Visualizing Consecutive Binary Integers

The following is a visual depiction of the binary integers 0 through 11111111:

Binary Integers 0-11111111

A nice pattern, right? I generated it based on the image found on page 117 of Stephen Wolfram’s “A New Kind of Science”. I’ll discuss its structure in detail in this article.

A Visual Table of Binary Integers

If you were to list the first 16 nonnegative binary integers in a table like this,

Binary Integers 0-1111

the 0s and 1s would form a pattern, although it may be hard to see.

If you were to simply color in the 1 cells, the pattern would become much more obvious:

Binary Integers 0-1111

If you were to extend the table, the pattern would become more obvious still. Here it is extended through 11111111, as depicted in the opening of this article:

Binary Integers 0-11111111

Powers of Two Sized Patterns

The overall pattern in the table is based on vertical and horizontal sub patterns. Each sub pattern is based on nonnegative power of two sized units, which is a consequence of individual binary numbers being made up of powers of two.

Vertical Patterns

The basic pattern in the table is vertical, in the columns. Each column alternates between 2ⁿ 0s (blanks are considered 0s) and 2ⁿ 1s, with 2ⁿ being the value of that column’s place. This gives 2ⁿ⁺¹ sized blocks of 0s followed by 1s. For example, the pattern in the ones column is 01; the pattern in the twos column is 0011; the pattern in the fours column is 00001111; etc.

Here are the patterns isolated for the ones through sixteens column:

Patterns in First 5 Columns of Table

These patterns explain the toggle frequency of a binary counter, as discussed in my article What a Binary Counter Looks and Sounds Like.

Horizontal Patterns

The predominant pattern in the table is horizontal, in power of two sized groups of rows. The beauty in this pattern is the nesting. The first two rows start the pattern; the next two rows repeat the first two rows and then prefix them with a 1; the next four rows repeat the prior four rows (with leading 0s filled in first) and then prefix them with a 1; etc.

Here are the patterns isolated for the first five groups of rows:

Patterns in First 32 Rows of Table

In other words, each new group of rows is a copy of all prior rows with a leading 1 attached. Expressed mathematically, the group of rows 2ⁿ to 2ⁿ⁺¹ – 1 is the group of rows 0 to 2ⁿ – 1 with 2ⁿ added to the number in each row.

Horizontal Symmetry

If you fold the table on itself at its halfway point — for the 256 entry table, this is at the border between decimal numbers 127 and 128 — each half will be the negative, or complement, or “dual” of the other. That is, they are the same, except white and black, or 0s and 1s, are interchanged.

Related Binary Art

Ivars Peterson analyzes a piece of artwork by Arlene Stamp called the Binary Frieze. The Binary Frieze incorporates the imagery above, but with two minor differences: black and white are interchanged, and the table is transposed so that numbers are in columns instead of rows:

Binary Frieze Imagery

This image does not appear exactly this way in the Binary Frieze, but is cut up and transformed into a piece of art. Here are three pictures of it: Binary Frieze (Stretch), Binary Frieze (Squeeze), and Binary Frieze (Swing).

As for black and white being interchanged, here’s a different interpretation: perhaps it represents a countdown from 255 to 0.