"Base infinity" number system

Base ten has ten symbols, base two has two symbols, base infinity logically has an infinite number of symbols, one for each number!  The first thing one might think of with base infinity would simply be something like a QR code, which you've probably seen before.

However, I wanted to make something in which the symbol would always be one continuously connected symbol, and also tried to make it obvious to the human eye where on the grid each bit is placed (neither of these things is guaranteed or expected with a QR code.)  There is a simple formula in my system for determining from a given symbol what the corresponding number is.  With these limits on what base infinity would look like, I made this:

More recently, I came up with another system.  In this system, the binary number swaps around parts of shapes, and it's more fractally.

 

One nice feature of this system is that when a number is written in binary, two numbers which differ by exactly one digit are identical in my system except that one "arm" is flipped around.  Another feature is that digits which are further to the left, and hence have more influence on the size of the integer, are generally represented by larger parts of the shape.  (The leading "1" is mostly ignored, which works since all positive integers start with a 1 anyway.)  Here are some more numbers in my system: 

Here's a large number, 3^3^3, or 7,625,597,484,987

There is actually a simpler way to display these numbers, which is to just mark the ends of the lines, instead of showing direction by having different sized lines.

But I think having using different sizes to mark the direction of the "arms" for one thing makes it easier to tell what order the digits are in when reading one of these shapes, and for another thing just looks cooler ;-)