1. What is the Pappus Chain?B A

2. What is the Pappus Chain Theorem?
is the height from line AC to the centre of circle
is the diameter of circle C B A

3. The Investigation of the Pappus Chain Theorem through Inversion
Grp 8-16
Scott Loo Yi He 4S1(19)
Ng Kai Jie 4S1(13)
Sun Longxuan 4S1(21)
Tan Zhi Heng 4S2(22)

4. 3rd century A.D.
Pappus created the Pappus Chain Theorem

5. Literature review
Created the Pappus Chain Theorem using solely Euclidean geometry
Pappus of Alexandria
Pappus of Alexandria created the Pappus chain theorem, with his proof using solely Euclidean geometry
As a result his proof was very long winded

6. 3rd century A.D. 1820 Pappus created the Pappus Chain Theorem
Jakob Steiner invented the method of geometric inversion

7. Literature review
Invented the powerful method of geometric inversion, which is applicable to the Pappus Chain
Invented the powerful method of inversion, which makes many difficult problems in geometry much more tractable
Jakob Steiner

8. 3rd century A.D. 1820 2017 Pappus created the Pappus Chain Theorem
Jakob Steiner invented the method of geometric inversion
???

9. 2017
We will extend on the Pappus Chain theorem using geometric inversion

10. Inversion
In Number Theory…

11. Inversion
In Combinatorics…

12. Inversion
In Algebra… f f

13. Geometric Inversion In Geometry…
It is a geometric transformation of an entire plane by mapping every point on the plane to a new point T r O P’ P

14. Geometric Inversion
In the plane, the inverse of a point P with respect to a reference circle with center O and radius r is a point P', lying on the ray from O through P such that
This is called circle inversion or plane inversion

15. Fields of Math

16. Terminology

17. Term
Definition
Arbelos
A plane region bounded by three semicircles connected at the corners, all on the same side of a straight line (the baseline) that contains their diameters.
Pappus Chain
A chain of tangent circles, starting from a circle that is tangent to the 3 circles in an Arbelos, of which all are tangent to one of the two small interior circles and to the large exterior circle in an Arbelos.

18. Term
Definition
Pappus Chain Theorem
In a Pappus Chain, the height, hn, of the center of the nth inscribed circle, iCn, above the line segment AC is equal to n times the diameter of iCn.
Inversion
In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion. These transformations preserve angles and map generalized circles into generalized circles, where a generalized circle means either a circle or a line

19. Term
Definition
Steiner Chain
Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles, where n is finite and each circle in the chain is tangent to the previous and next circles in the chain.
Soddy’s Hexlet
Soddy's hexlet is a chain of six spheres, each of which is tangent to both of its neighbors and also to three mutually tangent given spheres.

20. Objectives

21. Objectives 1
To prove the Pappus Chain Theorem using inversion

22. Objective 2
To generalise the Pappus Chain Theorem to higher dimensions

23. Objective 3
To find out how Pappus Chain Theorem is analogous to Steiner Chain and Soddy’s Hexlet

24. Research Questions

25. Research Question 1
How do we prove the Pappus Chain Theorem using circle inversion?

26. Research Question 2
How can we generalise the Pappus Chain theorem to the 3rd dimension and even higher dimensions?

27. Research Question 3
How are our findings on Pappus Chain Theorem related to Steiner Chain and Soddy’s Hexlet?

28. Methodology Read up and learn more about geometric inversion
Prove Pappus Chain Theorem using inversion
Research about Steiner Chain and Soddy’s Hexlet

29. Timeline

30. References
1. Andrew Olson. (2015, December 5). Chain Reaction: Inversion and the Pappus Chain Theorem. Retrieved March 18, 2017 from
2. Davis, T. (2013, March 23). Inversion in a Circle. Retrieved March 23, 2017, from
3. Harold P. Boas. (2006, March). Reflections on the Arbelos. Retrieved March 18, 2017 from
4. Hunter, R. (2003). The Shoemaker's Knife Problem - An Application of Inversion. Retrieved March 18, 2017, from
5. K. (2009, January 9). Circle Inversions and Applications to Euclidean Geometry. Retrieved March 23, 2017, from
6. Lamoen, F. (n.d.). Pappus Chain. Retrieved March 18, 2017, from

31. Thank you!

32. Q&A