Heavenly Mathematics & Cultural Astronomy (GEK1506)

Johann Adam Schall von Bell Astronomers at the Istanbul Observatory
Adam Schall (汤若望 [湯若望], Tāng Rùowàng, 1591-1666), Imperial Astronomer in Beijing. Designed the current Chinese calendar Astronomers at the Istanbul Observatory

Welcome to Heavenly Mathematics & Cultural Astronomy!

Objectives of the Module Topics to be Covered Practical Information and Assessment Course Schedule
IVLE Course Page with Discussion Forum Recommended Texts Astronomical Java Applets and Animations Astronomical Video Clips
Strobel's Astronomy Notes Stellarium Software    

Course Content

Astronomy and its History Calendars Observing the Sun and the Moon from Different Parts of the World The Equation of Time
Ancient Astronomical Instruments Astrology Navigation Cartography
Archaeoastronomy Cultural Astronomy Astronomy in Nature Bad Astronomy and Pseudoscience

Additional Information

Lecture Notes Homework and Tutorials Old Exams Summary
Past Homework Past Projects References Good-bye!
Helmer Aslaksen's home page      

Stonehenge

Objectives of the Module

The goal of this course is to study astronomy and mathematics in a cultural context. We will look at questions like: How is the date of Chinese New Year determined? Why do the Muslim and Chinese months start on different days? Will the Moon ever look like it does on the Singapore flag? What date of the year is the earliest sunrise in Singapore? How did ancient sailors navigate?

After taking this course you will become conscious of the motion of the Sun and the Moon and notice and question things you have earlier taken for granted. You will appreciate mankind's struggle through the ages and throughout the world to understand the mathematics of the heavens.

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

Back to top of page

Astronomical instruments in the Imperial Observatory in Beijing made by the Jesuit missionary Ferdinand Verbiest, 1670
Astronomical instruments in the Imperial Observatory in Beijing made by the Jesuit missionary Ferdinand Verbiest, 1670

Topics to be Covered

We start by discussing the motion of the Sun and the Moon. Most astronomy books are written from a high northern latitude point of view, but in this course we will take a "hemispherically correct", tropical point of view. We then look at applications with a cultural flavor. In Singapore we use the Gregorian, Chinese, Islamic and Indian calendars for determining the public holidays and you will learn the rules for these calendars. Most people expect the time of sunrise and sunset in Singapore to stay fixed throughout the year, but we will study the equation of time and the analemma to see why there is a half hour variation. Navigation has been important to people all over the world throughout history, and people at different latitudes developed fundamentally different methods. If we have time, we will look at ancient astronomical instruments, astrology, cartography, archaeoastronomy (the use of astronomy in archaeology), cultural astronomy and astronomy in nature.

Back to top of page

Brian Greig's astronomical instruments for sale at the Melbourne Arts Centre Sunday Market
Brian Greig's astronomical instruments for sale at the Melbourne Arts Centre Sunday Market

Practical Information and Assessment

Classes

There will be three hours of lectures and one hour of large group tutorial each week. The time slots in 2010/2011 Semester 1 are Monday and Thursday 8-10 a.m. in LT34. I take a 10 minutes break in the middle, and aim to finish by 15 to the hour. I realize that 8 am is not a good time, but that is what we were given. The module is very lecture intensive, so please do not take the module if you do not plan to attend lectures at 8 am.

I'm only supposed to give three hours of lectures so on Thursday I will use the last hour (9 to 10) as a tutorial starting in Week 3. Alternatively, there will be a small group tutorial, too. (Wednesday 5-6 in LT34, starting in Week 3.) The first two weeks I use all four hours for lectures because there is so much practical stuff to cover. The last week there will be small tutorial and I will use all four hours for lectures and review. Feel free to attend whichever tutorial fits your schedule. The small one tends to be highly interactive.

The timetable on IVLE lists the main tutorial as Thursday 10-11. This is WRONG! It will be Thursday 9-10, but is probably because it can't be listed at time because the lecture is listed at that time.

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!

