User blog comment:Clarrity/List of ordinals (up to omega^omega)/@comment-30118230-20180225131945/@comment-30754445-20180227105337

@无限计数器

Creating a picture like the one presented in this blog post can be easily done for any ordinal (as long as we know the fundamental sequences).

Step 1:

Pick the maximum ordinal we want to represent and put in at the center.

Step 2:

List the members of the FS of said ordinal at "12 o'clock" (the positions which the current picture has for 0, ω, ω2, ω3...)

Step 3:

We interpolate between any consecutive pair of ordinals already listed, in this manner:

(a) If the difference between the ordinals is1, we do nothing.

(b) If the difference betwee the ordinals is ω, we interpolate between them with a standard visualization of ω (which is a copy of the outer 360 degrees of the spiral seen in the original ω^ω diagram).

(c) Otherwise, we interpolate between any pair of ordinals given in the previous step by putting the appropraite FS member at positions 1/2, 3/4, 7/8, ...

Step 4:

Repeat step 3 for as often that is needed.

And that's it!

Of-course, if we make such a diagram for a very large ordinals, it won't look too nice. But for reasonably small ordinals it would still be clear. Try visualizing the ε₀-level diagram. I think it's actually quite attractive.