2010 Bridges Conference

# Hideki Tsuiki

## Statement

The "Imaginary Cube" is based on a simple geometrical idea. See the paper in the Bridges proceedings for the details. Through a mathematical study of imaginary cubes, the author arrives at an idea for an object of art. The 16 components of the sculptures all have different shapes but together they present uniform appearances when viewed from three orthogonal directions. Moreover, the 16 components are not arbitrary ones but they are exactly the representatives of all the 16 classes of minimal convex imaginary cubes. Imaginary cubes are also useful in mathematical education; the author held a lot of workshops on imaginary cubes with classes from elementary schools up to universities. Special thanks to Hiroshi Nakagawa; through his accurate woodworks, imaginary cubes become really artistic sculptures. Thanks are also due to Kei Terayama for his techniques in assembling paper models and Mako Mizobuchi for fine pictures.

## Artworks

The object is placed in the center of the picture, and the surrounding three are reflected images in mirrors. The object is also an imaginary cube composed of the 16 representatives of the minimal convex imaginary cube classes, but with a different layout of the 16 components. These two arrangements of the 16 imaginary cubes are obtained through the investigation of Latin squares of degree 4. This object has 4 holes of the shape of a triangular antiprismoid - also an imaginary cube - in the four blocks, and one hole of the shape of a regular octahedron in the middle.
These four pictures present different appearances of one and the same object. It is composed of the 16 imaginary cubes of the above artwork, that is, all the representatives of the minimal convex imaginary cubes. The object as a whole is also an imaginary cube, as this picture shows. The imaginary cube components are arranged according to the structure of the 2nd level approximation of the Sierpinski tetrahedron, and their assignment and orientation are carefully chosen so that they are connected at vertices. They are connected by threads which are glued from the inside of the polyhedra at vertices. This object is colored with 7 colors except for white; six colors are assigned to faces and edges of the 6 directions with square appearances, and those faces which compose the holes in the four blocks are colored black. These holes have the form of a triangular antiprismoid, which is also an imaginary cube.
Woodworks by Hiroshi Nakagawa (Gallery of Wooden Polyhedra, http://ww6.enjoy.ne.jp/~hiro-4/woodenpolyhedra30.html ). Imagine a three-dimensional object which has square appearances in three orthogonal directions just as a cube. We call such an object an imaginary cube. Among imaginary cubes, consider convex ones, and also among convex ones, minimal ones for a fixed surrounding cube. Such minimal convex imaginary cubes are divided into 16 equivalence classes and here are the representatives of them, made of wood. It is difficult to imagine that these polyhedra have this property if they are put solely, but once each of them is put in an acrylic resin box with one side open, one can easily find that it is an imaginary cube just by looking at it from the faces of the box. It is a good mathematical puzzle to put imaginary cubes in a box.