# Binary cubes representation

## Another abstract representation

The simplicity in applying the absorption and logic adjacency in Karnaugh Maps is derived from another way of representing binary functions: the binary cubes.

This representation is also key to understand the Quine McCluskey algorithm to minimize logic functions.

More information can be found in the section on Karnaugh Maps in [1]

### Functions of 1 variable

Any logic function of a single variable $f(x)$ can be represented by a line:

The points at both ends of the line indicate the values that the variable $x$ can take (0 or 1).

### Functions of 2 variables

In a similar way a function of two variables can be represented by a square in a two-dimensional space:

In the figure it is easy to see that (1,0) and (0,1) are not adjacent, while (1,1) and (0,1) are adjacent.

### Functions of 3 variables and more

This representation can be further extended to higher orders. For instance, a function with three variables would be represented by the following three-dimensional cube:

Again, the adjacencies can be identified by the lines connecting the corners of the cube. If there is not a line, those binary numbers are not adjacent.

The adjacency is key to group cubes in Karnaugh maps and in the Quine McCluskey method to obtain minimal expressions.

## References

• [1]G. Donzellini, L. Oneto, D. Ponta, and D. Anguita, Introduction to Digital Systems Design. Springer International Publishing, 2018 [Online]. Available at: https://books.google.cl/books?id=va1qDwAAQBAJ
1. Which shape is represented by an object having exactly 1 variable?
1. Line
• Circle
• Triangle
2. Which shape is represented by an object having exactly 2 variables?
1. Square
• Circle
• Triangle
3. Which shape is represented by an object having exactly 3 variables?
1. Cube
• Parallelogram
• Sphere