There are 8 distinct tetracubes.
Understanding Tetracubes and Polycubes
Tetracubes are a specific type of polycube, which are three-dimensional geometric shapes formed by connecting multiple identical cubes face-to-face. They are the 3D equivalents of polyominoes (shapes made from squares). The term "tetracube" specifically refers to polycubes made from four unit cubes.
The study of polycubes is a fascinating area of recreational mathematics and puzzle design. The number of unique polycubes increases significantly as the number of constituent cubes (or "order") grows.
The Number of Polycubes by Order
The quantity of distinct polycubes varies depending on their order. When considering shapes that are distinct even when rotated, reflected, or both (free polycubes), the numbers are as follows:
| Order | Polyform Name | Number of Distinct Polycubes |
|---|---|---|
| 1 | Monocube | 1 |
| 2 | Dicube | 1 |
| 3 | Tricubes | 2 |
| 4 | Tetracubes | 8 |
| 5 | Pentacubes | 29 |
| 6 | Hexacubes | 166 |
| 7 | Heptacubes | 1,023 |
As shown in the table, there is only one monocube (a single cube) and one dicube (two cubes joined). Tricubes introduce the first set of distinct shapes beyond simple lines. When it comes to tetracubes, there are exactly 8 unique configurations.
Examples of Tetracube Shapes
The eight tetracubes include various shapes, often visualized and used in spatial reasoning puzzles. They can be thought of as different ways to arrange four cubes so that each cube shares at least one face with another. These distinct shapes are crucial for understanding and solving various packing and construction puzzles involving polycubes.
For more information on polycubes and their fascinating properties, you can explore resources like Polyform Puzzler, which delves into the mathematics and applications of these shapes.
