Imaginary Cubes—Objects with Three Square Projection Images

Hideki Tsuiki
Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (2010)
Pages 159–166 Regular Papers


An imaginary cube is a three dimensional object which has square projections in three orthogonal directions just as a cube has. In this paper, we study imaginary cubes and present sculptures based on imaginary cubes. We show that there are 16 kinds of minimal convex imaginary cubes. The list of 16 representative minimal convex imaginary cubes contains, as well as a regular tetrahedron and a cuboctahedron, a hexagonal bipyramid imaginary cube, which is a double imaginary cube, and a triangular antiprism imaginary cube, which plays an important role in the construction of the sculptures. Then, we present two Imaginary Cube Sculptures. Each of them is composed of all the 16 representative minimal convex imaginary cubes and forms an imaginary cube as a whole. Thus, they reveal uniform overall structures though they are composed of different shapes.