The lectures will be NUScast. The goal of the webcast if to give you a chance to take another look at things you didn't catch, or to help you if arrive a couple of minutes late. They are not meant as a substitute for going to lectures. I'm afraid my lecturing style is not very suitable for webcast. I move around a lot, and I rely on physical demonstrations that probably will not show up in the webcast. Webcast is great for lecturers who read aloud from their lecture notes, which is not my style. Please do not take the module if you plan to rely on the webcast.

Assessment

The final exam counts 50% of your grade. There will be a midterm that counts 30%. There will also be a homework that counts 20% and is done in groups of four to six students.

There will be a midterm test on Monday 27/9/2010 at 8 am in MPSH1 Section B.

The homework is due at the beginning of class on Monday 25/10/2010.

The exam will be on Monday 22/11/2010, 9-11 am.

I want both hard copy and soft copy. Don't worry if the printed copy doesn't look nice, if for instance you don't have a color printer or if you have to do a printout of a web page you created.

Please submit the soft copy of the homework to the IVLE workbin. However, I don't want to get 50 files called Homework.doc. Please use the number of your group as the first two characters in the file name. For example “04 Homework.doc”.

I have a special system for group assessment. Each group can either choose to have everybody get the same grade (the default), or you can choose to grade the others in your group, by saying how many percent of the final mark the others deserve. If you manage to agree, I will follow what you agree on! Just gather everybody and come and see me. If you cannot agree, come and see me (separately, if necessary), and I will take the average. But I will not let that average count 100%. I will give it a weight of x%, and the uniform distribution will get weight (100-x)%. I will not tell you what x is, and I may even use different x for different groups. I encourage you to discuss this in a polite, friendly, reasonable, generous and compassionate way, and make every effort to reach a unanimous decision. My weightage may not be fair, so try to sort it out yourselves. At the same time, I want to have a way to protect people from unreasonable group mates. To sum up:

Groups

Please do the homework in a group of four to six people.

Don't worry if you have a hard time finding a group, you're probably not alone! Post a message on the discussion forum and you'll probably find other people in the same situation who you can form a group with.

Homework

I used to have two homework, but many students complained that it was too much work and too dependent on good weather, so I have simplified into one homework.

E-mail

I get a lot of e-mail, so it would be helpful if you try to observe the following.

Many of the questions involve geometry, and it is hard to discuss it by e-mail. If you want to make an appointment to see me, just mail me and tell me when you are free.

Back to top of page

Course Schedule

Week 1 First week of lectures, no lecture on Mon 9/8/2010
Week 2 Signing up for groups, no lecture Mon 16/8/2010
Week 3 Tutorial 1
Week 4 Tutorial 2
Week 5 Tutorial 3
Week 6 Tutorial 4
Mid-semester Break Yippee!!! :-)
Week 7 Midterm test Monday 27/9/2010 at 8 am in MPSH1 Section B, Tutorial 5
Week 8 Tutorial 6
Week 9 Tutorial 7
Week 10 Tutorial 8
Week 11 Homework due Monday 25/10/2010, Tutorial 9
Week 12 Tutorial 10
Week 13 Last lecture Thursday 11/11/2010 8-10, no tutorial
Reading Week Don't panic! Stay cool!
Exam Exam Monday 22/11/2010 9 am. Good luck!

Back to top of page

Royal Observatory Greenwich
Royal Observatory Greenwich

IVLE Course Page

I have a course page at IVLE, the Integrated Virtual Learning Environment. It contains several useful tools.

Otherwise, the current page is the main source of information.

Back to top of page

Cover of Kuhn's book

Recommended Texts

Unfortunately, there's no textbook for this course. Maybe I'll try to write one myself? The closest is The Ever-Changing Sky, A Guide to the Celestial Sphere by James B. Kaler. However, it goes into a lot more detail about astronomy than we will cover. The level of astronomy I expect you to learn is comparable to the first chapter and the appendix of The Copernican Revolution, Planetary Astronomy in the Development of Western Thought by Thomas S. Kuhn. For convenience, the main reference for the first part will be Strobel's Astronomy Notes on the web.

