Stellar navigation and mathematics
Aboriginal and Torres Strait Islander people also developed techniques to navigate using the stars in a variety of ways, using mathematical principles.
The Indigenous Polynesian and Micronesian people of the Pacific are well known for their excellent navigational knowledge. Aboriginal and Torres Strait Islander people also developed techniques to navigate using the stars in a variety of ways, having their own names for the cardinal points (North, South, East, and West) in different languages.1 For example, cardinal directions in the Wardaman language of the Northern territory are Jongon (North), Gorro (South), Yiyanggu (East) and Namanya (West).2 In the Torres Strait, people use the stars in the Western constellation Ursa Major to find North.3 In the Meriam Mir language of the eastern Torres Strait, this constellation is called Beizam - the celestial shark. The gills of the shark point northward. In a similar manner, Aboriginal and Torres Strait Islander peoples across Australia use the Southern Cross to find South, and the Sun to find East and West.
Many Aboriginal groups constructed stone arrangements that mark these positions. These sites often contain stone circles or lines that mark the cardinal points4, often pointing to North/South or East/West.5 It is important to note that while this unit describes concepts of cardinal directions, many Aboriginal concepts of direction can be quite different to those of non-Aboriginal people.
The Sky Dome
Imagine the sky is a dome above our heads (see Fig. 1), a view often described in Aboriginal and Torres Strait traditions. We can measure the positions and movements of celestial objects along this dome. For an observer standing on the Earth, the angle along the horizon is called the azimuth. It is measured in degrees and moves in a complete circle around us: North (0°), East (90°), South (180°), West (270°). The angle from the horizon up in the sky or down below the ground is the altitude. It is also measured in degrees. On the horizon, the altitude is 0°, going up to 90° at zenith - the point directly above us.
An observer can measure the position of a star using these two angles: the azimuth along the horizon, and the altitude in the sky. Stars always rise and set at a given azimuth, but their altitude changes as they move across the sky. Some stars rise Due East and set at Due West. Other stars (especially those in the far north) appear to rise very low in the sky and set soon after. Some stars in the far southern skies never rise or set – they are always in the sky, doing loops around the South Celestial Pole. These stars are circumpolar. The line connecting North and South, running along the sky across the zenith, is called the meridian. When a star crosses the meridian, that is its highest altitude.
Fig. 1: (Left) The Celestial Dome. Image: DateAndTime.com.6
The Earth rotates on its axis at an angle of 23.5°. If you draw a line from the poles into space, you would notice all the stars moving in a circle around those points, called the Celestial Poles. The star Polaris lies very close to the North Celestial Pole, which makes it useful for navigation (thus its popular name as the “North Star”). However, as there is no single bright star near the South Celestial Pole, so we must use other techniques to find it. This will form the basis of the student activities.
Beizam, the Shark (Torres Strait)
Torres Strait Islanders use the stars of Baidam/Beizam (aka “the Big Dipper” in Ursa Major) to find North. The gills of the shark are the stars known to Western astronomers as Dubhe (Alpha Ursa Major) and Merak (Beta Ursa Major). They point directly northward. Even though the North Celestial Pole is below the horizon (as is Polaris, the North Star), you can use these two stars to estimate Due North. When the stars are directly perpendicular to the horizon (meaning they are straight up and down) they are pointing down towards Due North. The angular distance from Dubhe to the North Celestial Pole is ~30°. If the stars are not pointing directly down, you can measure this distance, then move straight up to the horizon to find Due North (Fig. 2).
Fig. 2: Using the stars of Beizam/Big Dipper to find North. Image: Stellarium, modified by D.W. Hamacher.
Classroom activity - Mathematics Year 5
In these classroom activities, students will learn the basics of celestial navigation by measuring the positions of stars in the sky in degrees, using their hands as a protractor. They will also learn how to find cardinal directions (North, South, East, and West) and their latitude on Earth using the angular measurements of stars. They will then test their measurements using the Stellarium astronomy app7.
This resource addresses the following content descriptions from the Australian Curriculum:
- Estimate, measure and compare angles using degrees. Construct angles using a protractor (ACMMG112)
- Use a grid reference system to describe locations. Describe routes using landmarks and directional language (ACMMG113)
- Pose questions and collect categorical or numerical data by observation or survey (ACMSP118)
This resource addresses the following excerpts of from the achievement standard for Year 5 in Mathematics:
- solve simple problems involving the four operations using a range of strategies
- check the reasonableness of answers using estimation and rounding
- use a grid reference system to locate landmarks
- measure and construct different angles
Inquiry-based learning questions
- How can we use our hands and fingers as protractors to measure angles?
- How can we navigate at night using the stars and mathematics?
- How are mathematical principles utilised by Aboriginal and Torres Strait Islander people?
- How can we apply Indigenous Knowledge about cardinal points to find our way across the landscape?
Activity 1 - Measuring angles with your hands and fingers
Suggested timing for activity: 10-15 minutes
Required resources: (TV or computer is optional)
In this activity, students will learn how to use their hands and fingers as protractors to measure angular distances between objects in degrees. The first in-class activity will provide students with the tools they need to conduct their own experiment outside.
- Teach the students how to use their hands and fingers to measure angular distances in degrees. By holding your hand out at arm’s length, the width of your little finger is about 1°. For reference, the diameter of the Sun and Moon are about half a degree. Three fingers is about 5°, a fist is 10°, and so on using the following image:
If students have seen the Disney film Moana, the teacher can discuss how Moana used a similar technique in the film.8
- Students should develop their own mnemonic (memory aide) for remembering which hand measures what angles.
