The grouping order of p-adics is normally displayed using circles nested in circles using a fractal display pattern. Here is another way to display that order. It places new numbers around the perimeter of a circle while maintaining the concept of p-adic distances. It puts the p-adic numbers in order like on number line, which I think is superior. Whereas, existing diagrams place those numbers over a plane.
The location of any p-adic number is a rotation of it's digits in base p in reverse divided by the number of digits.
Progression of 3-adic number counting with bounded distances.
The location of a p-adic number is it's base p reflection over the decimal point.
update: I notice that these numbers revolve clockwise. The numbers should revolve counter clockwise, in the same direction as the imaginary unit raise to greater powers. Oops. Just imagine the numbers spin the other way around the 0 point at the right of the circle.
The above diagram can be revolved and is read off the right side of the circle. You need to imagine revolving the diagram to make other numbers read horizontally off that new right side of the circle. Each number has a decimal point. The number to the left of the decimal is the 3-adic number under consideration. The number including and after the radix point is the fraction of a full revolution that the number is placed on the circle relative to zero. The numbers are shown in base 3, because then we can see that the numbers and their locations are their reflections.
Also, in the above diagrams, the zeros on the outside of each number are aesthetic. Zeros to the left of whole numbers and to the right of base fractions are always ignored. So 0012.1200 is the same as 12.12.
Below are the 2-adic, 5-adic and 7-adic diagrams of distance boundaries. Numbers are not shown (They are there. I just didn't bother).
Questions:
Am I correct in saying that all p-adic numbers are properly grouped on a circle perimeter by a revolution of the fraction of their base p representation revolved around their radix point?
Why do we say that p-adic number can not go on a line? While the p-adic numbers are ordered on a circular number line as shown. It may not have total order across the line, but the order does remain relevant while new numbers are added to this line. So it is somewhat ordered. Why not display it? Is it too difficult for us to imagine that things ordered on a line may not have total order across all of that line?
In regard to question #1. The angular location of numbers are their digits swung over the decimal point as in this example:
15 in base 3 is 120₃
120₃ revolved around its radix is .021₃
.021₃ = 21₃/3³ = 7/27
Therefore, 15 appears at 7/27 of a full turn on a 3-adic diagram. The location will remain relevant to the location of all other number placed in the same way.
My thoughts:
I find this order interesting because before I noticed that the Surreal numbers also go on a circle in the same sort of way, but the binary digits are rolled instead of reversed.
But I am new to p-adics. I could be totally wrong about all of this. I'm trying to understand p-adics and found this other ordering. Then I wondered why nobody showed it to me. Instead I saw 3 dimensional fractal towers that looked nothing like the concept of counting out numbers and rather seemed like creating a structure. I just felt it was unnecessary to show me those beautiful but confusing drawings. I thought mine were better. It made it easier to understand how the grouping happens because the placements of numbers and their locations have a formula.
So maybe my diagrams are wrong and not better then existing ones. I'd like to hear why.