Rep-Cubes: Dissection of a Cube into Nets (original) (raw)

2018, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

Last year, a new notion of rep-cube was proposed. A rep-cube is a polyomino that is a net of a cube, and it can be divided into some polyominoes such that each of them can be folded to a cube. This notion was inspired by the notions of polyomino and reptile , which were introduced by Solomon W. Golomb. It was proved that there are infinitely many distinct rep-cubes. In this paper, we investigate this new notion and obtain three new results. First, we prove that there does not exist a regular rep-cube of order 3, which solves an open question proposed in the paper. Next, we enumerate all regular rep-cubes of order 2 and 4. For example, there are 33 rep-cubes of order 2; that is, there are 33 dodecominoes that can fold to a cube of size √ 2 × √ 2 × √ 2 and each of them can be divided into two nets of unit cube. Similarly, there are 7185 rep-cubes of order 4. Lastly, we focus on pythagorean triples that consist of three positive integers (a, b, c) with a 2 + b 2 = c 2. For each of these triples, we can consider a rep-cube problem that asks whether a net of a cube of size c × c × c can be divided into two nets of two cubes of size a × a × a and b × b × b. We give a partial answer to this natural open question by dividing into more than two pieces. For any given pythagorean triple (a, b, c), we construct five polyominoes that form a net of a cube of size c × c × c and two nets of two cubes of size a × a × a and b × b × b.