# Pappus' Sangaku

Few sangaku illustrate better the disparity of the independent development of mathematics during the Edo period (1603-1867) of self-imposed seclusion of Japan from the Western world than those related to arbelos and Pappus' chain of circles. The tool of inversion so useful in solving problems involving circles remained unkown to the Japanese until late 19 century.

Consider the sangaku 1.8.2 from the collection by Fukagawa and Pedoe:

Circles C_{1}(r) and C_{2}(r) touch at O in the circle O(2r) and also touch O(2r), C_{1}(r) touching at T. O_{1}(r_{1}) touches O(2r) internally and both C_{1}(r) and C_{2}(r) externally, O_{2}(r_{2}) touches O_{1}(r_{1}) and C_{1}(r) externally and O(2r) internally, and so on. Find r_{n}.

### References

H. Fukagawa, D. Pedoe,

*Japanese Temple Geometry Problems*, The Charles Babbage Research Center, Winnipeg, 1989Write to:

Charles Babbage Research Center

P.O. Box 272, St. Norbert Postal Station

Winnipeg, MB

Canada R3V 1L6

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Copyright © 1996-2018 Alexander Bogomolny

Circles C_{1}(r) and C_{2}(r) touch at O in the circle O(2r) and also touch O(2r), C_{1}(r) touching at T. O_{1}(r_{1}) touches O(2r) internally and both C_{1}(r) and C_{2}(r) externally, O_{2}(r_{2}) touches O_{1}(r_{1}) and C_{1}(r) externally and O(2r) internally, and so on. Find r_{n}.

With the help of inversion the solution is immediate. In the generalization of Archimedes' formula

r_{n} = rk / (k²n² + k + 1).

where k is the ratio between the radii of the two inner circles, i.e.

r_{n} = r / (n² + 2),

exactly the expression that appears in the Gothic Arc diagram.

In the very first volume (1826) of *Crelle's Journal*, J. Steiner proved several theorem concerning arbelos, the above being one of them. The sangaku however is dated 1801 (and was written in the Mie prefecture.)

## Sangaku

- Sangaku: Reflections on the Phenomenon
- Critique of My View and a Response
- 1 + 27 = 12 + 16 Sangaku
- 3-4-5 Triangle by a Kid
- 7 = 2 + 5 Sangaku
- A 49
^{th}Degree Challenge - A Geometric Mean Sangaku
- A Hard but Important Sangaku
- A Restored Sangaku Problem
- A Sangaku: Two Unrelated Circles
- A Sangaku by a Teen
- A Sangaku Follow-Up on an Archimedes' Lemma
- A Sangaku with an Egyptian Attachment
- A Sangaku with Many Circles and Some
- A Sushi Morsel
- An Old Japanese Theorem
- Archimedes Twins in the Edo Period
- Arithmetic Mean Sangaku
- Bottema Shatters Japan's Seclusion
- Chain of Circles on a Chord
- Circles and Semicircles in Rectangle
- Circles in a Circular Segment
- Circles Lined on the Legs of a Right Triangle
- Equal Incircles Theorem
- Equilateral Triangle, Straight Line and Tangent Circles
- Equilateral Triangles and Incircles in a Square
- Five Incircles in a Square
- Four Hinged Squares
- Four Incircles in Equilateral Triangle
- Gion Shrine Problem
- Harmonic Mean Sangaku
- Heron's Problem
- In the Wasan Spirit
- Incenters in Cyclic Quadrilateral
- Japanese Art and Mathematics
- Malfatti's Problem
- Maximal Properties of the Pythagorean Relation
- Neuberg Sangaku
- Out of Pentagon Sangaku
- Peacock Tail Sangaku
- Pentagon Proportions Sangaku
- Proportions in Square
- Pythagoras and Vecten Break Japan's Isolation
- Radius of a Circle by Paper Folding
- Review of Sacred Mathematics
- Sangaku à la V. Thebault
- Sangaku and The Egyptian Triangle
- Sangaku in a Square
- Sangaku Iterations, Is it Wasan?
- Sangaku with 8 Circles
- Sangaku with Angle between a Tangent and a Chord
- Sangaku with Quadratic Optimization
- Sangaku with Three Mixtilinear Circles
- Sangaku with Versines
- Sangakus with a Mixtilinear Circle
- Sequences of Touching Circles
- Square and Circle in a Gothic Cupola
- Steiner's Sangaku
- Tangent Circles and an Isosceles Triangle
- The Squinting Eyes Theorem
- Three Incircles In a Right Triangle
- Three Squares and Two Ellipses
- Three Tangent Circles Sangaku
- Triangles, Squares and Areas from Temple Geometry
- Two Arbelos, Two Chains
- Two Circles in an Angle
- Two Sangaku with Equal Incircles
- Another Sangaku in Square
- Sangaku via Peru
- FJG Capitan's Sangaku

- Arbelos - the Shoemaker's Knife
- 7 = 2 + 5 Sangaku
- Another Pair of Twins in Arbelos
- Archimedes' Quadruplets
- Archimedes' Twin Circles and a Brother
- Book of Lemmas: Proposition 5
- Book of Lemmas: Proposition 6
- Chain of Inscribed Circles
- Concurrency in Arbelos
- Concyclic Points in Arbelos
- Ellipse in Arbelos
- Gothic Arc
- Pappus Sangaku
- Rectangle in Arbelos
- Squares in Arbelos
- The Area of Arbelos
- Twin Segments in Arbelos
- Two Arbelos, Two Chains
- A Newly Born Pair of Siblings to Archimedes' Twins
- Concurrence in Arbelos
- Arbelos' Morsels

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Copyright © 1996-2018 Alexander Bogomolny

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