Anomalous Mirror Reflections Created by Symmetries and Their Application to Paper Crafts of Impossible Objects

There is a class of 3D illusion objects, called ambiguous cylinders, whose appearances change drastically in a mirror, so that we feel that the object is replaced with another, although we are looking at the same object from two viewpoints. In this paper, we concentrate on a special subclass of the ambiguous cylinders in which the illusion occurs due to the symmetry of the objects. This subclass includes the left-right reversal illusion, the translation illusion, and the whole-from-half illusion. We point out that the symmetry here guides us to design original optical illusion objects using paper in a simple manner, providing possible educational materials for children to learn about symmetry and optical illusions.