics behind ornamental patterns, using Ming and Peranakan ceramics as examples. There are three main types of ornamental patterns: Rosette patterns, frieze patterns and wallpaper patterns. Rosette patterns are individual motifs that have rotational or reflectional symmetry. Frieze patterns are obtained by covering a strip with copies of a fundamental motif, while wallpaper patterns are obtained by covering a surface with copies of a fundamental motif. These patterns are common in both art and architecture.

From a mathematical point of view, it turns out that there infinitely many types of rosette patterns, but only seven types of frieze patterns and 17 types of wallpaper patterns. We have found all seven frieze patterns on Ming ceramics, while so far we have only found six of them on Peranakan ceramics.

It is important to realize that what is important is the way the copies of the motif are spread out along the surface or strip, and not the motif itself. Different cultures have a preference for different symmetry types, and this can be used to determine the origin and date of artifacts.

In this talk we will start with a non-technical outline of the mathematical view of ornamental patterns, and how to distinguish between the different types. We will then apply this to the study of Chinese ceramics. The goal of the talk is to make you look at ceramics and ornamental patterns with different eyes.

Polyhedra are figures with polygonal faces. The most famous are the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron and icosahedron. But there are also many other important families, like the Archimedean solids and deltahedra.

These figures have fascinated people throughout the ages. The best way to appreciate them it so play with models, and I will demonstrate different ways of building such polyhedra, using either paper or plastic building kits. You can see examples of this on my web page on polyhedra. I will then use these models to give an introduction to some of the theory and history behind polyhedra.

Optional hands-on: Use the Zome System construction kit to make a model of the five Platonic solids inside each other!

In school you learn about the three regular polygonal tilings of the plane consisting of triangles, squares and hexagons. But what if you use more than one different type of tiles? In that case we get eight additional tilings. In this talk you will learn how to make them and study their mathematical structure.

Sudoku is a logic puzzle where you are given a 9×9 grid made up of nine 3×3 blocks. The goal is to place the numbers 1 through 9 into the cells in such a way that each row, column and box contains each number exactly once. Some of the cells are given, and this is done in such a way that there is a unique way to fill in the remaining cells. The puzzles can be of varying levels of difficulty. They can be easy enough to appeal to anybody, while a mathematician will immediately be fascinated by the more fiendish puzzles and start thinking about algorithms. I will describe some of the techniques for solving this puzzle and we will solve some puzzles together.

What can we expect from mathematical project work? How to select topics, find sources, supervise and assess? We will also look at samples of good topics.

Associate Professor Helmer Aslaksen was born in Oslo, Norway, and did his undergraduate at the University of Oslo. After receiving his Ph.D. at the University of California, Berkeley, he joined the Department of Mathematics at the National University of Singapore in 1989.

His interests include geometry, Lie theory, and the relationship between mathematics and astronomy and art.

He has been academic advisor for the exhibition "Art Figures: Mathematics in Art" at the Singapore Art Museum and "The Dating Game: Calendars and Time in Asia" at the Asian Civilisation Museum and for the TV series "Ancient Chinese Inventions" on the Discovery Channel. He was also on the Program Committee and a judge for "National Science Challenge", a TV science quiz for secondary school students. He was on the organizing committee of a topic study group at the International Congress on Mathematical Education in 2004. He has been invited to be a plenary speaker for the Mathematical Association of America. He has an extensive web site, including a highly ranked page on The Mathematics of the Chinese Calendar.

At the NUS he has introduced two General Education Modules, Heavenly Mathematics & Cultural Astronomy and Mathematics in Art and Architecture. In 2004 he was awarded the University's Outstanding Educator Award.

- Lectures on “The Mathematics of the Chinese, Indian, Islamic and Gregorian Calendars”, “The Mathematics of the Public Holidays of Singapore”, or “The Mathematics of the Chinese Calendar”:
- RI-RGS Advanced Mathematics Modules (series of six lectures), 2010.
- RI-HCI Astrigue 2009, organized by the Raffles Institution and Hwa Chong Institution Astronomy Clubs, 2009.
- Hwa Chong Science and Mathematics Talent Programme (series of three lectures), 2008.
- NUS Business Incubation of Global Organisations (BINGO), 2007.
- Sunburst Youth Camp, Singapore Technologies Endowment Programme, 2005.
- Science.05, a month of Science, Technology and Biomedicine, 2005.
- National University of Singapore Mathematics Society, 2005 and 2009.
- Outstanding Educator Award Public Lecture Series, NUS, 2005.
- Bay Area Mathematical Adventures, San Jose State University, USA, 2005.
- Singapore Malaysia Students' Association, University of California, Berkeley, USA, 2005.
- Lodestar Dinner, International House, Berkeley, USA, 2005.
- Berkeley Math Circle, University of California, Berkeley, USA, 2005.
- Evening lecture, Harvey Mudd College, USA, 2004.
- Winkelmann Mathematics and Society Lecture Series, Macalester College, USA, 2004. (An annual lecture series in mathematics, past speakers include Herb Wilf, Gilbert Strang and Tom Banchoff.)
- All Secondary Schools Mathematics Competition, National Junior College, 2003, 2007.
- Oslo katedralskole, Oslo, Norway, 2003. This lecture at my old high school was part of the activities surrounding the first Abel Prize in mathematics. Niels Henrik Abel was a student at this school, which celebrated its 850th anniversary a few months earlier.
- National University of Singapore Astronomical Society, 2002.
- National Library, Ngee Ann City Shopping Centre, 2001-2002.
- Science Focus, National University of Singapore, 2001, 2006.
- Singapore Mathematical Society, 2001.
- Singapore International Math Olympiad Training Camp, 2001.
- Astronomical Society of Singapore, 2001.
- Gifted Education Programme Mathematics Enrichment Camp, 2001-2002.
- Annual General Meeting of the Norwegian Business Association (Singapore), 2001.
- The Culture Society, 2001.
- Asian Civilisations Museum, 2000.
- National Science and Technology Board, 2000.
- Lee Kong Chian Art Museum, 2000.
- Science Millennium Lecture, National University of Singapore, 2000.
- Singapore Science Centre, 2000.
- National University of Singapore Science and Technology Camp, 1999-2000.

