Aesthetic Numbers

An Aesthetic or Esthetic Number is defined by Numbers Aplenty as a number \(n\)  such that the difference between adjacent digits is 1 when the number is written in base \(b\). Such numbers attracted my attention because 27564, my diurnal age yesterday, is an aesthetic number in base 8 or simply 8-aesthetic because:$$27564_{10}=65654_8$$Let's look firstly at aesthetic numbers in base 10 between 10 and 40000. There are 149 such numbers and they are (permalink):

10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 101, 121, 123, 210, 212, 232, 234, 321, 323, 343, 345, 432, 434, 454, 456, 543, 545, 565, 567, 654, 656, 676, 678, 765, 767, 787, 789, 876, 878, 898, 987, 989, 1010, 1012, 1210, 1212, 1232, 1234, 2101, 2121, 2123, 2321, 2323, 2343, 2345, 3210, 3212, 3232, 3234, 3432, 3434, 3454, 3456, 4321, 4323, 4343, 4345, 4543, 4545, 4565, 4567, 5432, 5434, 5454, 5456, 5654, 5656, 5676, 5678, 6543, 6545, 6565, 6567, 6765, 6767, 6787, 6789, 7654, 7656, 7676, 7678, 7876, 7878, 7898, 8765, 8767, 8787, 8789, 8987, 8989, 9876, 9878, 9898, 10101, 10121, 10123, 12101, 12121, 12123, 12321, 12323, 12343, 12345, 21010, 21012, 21210, 21212, 21232, 21234, 23210, 23212, 23232, 23234, 23432, 23434, 23454, 23456, 32101, 32121, 32123, 32321, 32323, 32343, 32345, 34321, 34323, 34343, 34345, 34543, 34545, 34565, 34567

Here are the frequencies for base 10 numbers being aesthetic in smaller bases in the range from 10 to 40000:

• base 9 - 165 e.g. 37825 --> 56787
• base 8 - 171 e.g. 33436 --> 101234
• base 7 - 164 e.g. 36881 --> 212345
• base 6 - 188 e.g. 38657 --> 454545
• base 5 - 182 e.g. 36119 --> 2123434
• base 4 - 188 e.g. 39867 --> 21232323
• base 3 - 134 e.g. 36905 --> 1212121212
• base 2 - 12 e.g. 21845 --> 101010101010101

There are 17, 32, 61, 116, 222, 424, 813, 1556, and 2986 10-aesthetic numbers with 2, 3, 4, 5, 6, 7, 8, 9 and 10 digits respectively. Table 1 shows a screenshot of the Numbers Aplenty website featuring an interesting formula:

Table 1

Polyominoes and Conway's Game of Life

I've been reading a free ebook (title shown above) that I downloaded from this site. To quote from a footnote in the book:

A polyomino is a pattern made up of orthogonally connected live cells, and a tetromino is a polyomino with 4 live cells. More generally, polyominoes with 2, 3, 4, ..., 8 live cells are called dominoes, triominoes, tetrominoes, pentominoes, hexominoes, heptominoes, and octominoes.

Once we exceed eight cells, the polyominoes are referred to as 9-ominoes, 10-ominoes, 11-ominoes etc. This is a formidable but comprehensive book of 494 pages that I'll try to make my way through gradually. It was published in 2022 with the dedication:

I've mentioned John Conway before in posts about:

• Free Fibonacci Sequences on July 18th 2021 

• RATS Sequence on September 26th 2020 

• Look and Say Sequence on February 19th 2017

More recently I've been mentioning him in posts about his Game of Life:

• The Game of Life on December 16th 2023

• 108 Meets Conway's Game Of Life on February 9th 2024

• Diurnal Age Meets Conway's Game Of Life on February 15th 2024

In this and probably subsequent posts I'll be pursuing the Game of Life from the perspective of polyominoes formed by the decimal digits when using a 5 high x 3 wide grid of pixels. See Figure 1:

Figure 1

In my post titled Diurnal Age Meets Conway's Game Of Life I looked at the sort of "ash" that was produced by a "soup" of pixels representing my diurnal age. The terms "soup" and "ash" are used in the book mentioned at the start of this post. However, I thought it would also be interesting to examine how the digits from 0 to 9 behave when not in close proximity to other digits. To this end, I'll begin with the digit 0.

In a 5 high x 3 wide grid of pixels, the digit 0 occupies 12 of those pixels. See Figure 2:

Figure 2: a 12-omino in the shape of the digit 0

The zero quickly becomes the famous pulsar. See Figure 3.

Figure 3: source

The digit 1 forms a pentomino and it quickly morphs into what is called a blinker. See Figure 4 and Figure 5.

Figure 4: pentomino in shape of the digit 1

Figure 5: the digit 1 becomes a blinker

The digit 2 is an 11-omino and it quickly transforms into two boats. A boat is an instance of a still-life. See Figures 6, 7, 8 and 9.

Figure 6

Figure 7: first step on the way to two boats

Figure 8: second step on the way to two boats

Figure 9: two boats (unchanging)

I'll look at the other digits in future posts. That's enough for now.