- Students then practice measuring angular distances (height, width, etc) in the classroom. They can use combinations of their fingers to measure the height or width of a door, window, chalkboard, building, etc. Students nearer the object being measured will measure larger values than students further away. This will enable them to estimate, measure and compare angles using degrees, using their hands as a protractor.
This introduces students to estimating, measuring, and comparing angles in units of degrees, using their hands and fingers as a protractor (ACMMG112).
Activity 2 - Finding the South Celestial Pole using the Southern Cross
Suggested timing for activity: 30-45 minutes
Required resources: Computer with the Stellarium9 program uploaded
Aboriginal and Torres Strait Islander people used a variety of techniques to find cardinal points. In this activity, students will learn how to find the South Celestial Pole using the Southern Cross, as Aboriginal people did in the past. It will require teachers to familiarise themselves with the Stellarium astronomy software program beforehand.
- Students will open the Stellarium program, setting the location to your town or city, and the time to after sunset when it’s dark.
- Ask students to find the Southern Cross constellation (there is a search function if they have difficulty).
- Open the “Sky and Viewing Options Window” and click “Markings”. Tick off “Celestial Poles (of date)”. This will show the South Celestial Pole (SCP).
- Students can measure the angular distance from Acrux (the brightest star of the Southern Cross) to the SCP by clicking the “angle” button. They then click on the star Acrux and drag the mouse over to the SCP.
- What is the distance? What combination of hands/fingers would you need to use to measure that distance? Is there anything in the classroom that measures that angle? (The distance between Acrux and the South Celestial Pole is ~27°. Students should note that the two stars do not perfectly line up to the SCP. It is an approximation. Have them figure out the best way to accomplish the most accurate measurement.)
- Once a student finds the SCP, they can see that they simply need to move straight down to the ground to find Due South.
This further develops students’ ability to estimate, measure, and compare angles using degrees, using the software-app as a protractor (ACMMG112). Students can increase the time to show that all the stars rotate around the CSP.
Activity 3 - Estimating latitude by measuring the altitude of the SCP
Suggested timing for activity: student’s own time
Required resources: Notebook and pen.
Students should use the tools they gained in class to find the SCP on their own at night. Aboriginal people travelled great distances across Australia for trade and ceremony. They would have noticed that the altitudes of constellations shifted as they moved to/from North/South.
- Ask students to see if they can measure the SCP. What is the orientation of the Southern Cross when they did this? It is rightside up, or upside down, or tilted at an angle?
- The student should then be asked to measure the altitude of the SCP and record it in their book. Ask them to take multiple measurements over the course of a week. Their measurements might be slightly different each time. They can use their recorded values to find their average measurement.
- Students will then learn that the altitude of the SCP is the same as their latitude. The further South you are, the higher in the sky (greater the altitude) the SCP will be. Students can check their estimates by looking up the latitude of where they live.
Using their average measurement and testing it with their actual latitude will help them understand collecting numerical data by observation (ACMSP118). The activity also further develops students’ ability to estimate, measure, and compare angles using degrees (ACMMG112)
Activity 4 - Using the Sun to Find Cardinal Directions
Suggested timing for activity: 30 mins to 1 hour
Required resources: A stick/stake.
Many Aboriginal stone arrangements are oriented to either North/South or East/West. One simple way to measure this is to drive a stick or stake into the ground on a sunny day.
- Place a stick vertically in the ground, and mark the tip of the stick’s shadow at two different points in time (say, a half hour to an hour apart). In the Southern Hemisphere, the sun’s shadow will move from left (Position 1) to right (Position 2) in Figure 4. Draw a line connecting these points. This line points to East and West. Draw a line 90° perpendicular to this, which gives the North/South line (Fig. 4).
Fig. 4: Finding Cardinal Directions using a stick and the Sun. Image: Darren Edwards.
- This will function as a grid, enabling students to measure North/South and East/West. They can see what landmarks are found in those directions and discuss how classmates are oriented relative to themselves. In Warlpiri traditions (Central Desert, NT) they refer to people and objects in terms of cardinal points. So rather than saying “Jack is sitting to the left of me”, they would say “Jack is sitting East of me” (if that’s the orientation).
This enables students to use a grid reference system to describe locations and routes using landmarks and directional language (ACMMG113) and further develops their ability to estimate, measure, and compare angles (ACMMG112).
1 Norris, R.P.; Harney, B.Y. (2014). Songlines and navigation in Wardaman and other Australian Aboriginal cultures. Journal of Astronomical History and Heritage, 17(2), 141–148.
2 Cairns, H. and Harney, B.Y. (2003). Dark Sparklers. Merimbula, NSW: H.C. Cairns.
3 Hamacher, D.W. (2013) A shark in the stars: astronomy and culture in the Torres Strait. The Conversation, 10 July 2013. https://theconversation.com/a-shark-in-the-stars-astronomy-and-culture-in-the-torres-strait-15850
4 Norris, R.P.; Norris, P.; Hamacher, D.W.; and Abrahams, T.R. (2013) Wurdi Youang: an Australian Aboriginal stone arrangement with possible solar indications. Rock Art Research, 30(1), 55-65.
5 Hamacher, D.W.; Norris, R.P.; Fuller, R.S.; (2012) Orientations of Linear Stone Arrangements in New South Wales. Australian Archaeology, 75, 46-54.
8 Hamacher, D.W. (2017) How far they’ll go: Moana shows the power of Polynesian celestial navigation. The Conversation, 15/2/2017. Retrieved from: https://theconversation.com/how-far-theyll-go-moana-shows-the-power-of-polynesian-celestial-navigation-72375