- Lectures on “Observing the Sun
and the Moon from Different Parts of the World” or “When Does the Sun Rise in Singapore and What is ‘Wrong’ with the Singapore Flag?”:
- San Jose Math Circle, San Jose State University, USA, 2005.
- Berkeley Math Circle, University of California, Berkeley, USA, 2004.
- Open House, Department of Mathematics, National University of Singapore, 2003.
- Singapore Mathematical Society Annual Prize Presentation Ceremony, Chinese High School, 2003.

- Lectures on “Symmetry in Art and
Architecture”, “The Mathematics of Symmetry: From Wallpaper to Islamic Art” or “The Mathematics of Ceramics”:
- Open House, Department of Mathematics, National University of Singapore, 2003.
- Singapore Mathematical Society, 2003.
- National University of Singapore Museums, 2002.
- Asian Civilisations Museum, 2002.
- Gifted Education Programme Mathematics Enrichment Camp, 2001.

- Lecture on “The Beauty of Polyhedra”:
- Gifted Education Branch Primary Mathematics Masterclass, 2008.
- Gifted Education Programme Mathematics Seminar, 2003.
- Gifted Education Programme Mathematics Enrichment Camp, 2003.
- Singapore Mathematical Society, 2002.

- Lecture on “The Mathematics of Sudoku”:
- Singapore Mathematical Society-Royal Institution Masterclasses, 2007-2009.
- Gifted Education Programme Mathematics Enrichment Camp, 2006-2009.
- People's Association, 2006.
- Open House, Department of Mathematics, National University of Singapore, 2006.
- Mathematics Enrichment Camp, National University of Singapore, 2005-2010.

- National University of Singapore Open House, Faculty Lecture, 2003.
- Lecture for the guides and public lecture at the Singapore Art Museum in conjunction with the exhibition “Art Figures: Mathematics in Art”, 2002.
- Series of two lecatures for the guides and three public lectures at the Asian Civilisations Museum in conjunction with the exhibition “The Dating Game -- Calendars and Time in Asia”, 1999-2000.
- Lectures in the Gifted Education Programme Science Camp, 1991, 1993-1994.

- Keynote speaker in “The Mathematics of the Public Holidays of Singapore” at Mathematics Symposium at Centre of Excellence at Zone East, 2008.
- Lecture at Knowledge and Inquiry Symposium on New Technologies and Knowledge Construction, Curriculum Planning and Development Division, Ministry of Education, Singapore, 2007.
- Lecture at in-service course on the history of mathematics for high school teachers, University of Oslo, Norway, 2006.
- Workshop on “Proposing, Supervising and Assessing Student Research Projects in Mathematics”, Raffles Girls' School (Secondary), 2005.
- Lecture on “Proposing, Supervising and Assessing Student Research”, Raffles Institution, 2004.
- Lecture on “The Mathematics of the Public Holidays of Singapore” at the Mathematics Workshop for Primary School Teachers in conjunction with the Singapore Mathematical Olympiad for Primary Schools, Chinese High School, 2004.
- Lecture on “When Does the Sun Rise in Singapore and What is ‘Wrong’ with the Singapore Flag?” at the W1 Cluster Schools Good Practices Day for Mathematics Teachers, Anglo-Chinese Junior College, 2003.
- Seminar for Gifted Education Programme Primary Science teachers on “The Mathematics of the Public Holidays of Singapore” and “When Does the Sun Rise in Singapore and What is ‘Wrong’ with the Singapore Flag?”, Anglo-Chinese School (Primary), 2003.
- Lecture on “The Mathematics of the Public Holidays of Singapore”, West Zone Maths Committee, Fajar Secondary School, 2002.

In Singapore, junior college is grade 11 to 12 and secondary school is grade 7 to 10. High school is an unofficial name, which usually means 7 to 10, but can also mean 7 to 12. I've visited 14 of the 22 pre-university centres in Singapore (Junior Colleges and Integrated Programmes).

- Anderson Junior College, 2002, 2003.
- Anglo-Chinese School (Independent), 2007.
- Catholic Junior College, 2004.
- Dunman High School, 2002, 2006.
- Fuchun Secondary School, 2007.
- Hwa Chong Institution (The Chinese High School/Hwa Chong Junior College), 2000, 2001, 2002, 2003 (twice), 2008 (three times), 2010.
- Nan Hua Secondary School, 2002.
- National Junior College, 2003, 2007.
- NUS High School of Math and Science, 2005, 2008 (twice).
- Outram Secondary School, 2002 (twice).
- Paya Lebar Methodist Girls' School, 2007.
- Pioneer Junior College, 2001.
- Raffles Girls’ School (Secondary), 2003 (twice), 2007, 2010 (six times).
- Raffles Institution, 2009.
- School of the Arts, 2008.
- Singapore Chinese Girls’ School, Singapore, 2004 (twice).
- St. Andrew’s Junior College, 2002, 2003, 2007.
- Tampines Junior College, 2006.
- Temasek Junior College, 2001, 2008.
- Victoria Junior College, 2002.
- Yishun Junior College, 2002.

Department of Mathematics

National University of Singapore

aslaksen@ nus.edu.sg

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