I have compiled a list of additional references. In addition to the books by Kaler and Kuhn, I've also placed The History and Practice of Ancient Astronomy by James Evans and Calendrical Calculations: The Millennium Edition by Nachum Dershowitz and Edward M. Reingold on RBR in the Science Library.

Back to top of page

Orrery by Brian Greig
Orrery by Brian Greig

Astronomical Java Applets and Animations

Together with Tey Meng Khoon and Frederick H. Willeboordse of CITA (Centre for Information Technology and Applications), I have developed interactive Java applets that I hope will help you understand the motion of the Earth, the Sun and the Moon.

You will notice that a lot of the applets are somewhat similar. They show the same thing, but we're trying to emphasis different aspects or points of view in each one.

You can get the Java Runtime Environment (JRE) at java.com. If you prefer to view the applets off-line, you can download them. Just unzip astro-applets.zip, and open astro-applets.html in your browser.

The last two applets are related to What Does the Waxing or Waning Moon Look Like in Different Parts of the World?

Back to top of page

Equatorial armillary sphere Ecliptic armillary sphere
Equatorial (left) and ecliptic (right) armillary spheres in the Imperial Observatory in Beijing made by the Jesuit missionary Ferdinand Verbiest, 1670

Video Clips

Together with Keith Phua Kuan Wee of CITA (Centre for Information Technology and Applications), I have recorded video clips to help you understand the material. The clips are available in three formats:

I have recorded video clips to explain the Java applets and Strobel's Astronomy Notes.

Back to top of page

Strobel's Astronomy Notes

Nick Strobel of Bakersfield College has written a wonderful set of Astronomy Notes for his introductory astronomy course. His chapter on Astronomy Without a Telescope is very relevant background for this module. I have recorded video clips with annotations to some of the sections of the first chapter of Strobel's notes.

Here is the table of content for the first chapter of Strobel's Astronomy Notes with links to my video clips.

Back to top of page

Circumpolar stars

Stellarium Software

I recommend the wonderful freeware Stellarium Astronomy Software (or from Sourceforge) available for Windows, Linux/Unix and MacOSX. I have written a Help Page.

Back to top of page

Astronomy and its History

How can we use the celestial sphere to understand the movement of the Sun, Moon and stars across the sky? Observing the sky has always been an important part of human civilization. We give a summary of some basic facts from spherical astronomy. One of my students, Viduranga Yashasui Waisundara, has written a beautiful poem about this, called If I were God.

There are a lot of Myths about the Copernican Revolution. Many people whose main interest is philosophy of science, but with little knowledge of astronomy or history of science, have written extensively about it. Unfortunately, much of what they say is incorrect.

Tycho Brahe (1546-1601)
Tycho Brahe (1546-1601)

Multimedia

Course pages

The Meridian Line at Greenwich
The Meridian Line at Greenwich

Astronomical Institutions

Skywatching

General Astronomy Sites

Singaporean Astronomy Related Sites

Back to top of page

Calendars

How is the date for Chinese New Year, Eid ul-Fitr (Hari Raya Puasa) and Deepavali determined? Singapore is unique in that we use four different calendars for determining the date of the public holidays. Seven of our eleven public holidays move, and we study the rules for Chinese New Year, the two Eid's (Hari Raya's), Deepavali, Vesak Day and Good Friday. The exact rules are very complex, but the basic ideas are simple. We give two simple rules of thumb that determine the date for Chinese New Year with a margin of error of one day. I have written a set of lecture notes on The Mathematics of the Public Holidays of Singapore.

Cover of book by Dershowitz and Reingold

You will also see why both Eid ul-Fitrs (Hari Raya Puasas) in 2000 were celebrated one day earlier than they would have been if MUIS had relied on sightings or scientific criteria.

Why is it difficult to determine the Islamic prayer times in Northern Europe? This is an interesting astronomical problem, which very few people understand.

What are leap seconds? This requires understanding some delicate points in the definition of atomic time.

