The South Pointing Chariot

Xin Yan Loh

Replica as displayed in the Science Museum in London, By Andy Dingley - Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=4944626

Introduction The south pointing chariot, invented in ancient China, is a chariot with a pointer mounted on the top; utilising differential gears, the pointer continues to point towards the same direction regardless of how the chariot moves, and thus could serve as a navigational device without using the magnetic compass. The cultural and historical aspects, teaching activities based on the chariot, and the rationale and links to the Australian Mathematics Curriculum are discussed below. Cultural and Historical Aspects Legend has it that the south pointing chariot was invented by the Yellow Emperor (Huang Di, 黄帝), more than 4000 years ago, around 2650 BCE3,4,5. During that time, there were three major rulers in China- the Yellow Emperor, Yan Di (炎帝), and Chi You (蚩尤). Chi You was a tyrannical ruler, and the people under his rule were suffering. In order to liberate those people, Yan Di decided to do battle with Chi You, but suffered a decisive lost. Yan Di was forced to retreat, and went to the Yellow Emperor for help. Seeing all the suffering caused by Chi You, the Yellow Emperor agreed to assist Yan Di in overthrowing Chi You, and their forces clashed at the Battle of Zhuolu (涿鹿之战). During this battle, Chi You created a magical fog to confound and misdirect the Yellow Emperor’s army. The fog blocked out the skies and stars and reduced visibility to less than a mile, crippling the Yellow Emperor’s army’s ability to navigate. In a stroke of brilliance, the Yellow Emperor invented the south pointing chariot, a cart with a wooden figure mounted on top that would always point in the same direction regardless of how the cart was moved. Using this new invention, the Yellow Emperor successfully navigated the magical fog, and went on to defeat Chi You and unite all of China. Like most things based in ancient Chinese history, the narrative above has heavy elements of legend and myth (such as the magical fog, and the seemingly instant invention of the chariot), thus is unlikely to be an accurate account of how the south pointing chariot was invented. However, it is plausible that the chariot was invented such a long time ago- to construct the chariot, the inventor would need access to simple machines such as the wheel and axel, and gears. Whilst there is debate on when the wheel and axel and gears were first invented, some of the earliest inventions date to more than 4000 BCE2. Thus it is entirely possible for the chariot to have been invented around 2650 BCE, as the components needed to construct the chariot were accessible. However, whether or not the Yellow Emperor was the inventor would be hard to ascertain, as it was (and still is) customary in China to attribute marvellous inventions to a legendary figure in order to add weight to the invention. Historical records show that a functioning replica of the south pointing chariot was made as early as 250 CE5. Ma Jun (马钧), a politician and engineer at the time, constructed a south pointing chariot using differential gears to eliminate doubt from his contemporaries that such a device could not exist. Thus even though the south pointing chariot may not have been invented by the Yellow Emperor 4000 years ago, it was still invented really early in China’s history, and pre-dates the first use of differential gears by ‘Western’ engineers (in 1720 CE) by more than 1000 years. 5 Most myths contain an element of truth- it could be possible that the south pointing chariot was invented during the Yellow Emperor’s time to serve a particular navigational need during war, and this was later mythicized and turned into the story of the magical fog of Chi You. Regardless of whether or not the narrative of the Yellow Emperor inventing the chariot is true, the story can still serve as a great hook for students to start investigating the variety of mathematics associated with the chariot. Students can be shown how to construct a replica of the chariot out of Lego. Here is a link to a video that shows what a Lego chariot looks like, as well as the essential parts required to construct them https://www.youtube.com/watch?v=s27xyrTOcGg Alternatively, parts used to construct the chariot can be 3D printed, using the files found on this website https://pinshape.com/items/33086-3d-printed-south-pointing-chariot Video of the 3D printed chariot can be seen here: https://www.youtube.com/watch?v=YzUKr621gr4&feature=youtu.be

