Visual Display of Numbers: Part 1

In my previous post titled Binary Display, I borrowed an idea from Numbers Aplenty about the visual display of a positive integer expressed in terms of its binary digits. This gave me the idea for a display of the number in terms of its decimal digits. I got Gemini to develop an algorithm to achieve this and Figure 1 shows the result for 28338, my diurnal age today (permalink):

Figure 1

There is a small text-based display:

  | 2   8   3   3   8  

--+--------------------

9 |                    

8 |     ██          ██ 

7 |                    

6 |                    

5 |                    

4 |                    

3 |         ██  ██     

2 | ██                 

1 |                    

0 |                    

The larger diagram below this in the output can then be screen-shot and annotated as shown in Figure 2.

Figure 2

On this basis, the number 28338 might be associated with Twin Peaks (with an intervening valley). This sort of depiction of the digits of a number can be used to visually depict so called \( \textbf{mountain numbers}\) that belong to OEIS A134941. These numbers are defined as:

The first digit is 1. The last digit is 1. The first digits are in increasing order. The last digits are in decreasing order. The numbers only have one largest digit. This sequence is finite. The last term is 12345678987654321. The total number of terms is 21846.

19871 is an example of such a number:

  | 1   9   8   7   1  

--+--------------------

9 |     ██             

8 |         ██         

7 |             ██     

6 |                    

5 |                    

4 |                    

3 |                    

2 |                    

1 | ██              ██ 

0 |                    

There are also generalized mountain numbers described in OEIS A134853 as:

There is only one largest digit which represents the top of the mountain. This sequence is finite. The last member is 123456789876543210. The sequence is a supersequence of A134941, because the restriction that both feet of the mountain are at "sea level" (first and last digit equal 1) is dropped here. There are 173247 terms in this sequence.

28320 is an example of such a number:

  | 2   8   3   2   0  
--+--------------------
9 |                    
8 |     ██             
7 |                    
6 |                    
5 |                    
4 |                    
3 |         ██         
2 | ██          ██     
1 |                    
0 |                 ██  

This is just a rough start for a general theme of how to represent a decimal number. I've addressed this issue before in a post titled What Does A Number Look Like from July of 2018. Using the site Numbermatics, the representation of the number 28338 is shown in Figure 3:

Figure 3

The commentary on the display is as follows:

In this visualization, the outer ring segments show the relative proportions of its three prime factors. Inside, the large and small discs symbolize the relationship between these factors (center) and eight divisors.

This representation is more informative and very different to the display shown in Figure 1. I've gotten Gemini to investigate visual representations more using this prompt:

I'm interested in exploring different ways to represent a positive integer such as 28338. Numbermatics (https://numbermatics.com/n/28338) shows one way and the display that I've just gotten Gemini to create shows another. What are some other visual ways to represent numbers that ideally show information not just about the digits but also its factors, divisors etc. Maybe you suggest modifications or improvements to existing methods. See what you can come up with. Links to other sites like Numbermatics can naturally be included in your report.

Using Gemini's Argon set to high level thinking, the response that I got was very comprehensive and I won't attempt to summarise it in this post and I'm still digesting what it came up with. I'll reserve further comment to my next blog post.