Chinese astronomers determining the summer solstice
Chinese astronomers determining the summer solstice

General Calendar Info

Chinese sundial from the Adler Planetarium Chinese sundial from the Adler Planetarium
Chinese sundials from the Adler Planetarium

The Chinese and other East Asian Calendars

MoonCalc image
MoonCalc image

The Islamic Calendar

Qibla - the direction to Mecca

Determining the direction to Mecca was together with determining the first visibility of the lunar crescent and computing prayer times one of the central problems in Islamic astronomy. It requires a clear understanding of spherical trigonometry and knowing how to find longitude. However, you would think that at the moment it would be a trivial computation. I was therefore quite surprised when I started to receive e-mails from Muslims in the US and Canada asking me about it. I then realized that it is a major controversy among Muslims in North America.

All the famous Muslim astronomers and mathematicians in the past, like al-Khwarizmi (780-850), al-Battani (858-929) and al-Tusi (1201-1274) agreed that qibla should be measured by great circle. However, some people who are used to looking at Mercator maps and don't understand that lines of sight are represented by great circles argue that you should draw straight lines on a Mercator map, i.e., follow rhumb lines. They strongly attack people who follow the traditional method of great circles. For examples of this, see Direction of Al-Qiblah, Determining the direction of the Qibla and Qiblah.us. For a scientifically correct discussion, please see moonsighting.com, the excellent paper The Correct Qibla by Kamal Abdali or Which Way Is Jerusalem? Which Way Is Mecca? The Direction-Facing Problem in Religion and Geography by Daniel Z. Levina.

Prayer Times

The Indian Calendar

Easter

Time

Back to top of page

Observing the Sun and the Moon from Different Parts of the World

This is one of my favorite topics. I have some lecture notes about this, which I'm trying to expand on. 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.

I have a number of web pages about astronomical topics from a Singaporean point of view.

Some of my students have contributed greatly to this. One of my favorites is the following e-mail from one of my students, LIEW Wen Hwee.

Subject: The theory of hanging clothes to dry
Date: Wed, 18 Dec 2002 12:44:51

Hi Sir,

Ohh my. I just discovered the theory behind my mum's “theory of hanging clothes to dry”.

My mum have been telling me since I was a kid that for half of the year that she will have to hang the clothes out in the kitchen and the other half of the year in the corridor for them to dry. I was perplexed by her “theory” for a long time. I thought “wasn't the sun suppose to rise East always?”

And your lesson cleared my doubts!!! My flat lies exactly (ermmm maybe not that exact. didn't notice tat until I used the compass. Was almost scared by a faulty compass that points “west” in the direction where the sun rises!!!) in the East- west plane. And so for half of the year, the sun be in the South-east and the other half in the North-east. MY MUM IS AN ASTRONOMER !!!(ha.).

Sir, what u have taught is of practical purpose!!! (Maybe u could tell the next batch of students abt tat and they could advise their mum on how to better dry their clothes. =))

Thanks a lot, Sir!!!

Sun and Moon info

Singapore weather and tide info

Back to top of page

The Equation of Time

I have a separate web page on Which Day Does the Sun Rise Earliest in Singapore? Singapore lies almost on the equator, so most people would expect the Sun to rise at more or less the same time each day of the year. In fact, the sunrise time varies between 6:46 a.m. and 7:17 a.m., with the earliest sunrise on November 1 and the latest on February 9. In the same way, the sunset time varies between 6:50 p.m. and 7:21 p.m., with the earliest sunset on November 5 and the latest on February 13. The difference between the earliest and latest sunrise in 30 minutes, but the difference between the longest and shortest day is only 8 minutes. The key to understanding this is the analemma, which is a graphical representation of the equation of time.

Analemma picture by Dennis di Cicco
Analemma picture by Dennis di Cicco

The famous picture by Sky & Telescope's Dennis di Cicco records the Sun's position in the sky at the same time of day on 45 different dates throughout the year.

Time zones map

This is also related to the concept of time zones. As you can see from the map, Singapore and West Malaysia are in the “wrong” time zone! For more details, please see my page Why is Singapore in the “wrong” time zone?

Sundial in Enschede, Netherlands, 1836, photo by Fer J. de Vries
Sundial in Enschede, Netherlands, 1836, photo by Fer J. de Vries.