The following mathematical concepts can be explored by students during the construction of the chariot: • Ratios- by looking at the relationship between the number of teeth and the number of revolutions made in interconnected gears • Circumference- by getting students to measure the length travelled by one revolution of a wheel and comparing it to the diameter of the wheel. This can be done using many different sized wheels from the Lego set to discover the formula for circumference • Applications of the circumference formula and algebra- because the south pointing chariot is a purely mechanical device, if there are small differences between certain components, these small differences would lead to errors that accumulate over time. For example, for the chariot to function, it is assumed that both the wheels have the exact same diameter, resulting in both of them travelling the exact same distance when travelling in a straight line. However, this is almost impossible to achieve in practice, and there would be slight differences between the diameters of the wheels. This would mean that over a long distance (say 100 times the difference between the diameters), the wheels would not have made the same number of revolutions, resulting in the pointer no longer pointing in the same direction. This can be a problem that students can investigate, and they would need to apply the circumference formula and some algebra in their investigation. These activities are detailed in the teaching resource below. The explicit links to the Western Australian Curriculum are also shown in the rationale at the end of the document.

Teaching Resource Build a South Pointing Chariot Use the legend of the Yellow Emperor as a hook to introduce the south pointing chariot. Show students a constructed model of a south pointing chariot out of Lego (example can be seen at https://www.youtube.com/watch?v=s27xyrTOcGg). Suitable Lego kits include the Simple and Powered Machine set (https://education.lego.com/en-au/product/machines- and-mechanisms) and also the Mindstorm EV3 sets (https://education.lego.com/en- au/product/mindstorms-ev3). Ensure that differential gears are included in the set (such a piece is shown in the Youtube video above), otherwise students would need to build their own differential gear, which is extremely difficult (see Figure 1 for my attempt at building one that barely worked and was prone to breaking).

Gear C

Differential Gear Box

Gear B

Gear A

Figure 1- Self-made Differential Gear Box (that doesn’t quite work)

Based on the Lego set available, develop a set of instructions that gives a step-by-step guide on how to build the chariot. For the chariot to function, the gearing ratio from the wheel to the pointer must be 1:1, and the distance between the wheels must be the same as the wheel diameter (see Figure 2). Alternatively, the gearing ratio must be 1:2 if the distance between the wheels is twice their diameter etc.

Figure 2- Ideal proportions of the Chariot- Not achieved in this model due to lack of materials

Alternatively, parts to build the chariot can be 3D printing, using the files found at the following link: https://pinshape.com/items/33086-3d-printed-south-pointing-chariot. Video of the 3D printed chariot can be seen here: https://www.youtube.com/watch?v=YzUKr621gr4&feature=youtu.be

Figure 3- 3D printed chariot, source: https://pinshape.com/items/33086-3d-printed-south-pointing-chariot As part of the building activity, students can explore ratios by answering a series of questions such as: 1. How many teeth does Gear A have? (see Figure 1) 2. How many teeth does Gear B have? 3. If Gear A turns one revolution, how many turns does Gear B turn? 4. If Gear B turns one revolution, how many turns does Gear A turn? Students can explore those answers enactively by physically turning the gears and counting the number of revolutions, and ratios can be introduced later by comparing the number of teeth between the gears and the number of revolutions made when one of the gears complete one revolution. Students can also be extended by calculating how many revolutions a gear at the end of a chain of multiple gears would turn if the starting gear turns one revolution (e.g. if Gear A turns one revolution, how many revolutions would Gear C turn). Once students have a grasp of ratios, further problem solving questions could include: 1. If Gear A is replaced with another gear with double the amount of teeth, how many revolutions would Gear B turn if Gear A turns once? 2. If Gear B is replaced with another gear with triple the amount of teeth, how many revolutions would Gear B turn if Gear A turns once? Students can be scaffolded with questions such as: If Gear A has more teeth, do you think Gear B will turn more or less if Gear A is turned once? Students can then verify their answers by physically replacing the gears and observing the results. Investigating Circumference Continuing with the theme of looking at ratios, students can now look at the ratio between the circumference of a wheel and its diameter. This can be done by marking a point on the wheel, and measuring the horizontal distance travelled when the marked point travels one revolution. Because the chariot is made of Lego, students can easily substitute the wheels with wheels of different diameters, and perform the experiment again. Once students have a set of results using different sized wheels, they can then divide the distance travelled by the diameter, and write down what they notice. Students can then be introduced to the formula for circumference, replacing the data they were working with using symbolic representations, � = ��. Typical scientific investigation processes such as doing three measurements using the same wheel and taking the average can also be introduced as part of this process (cross-curriculum link to science). Students could also plot their results for difference sized wheels on a graph and draw a line of best fit, similar to what they would do in a science experiment, and determine pi empirically by measuring the gradient of the line.

