A Sushi Morsel

Twice in his books Ross Honsberger discussed the following problem:

A circle C0 of radius R0 = 1 km is tangent to a line L at Z. A circle C1 of radius 1 mm is drawn tangent to C0 and L, on the right-hand side of C0. A family of circles Ci is constructed outwardly to the right side so that each Ci is tangent to C0, L and the previous circle Ci-1. Eventually the members become so big that it is impossible to enlarge the family further. How many circles can be drawn before this happens?

The first solution in [Gems] is rather a brute force use of coordinate geometry. The second solution [Morsels], which Honsberger attributes to C. Stanley Ogilvy, is based on the observation that, if L is seen as the circle of infinite radius, the problem is reduced to the study of the chain of circles in arbelos or as a particular case of Steiner's chains. The problem is then elegantly reduced to the sequence of circles in a strip of paper by inversion in the circle centered at the point of tangency of C0 with L and of radius 2R0.

The earlier discussion comes with no reference; in the later one, Honsberger points to the book Mathematical Games and Pastimes by A. P. Domoryad (Pergamon Press, 1964, problem 19, p. 242).

In a recent article published in The Mathematical Gazette, M. Hajja gives an elementary solution to the problem that makes use of a known result concerning three mutually tangent circles standing on a straight line:

Assuming the radii of the three circles in the diagram are successively R0, R1, R2 we have

(1) 1/R2 = 1/R1 - 1/R0.

By induction, then

(2) 1/Rn = 1/R1 - (n - 1)/R0.

Denoting 1/Rn = un, we convert (2) to a simple linear relation

(3) un = u1 - (n-1)u0, n > 1.

Since, for all n, un > 0, (3) shows that the process may continue as long as u1 - (n-1)u0 > 0, o, in other words, while

(4) n < 1 + u1 / u0, i.e, n ≤ u1 / u0

For R0 = 106mm and R1 = 1 mm, u1 / u0 = 103. Which means that it is impossible to construct circle C1001 and C1000 is the last one, thus solving the problem. (The solution from [Morsels] shows also that R1000 = R0, giving another reason why the process can't continue beyond n = 1000.)

The point of this note is to observe that the problem and the simple solution above both have a long history. Relation (1) is the subject of an 18th century sangaku. Relation (2) appeared in its generality in a 1789 sangaku and in several modifications during the 19th century. This may be one of those cases where Wasan - the traditional Japanese geometry - beat the West to a punch.

References

  1. M. Hajja, On a Morsel of Ross Honsberger, The Math. Gazette, v. 93, n. 527, July 2009, pp. 309-312
  2. R. Honsberger, Mathematical Gems, MAA, 1973, pp. 52-53, 153-155
  3. R. Honsberger, Mathematical Morsels, MAA, 1978, pp. 218-219

Sangaku

  1. Sangaku: Reflections on the Phenomenon
  2. Critique of My View and a Response
  3. 1 + 27 = 12 + 16 Sangaku
  4. 3-4-5 Triangle by a Kid
  5. 7 = 2 + 5 Sangaku
  6. A 49th Degree Challenge
  7. A Geometric Mean Sangaku
  8. A Hard but Important Sangaku
  9. A Restored Sangaku Problem
  10. A Sangaku: Two Unrelated Circles
  11. A Sangaku by a Teen
  12. A Sangaku Follow-Up on an Archimedes' Lemma
  13. A Sangaku with an Egyptian Attachment
  14. A Sangaku with Many Circles and Some
  15. A Sushi Morsel
  16. An Old Japanese Theorem
  17. Archimedes Twins in the Edo Period
  18. Arithmetic Mean Sangaku
  19. Bottema Shatters Japan's Seclusion
  20. Chain of Circles on a Chord
  21. Circles and Semicircles in Rectangle
  22. Circles in a Circular Segment
  23. Circles Lined on the Legs of a Right Triangle
  24. Equal Incircles Theorem
  25. Equilateral Triangle, Straight Line and Tangent Circles
  26. Equilateral Triangles and Incircles in a Square
  27. Five Incircles in a Square
  28. Four Hinged Squares
  29. Four Incircles in Equilateral Triangle
  30. Gion Shrine Problem
  31. Harmonic Mean Sangaku
  32. Heron's Problem
  33. In the Wasan Spirit
  34. Incenters in Cyclic Quadrilateral
  35. Japanese Art and Mathematics
  36. Malfatti's Problem
  37. Maximal Properties of the Pythagorean Relation
  38. Neuberg Sangaku
  39. Out of Pentagon Sangaku
  40. Peacock Tail Sangaku
  41. Pentagon Proportions Sangaku
  42. Proportions in Square
  43. Pythagoras and Vecten Break Japan's Isolation
  44. Radius of a Circle by Paper Folding
  45. Review of Sacred Mathematics
  46. Sangaku à la V. Thebault
  47. Sangaku and The Egyptian Triangle
  48. Sangaku in a Square
  49. Sangaku Iterations, Is it Wasan?
  50. Sangaku with 8 Circles
  51. Sangaku with Angle between a Tangent and a Chord
  52. Sangaku with Quadratic Optimization
  53. Sangaku with Three Mixtilinear Circles
  54. Sangaku with Versines
  55. Sangakus with a Mixtilinear Circle
  56. Sequences of Touching Circles
  57. Square and Circle in a Gothic Cupola
  58. Steiner's Sangaku
  59. Tangent Circles and an Isosceles Triangle
  60. The Squinting Eyes Theorem
  61. Three Incircles In a Right Triangle
  62. Three Squares and Two Ellipses
  63. Three Tangent Circles Sangaku
  64. Triangles, Squares and Areas from Temple Geometry
  65. Two Arbelos, Two Chains
  66. Two Circles in an Angle
  67. Two Sangaku with Equal Incircles
  68. Another Sangaku in Square
  69. Sangaku via Peru
  70. FJG Capitan's Sangaku

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