The Equation of Time

Back to top of page

Ancient Astronomical Instruments

Why do sundials look different in Singapore and Beijing? We study the theory behind sundials and show how to construct an accurate sundial for any latitude. There are several nice sundials in Singapore.

Sundial in Jurong birdpark
Sundial in Jurong Bird Park

There is one in the Jurong Bird Park. one in the Botanic Gardens and one in Marina City Park. As part of their Science is everywhere program, The Singapore Science Centre together with the National Parks Board recently put up a series of sundials representing the planets. With Science Centre representing the sun, unique sculptures are placed at various locations around Singapore. The relative distance of these sculptures from the Science Centre represents the distances of each planet from the Sun. Please check out the map!

Four common types of sundials are the equatorial, armillary, horizontal and vertical sundials.

Equatorial sundial Sundial at City Hall, Ruurlo, photo by Frans W. Maes
Equatorial sundial Armillary Sundial at City Hall, Ruurlo, photo by Frans W. Maes.
Horizontal sundial by Harriet James Vertical sundial by Harriet James
Horizontal sundial by Harriet James Vertical sundial by Harriet James

For telling the time at night, you can use a nocturnal.

Nocturnal Nocturnal
Nocturnal Nocturnal

The quadrant is a simple tool for determining latitude.

Quadrant
Quadrant

The astrolabe was made famous by Arab astronomers, and can be thought of as an analog astronomical computer. We discuss the difference between an astrolabe and a mariner's astrolabe.

Astrolabe by Norman Greene Mariner's astrolabe by Norman Greene
Astrolabe by Norman Greene Mariner's astrolabe by Norman Greene

There's a lovely Persian astrolabe from 17th century at the Asian Civilisations Museum in Singapore.

Brass astrolabe, 17th century, Persia, Asian Civilisations Museum's collection, Singapore
Brass astrolabe, 17th century, Persia, Asian Civilisations Museum's collection, Singapore

One of the most fascinating of all astronomical instruments is the Antikythera mechanism. The Antikythera Mechanism is the name given to an astronomical calculating device, measuring about 32 by 16 by 10 cm, which was discovered in 1900 in a sunken ship just off the coast of Antikythera, an island between Crete and the Greek mainland. Several kinds of evidence point incontrovertibly to around 80 B.C. for the date of the shipwreck.

Geoffrey Chaucer, the author of Canterbury Tales wrote a Treatise on the Astrolabe in 1391.

Chaucer

We then look at the cross-staff and the back staff.

Cross-staff Backstaff
Cross-staff Backstaff

We will compare the relative advantages of the quadrant, mariner's astrolabe and cross-staff.

Quadrant Mariner's astrolabe Cross-staff
Quadrant Mariner's astrolabe Cross-staff

The Chinese astronomer Guō Shǒujìng (郭守敬,1231-1314) designed many outstanding instruments. We will look at some of the famous observatories in China and India.

Dengfeng Observatory, designed by Guo Shoujing in 1276 Armillary sphere at the Purple Mountain Observatory in Nanjing designed by Guo Shoujing (1231-1314) Armillary sphere at the Purple Mountain Observatory in Nanjing designed by Guo Shoujing (1231-1314)
Dengfeng Observatory, designed by Guō Shǒujìng (郭守敬,1231-1314) in 1276 Armillary spheres at the Purple Mountain Observatory in Nanjing designed by Guō Shǒujìng (郭守敬,1231-1314)

We study the armillary spheres made by the Danish astronomer Tycho Brahe (1546-1601). (My second cousin lives in a castle built by a nephew of Tycho Brahe!) You will learn the difference between an ecliptic armillary sphere and an equatorial armillary sphere. I also have a separate page about “Fake” Armillary Spheres.

Ecliptic armillary sphere Equatorial armillary sphere
Ecliptic armillary sphere Equatorial armillary sphere

How were cathedrals used as solar observatories? We study meridian lines in European churches. I have created a separate page on meridian lines, A Guide to “The Sun in the Church” by J.L. Heilbron.