Applying the Circumference Formula to Investigate the Practicality of the South Pointing Chariot Once students have explored the concepts of ratios and the formula for circumference, they can use that knowledge to investigate the practicality of the south pointing chariot. Because the south pointing chariot uses differential gears, if there was a slight difference between the diameters of the two wheels, that difference would cause an accumulation of small errors, resulting in the pointer not pointing in the same direction that it started with after it has travelled some distance. Considering that it is almost impossible to make two wheels exactly the same, this would have been a definite problem when using the chariot for navigation. Students can investigate this error by running their chariot over a long distance. This activity is most suitable with the Mindstorm sets, as it includes motorised components that can be used to drive the chariot. The procedure for this activity is as follow: 1. Set the pointer to point south using a magnetic compass. 2. Drive the chariot over a long distance (say 100m) at the school oval, courtyard, or other suitable locations. 3. While the chariot is travelling, mark down how many revolutions the wheels have turned. This can be done by marking a point on the wheel, and tallying how many times that point ‘touches the ground’ 4. At the end of the straight path, compare the direction of the pointer with a magnetic compass again. Note down the discrepancy in degrees, measured using a protractor or counting how many teeth the gear has turned 5. Recreate the discrepancy by lifting the chariot up and only spinning one of the wheels to get the desired change in pointer direction. 6. Note down how many revolutions it took to get the desired discrepancy, as well as the direction of the wheel (same or opposite direction to the direction of travelling) 7. Repeat steps 1-6 multiple times (three times to link in with experimental procedures typically used in science) and take the average 8. Scaffold students through the calculation below to allow them to calculate the difference in diameter between the wheels Let: n = number of revolutions made by wheel 1 a = additional revolutions on wheel 2 required to achieve the error, +ve for same direction, -ve for opposite direction D = distance travelled r1 = radius of wheel 1 r2 = radius of wheel 2

For wheel 1, � = 2��!. �, thus � � = ! 2�� Similarly, for wheel 2, � = 2��!. � + � , thus � � = ! 2� � + � Substituting the results obtained from the experiments, students should be able to calculate the radius of both wheels, and compare the difference. If the difference is too small to make a difference over a hundred metres, the difference can be artificially enlarged by wrapping a few layers of sticky tape over one of the wheels. This method could also be used to reduce the distance travelled, allowing students without motorised sets of Lego to complete the experiment. The following questions can then be used to help students evaluate the practicality of the south pointing chariot: 1. If your chariot with a wheel diameter of 5cm achieved this error by travelling 100m, how far would a life-sized chariot have to travel to achieve the same error? (This question might need some scaffolding in applying ratios to work out scale- result should be something along the lines of chariot will lose more than 10o of accuracy after travelling a few kilometres, which only makes it useful for navigating short distances) 2. Is it practical to use the south pointing chariot to keep direction? 3. Based on what you found, do you think the Yellow Emperor could have used a south pointing chariot to lead his army out of the fog? 4. Could the south pointing chariot still be useful? (Yes, over short distances) 5. How could you improve the design of the chariot? 6. The gear arrangement we used in the south pointing chariot is called ‘differential gears’. Are there other applications of differential gears besides using it to keep direction? Do some research to find out.

