# The Mathematics and Astronomy of the Singapore Flag

Disclaimer: I totally recognize the “artistic license” in the design of the Singapore flag. I do not mean any disrespect towards the Singapore flag. The purpose of this page is only to make people more conscious of the mathematics around them.

There are two astronomical problems with the Singapore flag. Near the equator, a young (waxing) Moon will not look like the crescent on the flag. It will look like the crescent on the coat of arms! In the northern hemisphere, a young Moon will be a right crescent while an old (waning) Moon will be left crescent. In the southern hemisphere a young Moon will be a left crescent while an old Moon will be a right crescent. This is because in the northern hemisphere the Moon moves clockwise across the sky, while in the southern hemisphere it moves counterclockwise across the sky. So what happens near the equator? The Moon will move in a straight line when seen from above, or straight up and down along an east to west line when seen from the ground. So both the old and the young Moon will be a bottom crescent! In order to tell them apart, you have to think about whether the Moon is in the east or the west and what time it is. Notice that the crescent faces the Sun, so there can never be a top crescent.

If you look at a young Moon in Singapore, you will see that it is not always straight down. There are two reasons for this. The Moon is on the ecliptic, and when it crosses the horizon, the angle between it and the daily path of the Sun, which is a small circle parallel to the celestial equator, can be up to 23.5 degrees. In addition the latitude of the Moon can be up to five degrees. I will put up a more detailed explanation of this on the page for my course Heavenly Mathematics: Cultural Astronomy.

Together with Tey Meng Khoon and Frederick H. Willeboordse of CITA (Centre for Information Technology and Applications), I have developed interactive Java applets to explain What Does the Waxing or Waning Moon Look Like in Different Parts of the World?

And of course, you will never be able to see stars “inside” the Moon. They will be behind the Moon!

The stars are pentagrams and form a beautiful pentagon. Some Singaporean flags made outside of Singapore put the pentagon upside down! Such a flag was used at the award ceremony at the Asian Games in Busan, Korea in 2002! I once saw such a flag for sale in Singapore, but I didn't buy. If you see it, please get one for me!

A pentagram is a pentagonal star. If you like them, I have made some nice models of dodecahedra whose faces are pentagons or pentagrams. For more information about this, please see the page for my course Mathematics in Art and Architecture.

Notice also that the Moon and stars motif in the coat of arms is not just a rotation of the motif on the flag. Look carefully at the position of the stars.

You may want to compare with the Malaysian Flag. From an astronomical point of view, I like the Kelantan flag, but not the Johor flag.

The US flag has gone through a lot of changes in the course of its history. For a detailed description see the page at flagspot.net or the page on Historical U.S. Flags.

You may also enjoy Binary Numbers and the South Korean Flag by Jill Britton.

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

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