Meridian Line. S. Petronio, Bologna, Calter Photo Solar eclipse at the Meridian line at S. Maria degli Angeli in Rome by Mario Catamo
Meridian Line. S. Petronio, Bologna, Calter Photo Solar eclipse at the Meridian line at S.Maria degli Angeli in Rome by Mario Catamo

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.

The reason why I include this painting on this page, is because it contains an incredible collection of astronomical and mathematical instruments! Please check out my page about 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)

Ancient Astronomical Instruments

Tycho Brahe (1546-1601)

Sundial

Sundials

Analemmatic Sundials

Moondials

Nocturnals

Celestial globe made for the Beijing Observatory by Ferdinand Verbiest (1673) Celestial Star Globe
Celestial globe made for the Beijing Observatory by Ferdinand Verbiest (1673) Celestial Star Globe

Celestial Globes

Astrolabes

The Antikythera mechanism

The large Samyat Yantra, equatorial sundial, at Jaipur Drawing of the small equatorial sundial showing the shadow, quadrants, gnomon, and Sun rays
The large Samyat Yantra, equatorial sundial, at Jaipur Drawing of the small equatorial sundial showing the shadow, quadrants, gnomon, and Sun rays

Jaipur Observatory

Back to top of page

Astrology

Does astrology make sense? I'm personally skeptical. In the past, however, there was a close relationship between astronomy and astrology, and astronomy has great historical interest. We look at the mathematical foundation of some of the key concepts in both Western and Chinese astrology.

15th century French woodcut
15th century French woodcut

What do astrologers mean when they say that somebody has an Aries ascendant? In order to understand this, we study the division of the ecliptic into houses. This is a very difficult issue, especially in high latitudes.

Chinese zodiac

Astrology

Back to top of page

Why is it harder to find your position at sea than at land? We study how ancient sailors in different parts of the world navigated in different ways, and how technological advances opened up new possibilities for discoveries and trade.

Cross-staff used for computing lunar distance and height of buildings
Cross-staff used for computing lunar distance and height of buildings on the cover of Introductio geographica by Petrus Apianus, Ingolstadt, 1533

Why is it harder to determine the longitude than the latitude? In 1714 the British Parliament offered 20,000 pound to anybody who could come up with a practical method to determine the longitude at sea. That was the equivalent of US$12 million in modern money! The Astronomer Royal was only paid 100 pounds a year, and had to pay for his own instruments.

Longitude

We will watch the movie Longitude, starring Jeremy Irons, which is based on the best-selling book Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time by Dava Sobel. The movie is very interesting, and includes famous names like Jeremy Irons, Michael Gambon, Bill Nighy, Ian Hart and Gemma Jones.

Why is it harder to find your position in an air plane than at sea? We look at some modern methods, including GPS.

Sextant

General Navigation

Harrison's 4th Timekeeper and Longitude prize-winner
Harrison's 4th Timekeeper and Longitude prize-winner

Finding the Longitude

General History of Navigation

Danish high school students working on a navigation project
Danish high school students working on a navigation project

On-line Navigation Tools

Back to top of page

Cartography

Most capital cities have a geographical origin of coordinates, which is used when making maps for the country. All cities also have a zero point for road distances from the city center to other points. Why do capital cities need an origin of coordinates? In the past, finding coordinates, especially the longitude, was so hard that the best way was to pick a local origin, do astronomical observations at that point, and then do triangulation for the rest of the country. That's why the origin of coordinates is often at observatories.

In London the origin of coordinates is at Greenwich, while the zero point for road distances is at Trafalgar Square. In Oslo the origin of coordinates is at the Observatory, while the zero point for road distances is at Stortorvet. But what about Singapore? After a bit of detective work, and a lot of help from various people, I found that the origin of coordinates is at Empress Place and the zero point for road distances is at the old General Post Office, now the Fullerton Hotel. For more details, see my page on Where is the Geographical Origin of Coordinates of Singapore?

Empress Place

How big is the Earth? We look at Eratosthenes's solution to this problem.

Did people in the Middle Ages believe that the Earth was flat?