Rationale The use of physical, hands on activities to teach mathematics is supported by Bruner’s notion of teaching and learning enactively, before transitioning to iconic and then symbolic representations1. The use of different sized gears required to construct the chariot can be used to explore the concepts of ratios (how many turns will this gear make when you turn the other gear once?), and links to the following Western Australian Curriculum Outcomes: ACMNA156- recognise and solve problems involving simple ratios (Year 7), and ACMNA188- solve a range of problems involving rates and ratios, with and without digital technologies (Year 8). As a follow on, students could continue to explore ratios, this time investigating the ratio between the distance travelled by one rotation of the wheel and the diameter of the wheel. By changing the size of the wheel on the chariot (different Lego pieces) and comparing the distance travelled with the diameter of the wheel, students can explore the relationship between the circumference and diameter, and this is linked to ACMMG197- Investigate the relationship between features of circles such as circumference, area, radius, and diameter (Year 8). By establishing the formula for the circumference, the irrational number π is introduced, which satisfies ACMNA186- investigate the concept of irrational numbers, including π (Year 8). Once students have established the formula for circumference, they can apply this formula to investigate the accuracy of the south pointing chariot. When there are slight differences in the wheel diameter resulting in slightly different distances travelled between the two wheels, small errors will accumulate resulting in the chariot no longer pointing in the same direction that it started with after some distance travelled. By measuring the error and using the formula for the circumference of a circle, students can calculate the difference between the diameters of the wheels (see Teaching Resource section for details). This links to the ratio outcomes listed above (ACMNA156, ACMNA188), and the second part of ACMMG197 use formulas to solve problems involving circumference and area. Additionally, outcomes in algebra such as ACMNA193 plot linear relationships on the Cartesian plane (Year 8), and ACMNA194 solve linear equations using algebraic and graphical techniques (Year 8), are also satisfied throughout the investigation process. Finally, by comparing the Lego model to a life size model, the practicality of the south pointing chariot can be investigated, and this links with ACMMG221 solve problems using ratio and scale factors in similar figures (Year 9). In addition to the mathematics outcomes, the historical and cultural aspects of the south pointing chariot could be used to engage and empower students. The legend of the Yellow Emperor and the use of a non-magnetic compass could serve as a hook for students, and facts such as the early application of differential gears by the Chinese can also be empowering to students from that culture, and serves to show the multicultural aspects in the development of mathematics and science throughout history. Combined with the amount mathematics involved, using the south pointing chariot as a theme becomes an engaging way to teach multiple mathematical concepts to Year 7 and 8 students.

Links to Western Australia Curriculum Activity Curriculum Links • ACMNA156- recognise and solve problems involving simple ratios (Year 7) Exploring ratios in gears • ACMNA188- solve a range of problems involving rates and ratios, with and without digital technologies (Year 8) • ACMMG197- Investigate the relationship between features of circles such as Exploring circumference of a circle circumference, area, radius, and diameter (Year 8) • ACMNA186- investigate the concept of irrational numbers, including pi (Year 8) • ACMMG197- Use formulas to solve Investigating the practicalities of the chariot, by: problems involving circumference and area • Applying the formula for circumference (Year 8) ACMNA193- plot linear relationships on • Using algebraic techniques such as • solving equations and plotting data on the Cartesian plane (Year 8)

graphs • ACMMG221- solve problems using ratio and scale factors in similar figures (Year 9) Table 1- Curriculum Links to the activities based on the south pointing chariot References 1. Bruner, J. (2006). The perfectibility of intellect. In J. Bruner (Ed.), In search of Pedagogy. Volume I: The selected works of Jerome Bruner, 1957-1978 (pp. 105–114). Florence: Taylor and Francis. 2. Bulliet, R. (2016). The wheel : Inventions and reinventions. New York: Columbia University Press. 3. Linet, B. (2009). The south-pointing chariot on a surface. Ithaca: Cornell University Press. 4. Rossi, C. (2016). Ancient engineers’ inventions : Precursors of the present (2nd ed.). Cham: Springer. 5. Sawin, S. (2015). South Pointing Chariot: An Invitation to Differential Geometry. Ithaca: Cornell University Press.

Recommended Further Reading http://www.3dprinterclocks.com/page20.html https://en.wikipedia.org/wiki/South-pointing_chariot https://en.wikipedia.org/wiki/Yellow_Emperor#Battles