The Fundamental Theorem of 3-Bar Motion
What Is It?
A Mathematical Droodle


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Copyright © 1996-2017 Alexander Bogomolny


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The Fundamental Theorem of 3-Bar Motion is due to the nineteenth-century British mathematicians William Kingdom Clifford and Arthur Cayley. The theorem describes a linkage of 15 rods and 10 swivel-joints. The simplest configuration that is of course distorted by moving the rods is that of a triangle ABC with a point 7 through which three lines are drawn parallel to the sides of the triangle. The points of intersection are denoted 1, 2, 3, 4, 5, and 6. The lines form three triangles 127, 347, 567, all similar to ΔABC, and three parallelograms. While the whole configuration may change, triangles 127, 347, 567 remain the same in shape and size. In the deformation, the parallelograms always remain parallelograms, because their opposite sides remain equal, but the angles change.

Honsberger's proof of the Clifford-Cayley theorem makes an elegant use of spiral similarities (see also [Yaglom]).

Let S be the spiral similarity that maps, say, 56 onto 57. S is an affine mapping of the plane that rotates the plane as a whole such that all lines form the same angle with their respective images, namely, ∠657, which is equal to the original angle A. All segment lengths are modified by S by the same factor. Thus S induces an operator S on vectors in the plane.

We have the following vector identities: AB = A5 + 56 + 6B and AC = A4 + 43 + 3C. Apply S to AB:

S(AB)= S(A5 + 56 + 6B)
 = S(A5) + S(56) + S(6B)
 = S(47) + 57 + S(71)
 = 43 + A4 + 72
 = A4 + 43 + 3C
 = AC,

which exactly means that the angle BAC at A is equal to ∠657 and, also, 56/57 = AB/AC, or that the triangles 567 and ABC are similar.

References

  1. R. Honsberger, In Pólya's Footsteps, MAA, 1999
  2. I. M. Yaglom, Geometric Transformations II, MAA, 1968

Linkages

  1. What Is Linkage?
  2. Peaucellier Linkage
  3. Hart's Inversor
  4. The Fundamental Theorem of 3-Bar Motion
  5. Pantograph
  6. Watt's and Chebyshev's Linkages

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Copyright © 1996-2017 Alexander Bogomolny

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