Why is there no perfect map? We define the curvature of a surface and show that the curvature of the Earth is positive while the curvature of the plane is zero.

Mercator's 1569 world map Mercator map of the world attributed to Edward Wright (1599) Mercator map
Mercator's 1569 world map Mercator map of the world attributed to Edward Wright (1599) Modern Mercator map

What makes the Mercator projection so important in navigation? We study the geometry of rhumb lines. We also show how it is related to the invention of logarithms.

Cartography

This is NOT a medieval woodcut!
This is NOT a medieval woodcut!

The Flat Earth Myth

Back to top of page

Archaeoastronomy

Was Stonehenge an observatory? When were the pyramids of Egypt built? Archaeoastronomy is a very active area these days. Many ancient monuments reveal an astronomical background.

Newgrange Stonehenge
Newgrange Stonehenge

Stonehenge

Winter solstice sunrise at Newgrange Winter solstice sunrise at Maeshowe
Winter Solstice Sunrise at Newgrange Winter Solstice Sunrise at Maeshowe

Newgrange and Maeshowe

Pyramid

Pyramids

General Archaeoastronomy

Back to top of page

Astronomical instruments in the Imperial Observatory in Beijing made by the Jesuit missionary Ferdinand Verbiest, 1670
Astronomical instruments in the Imperial Observatory in Beijing made by the Jesuit missionary Ferdinand Verbiest, 1670

Cultural Astronomy

What is “wrong” with the Singapore flag? Why is the coat of arms more correct? I have a page on The Mathematics and Astronomy of the Singapore Flag.

The Singapore Flag The Singapore Coat of Arms
The Singapore Flag The Singapore Coat of Arms

What is a Harvest Moon? What is a Blue Moon? What is Groundhog Day? There are a lot astronomical terms that people use but don't really understand.

What was the Star of Bethlehem? How can eclipses be used to date historical events?

Chaucer wrote a book on the astrolabe, and the Canterbury Tales are filled with astronomical references.

Celestial Navigation Net has a very interesting section on Dante and Celestial Navigation!

Donald Olson and others have championed a new field called forensic astronomy. A nice example of this is Astronomers turn detectives to pinpoint date of Ansel Adams image.

Sun and Moon

What is a Blue Moon?

What is a Harvest Moon?

The Star of Bethlehem

Phil the groundhog
Phil the groundhog

Groundhog Day

Back to top of page

Astronomy in Nature

I have a separate page on What Does the Waxing or Waning Moon Look Like in Different Parts of the World? This is a complex question that confuses many people.

Moonrise
Moonrise from Sky & Telescope

What does the orbit of the Moon around the Sun look like? Most people, even almost all mathematicians I've asked this question, tend to believe that it will have loops and look something like the picture below.

400

In fact it looks like this picture!

400

It is not a circle, but is close to a 12-gon with rounded corners. It is locally convex in the sense that it has no loops and the curvature never changes sign. I have a page that explains why The Orbit of the Moon around the Sun is Convex!

Halls Harbour, Nova Scotia, am Halls Harbour, Nova Scotia, pm
Halls Harbour, Nova Scotia, six hours apart

Why are there two high tides each day? This is very frequently asked question! I used to think that the answer was simple, but after looking around in the library and on the web, I discovered several different attempts at explaining it. Newton's equilibrium theory from 1687 used the differential of the gravitational force, but some people (especially oceanographers) also consider a centrifugal force caused by the rotation of the Earth around the Earth-Moon barycenter. Unfortunately, it seems to me that the methods using a centrifugal force are unnecessarily complicated, if not outright wrong! I will describe some of the attempted explanations I found.

How often can we see lunar and solar eclipses in Singapore? It turns out that it is fairly easy to make reasonably accurate eclipse predictions. There are several solar eclipses each year, but at a given place, they occur on average only every 360 years. Solar eclipses, Singapore lists all solar eclipses visible from Singapore between 1700 and 2100. The last total eclipse was in 1821, and there will be annular eclipses in 2019 and 2053, but I'm afraid there will not be any total eclipses in Singapore before 5 July 2168.

