This paper introduces a new concept of mirror symmetry, called “anomalous mirror symmetry”, which is physically impossible but can be perceived by human vision systems because of optical illusion. This symmetry is characterized geometrically and a method for creating cylindrical surfaces that create this symmetry is constructed. Examples of solid objects constructed by a 3D printer are also shown.

A new class of objects, called topology-disturbing objects, is presented. These objects appear to disturb the topological properties when they are seen from two specific viewpoints; for example, two objects appear to be separated when seen from one viewpoint while they appear to be intersecting when seen from the other viewpoint. This cannot happen physically, but due to the phenomenon of an optical illusion, it can be perceived to occur. A general method for designing this class of objects is presented and examples are given. These objects might provide a new resource for arts and entertainment.

Objects represented by anomalous pictures are usually considered unrealizable in a three-dimensional space, but some of them are actually realizable. This paper characterizes a class of realizable anomalous pictures from a mathematical point of view. Distribution of degrees of freedom in the choice of depths of the vertices of a polyhedron represented by a picture is studied, and a decomposition of a polyhedron into components with the minimum degrees of freedom is proposed. According to this decomposition, a class of realizable anomalous pictures is characterized.