1. Constructing A Sundial

2. Content What is a sundial? Sundial history Types of sundial Experiment & results Math properties of our sundial Other application of sun angle

3. introduction Aim of our project: To investigate the theories and principles that makes a sundial work, and thus to construct a functioning sundial in NUS High School campus.

4. Introduction What is a sundial? “The earliest type of timekeeping device, which indicates the time of day by the position of the shadow of some object exposed to the Sun's rays. As the day progresses, the Sun moves across the sky, causing the shadow of the object to move and indicating the passage of time.” – Encyclopædia Britannica

5. Introduction Components of a sundial: –Dial face –Gnomon dial –Dial calibration

6. History of sundial development Rudiment: 5000-3500 BC First known sundial with calibration: 800 BC, Egypt 250 BC onwards: more complex sundials were built by the Greeks –correct for season change –Portable The geometry knowledge was firstly applied in constructing these sundials.

7. History of sundial 100 AD: people found out that a slanting gnomon is more precise than a vertical gnomon 150 AD: trigonometry was introduced by the Greek. –Trigonometry is much easier than geometry. Sundial’s fate after mechanical clock

8. Types of sundial Equatorial sundial Horizontal sundial Vertical sundial

9. Equatorial sundials Gnomon in the center of the plate Parallel to the Earth's axis and points to the north celestial pole

10. Horizontal sundial Flat horizontal dial plate with hour lines O towards south and A towards north The shadow of gnomon placing on the hour lines indicates the time

11. Mathematics behind OP is pointing to the pole, P. PNS is the meridian, NPT is the hour angle TON is the shadow angle.

12. Mathematics behind cos NP cos PNT = sin NP cot TON - sin PNT cot NPT

13. Mathematics behind cos NP cos PNT = sin NP cot TON - sin PNT cot NPT In which, PNT = 90º, NP = Ø and TON = Since cos PNT = 0, sin PNT = 1, therefore 0 = sin Ø cot - cot (HA) tan = sin Ø tan (HA) Where, is the Shadow Angle for a given time. Ø is the Latitude of the sundial.

14. Mathematics behind tan = sin Ø tan (HA) HA is the hour required. –multiplied by 15 –24 hour solar day –Sun appears to go once around the Earth (360°). –This means that 360° is equivalent to 24 hours –making 15° equivalent to one hour (360 / 24 = 15).

15. Example The calculated shadow angles is at latitude 51ºN.

16. Types of sundial Vertical sundials can be divided into five groups due to the direction they face. 1.Vertical direct north sundials - early morning and late evening hour 2.Vertical direct south sundials - greater duration of time 3.Vertical direct east sundials - the morning hours 4.Vertical direct west sundials - the afternoon hours 5.Vertical declining sundials - Southwest decliners, Southeast decliners, Northwest decliners and Northeast decliners

17. Vertical direct south sundials - greater duration of time - hour lines run anti-clockwise

18. Vertical direct east sundials Greater duration of time Dial plate lies in the meridian Gnomon is parallel to the dial plate, thus parallel to the Earth's axis

19. Vertical direct west sundials Only the afternoon hours Can be used on any latitude Dial plate lies in the meridian Gnomon is parallel to the dial plate, thus parallel to the Earth's axis.

20. Experiment Aim: To construct a functional sundial locating in NUS High School campus. Mathematical model used:

21. Underlying assumptions Since Singapore is near to the earth equator(1 °22 ' N, 103° 48' ) if we put the gnomon parallel to the earth axis, approximately the path of the sun relative to the gnomon will be a semi-circle in the equatorial plane Assuming the angular velocity of the sun relative to a point on earth is constant

22. Model

23. Data collected Clock timeTime / minLength of shadow / cmtanθθ / ° 0745465173.00.1619.129 0800480127.50.21812.300 0815495100.10.27815.512 083051081.10.34318.921 084552566.80.41622.596 091555548.50.57329.821 093057045.30.61431.537 094558543.50.63932.582 101561530.30.91742.536 103063025.91.07347.026 104564523.11.20350.276 110066020.81.33753.196 111567518.61.49556.215 114570513.82.01463.600 120072011.72.37667.176 12457655.35.24579.206 13007803.28.68883.434 13157950.0Undefined90.000

24. Analysis After some analyze of the data, we had observed there is a relationship between time and θ. By plotting time vs. θ graph, we get the following result:

25. (7:30a.m.)

26. Trend Linear Exponential Introducing regression line Used to depict the relationship between the independent variable and the dependent variable

27. Regression line To determine the best fitting line, we need to calculate the correlation coefficient and coefficient for each line. where y is the value predicted by the graph

30. Discussion Inaccurate measure of shadow length Weather condition The position of the shadow of the gnomon has a displacement of zero from the origin at 1:15p.m. Interpolation and extrapolation

31. Conclusion Hypothesis is valid Further application that we can explore

32. Q & A Any questions? If not….

33. The End Thank you for your attention