How is astronomy related to the ice ages? It turns out that changes in the Earth's movement is related to the ice ages.

Why does the Moon look bigger when it is near the horizon?

Can you balance an egg on the Spring equinox? This turns out to be related to the Chinese calendar!

Eclipses

Stonehenge with Solar Eclipse

In October 1995 I went with Professor John Parkinson to Sabah to watch a total solar eclipse.

Astronomy and Climate

What's Wrong With This Moon?
What's Wrong With This Moon?

The Moon

The Moon Illusion

Back to top of page

Bad Astronomy and Pseudoscience

The truth may be out there, but there's a lot of crap, too! I will discuss some common misconceptions and hoaxes. One of my favorites is the “Mars will be as big at the Full Moon” e-mail. It has several variations, but here is a common one.

The Red Planet is about to be spectacular! This month and next, Earth is catching up with Mars in an encounter that will culminate in the closest approach between the two planets in recorded history. The next time Mars may come this close is in 2287. Due to the way Jupiter's gravity tugs on Mars and perturbs its orbit, astronomers can only be certain t! hat Mars has not come this close to Earth in the Last 5,000 years, but it may be as long as 60,000 years before it happens again.

The encounter will culminate on August 27th when Mars comes to within 34,649,589 miles of Earth and will be (next to the moon) the brightest object in the night sky. It will attain a magnitude of -2.9 and will appear 25.11 arc seconds wide. At a modest 75-power magnification Mars will look as large as the full moon to the naked eye.

Mars will be easy to spot. At the beginning of August it will rise in the east at 10p.m. and reach its azimuth at about 3 a.m. By the end of August when the two planets are closest, Mars will rise at nightfall and reach its highest point in the sky at 12:30a.m.

That's pretty convenient to see something that no human being has seen in recorded history. So, mark your calendar at the beginning of August to see Mars grow progressively brighter and brighter throughout the month. Share this with your children and grandchildren. NO ONE ALIVE TODAY WILL EVER SEE THIS AGAIN!

Does this make you think that Mars will be as big as the Full Moon in the sky? That's NOT what they're saying, but you have to read carefully to not get tricked.

Balancing Eggs on the Spring Equinox

Kepler's Mystery of the Cosmos (Mysterium Cosmographicum), 1596
Kepler's Mysterium Cosmographicum, 1596

Back to top of page

Lecture Notes

The first part of the course gives a basic introduction to spherical astronomy. Here are the slides from the lectures.

The second part of the course is about calendars.

The last part of the course is about time and navigation.

Back to top of page

Homework and Tutorials

Homework

Tutorials

Back to top of page

Old Exams

Back to top of page

Summary

A brief summary of some of the main points.

Armillary sphere at the Purple Mountain Observatory in Nanjing designed by Guo Shoujing (1231-1314) Armillary sphere at the Purple Mountain Observatory in Nanjing designed by Guo Shoujing (1231-1314)
Armillary sphere at the Purple Mountain Observatory in Nanjing designed by Guō Shǒujìng (郭守敬,1231-1314) Armillary sphere at the Purple Mountain Observatory in Nanjing designed by Guō Shǒujìng (郭守敬,1231-1314)

Back to top of page

Past Homework

I have a separate page for past homework.

Back to top of page

Past Projects

Before 2009, students did a project. I have a separate page for past projects.

Back to top of page

Astrolabe

References

I have a separate web page with references.

Back to top of page

Johann Adam Schall von Bell Armillary sphere
Adam Schall (1591-1666), Imperial Astronomer in Beijing Armillary sphere

Good-bye!

Congratulations! You've made it to the end of my page! Thanks for your patience! But before you go, it's time for a little quiz. In the picture of Adam Schall above, can you name the five astronomical and three geometrical instruments in the picture? Can you tell what is wrong with one of them? Hint: Look at the picture on the right. I have included answers.

I hope you have found something of interest. That makes my efforts worthwhile. Feel free to send me a message.

And remember, if life is hard, do like the sundial: Count only the bright hours!

Back to top of page


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.