Shepard's Parallelogram Illusion

A few words

Copyright © 1996-2009 Alexander Bogomolny

The two parallelograms shown appear at times different. However, they are always congruent and can be obtained from one another by rotation. This illusion has been published by Roger Shepard in 1981 and later in 1990 [Shepard, p. 48] The parallelograms look especially different when one is rather vertical while the other more horizontal. According to Shepard, we are commonly fooled by our depth perception. Almost in all cases the effect is enhanced when the parallelograms grow legs to resemble the table tops.

Jacobs makes a delightful use of this illusion to discuss a property of rotations. A rotation f can be determined by two arbitrary points P and Q and by their images f(P) and f(Q), provided neither is fixed. In the applet we think of E as being the image of A and H the image of D. The center of rotation O is bound to lie on the perpendicular bisectors of AE and DH and hence at their intersection. (In principle, the two two lines may coincide. Then the lines AE and DH intersect at the center of rotation.)

So assume the parallelograms ABCD and EFGH are equal and that O has been found as just described. In triangles ADO and EHO, AO = EO and DO = HO by construction, and AD = EH by our assumption. The triangles are equal by SSS. In particular, their angles at O are equal. If we add to each the angle DOE, we arrive at

AOE = DOH,

So that indeed when A is rotated around O to coincide with E, D rotates onto H. The rest of the parallelogram ABCD comes along as a rigid body.

References

1. H. R. Jacobs, Geometry, 3^rd edition, W. H. Freeman and Company, 2003
2. R. N. Shepard, Mind Sights, W. H. Freeman and Company, 1990
3. I.M. Yaglom, Geometric Transformations I, MAA, 1962

Copyright © 1996-2009 Alexander Bogomolny