I wanted to share two pictures illustrating a knight starting on a central square of an infinite chessboard jumping to every other square of that board without ever landing on the same square twice.
The first is what you usually see when researching the subject. A central 5-by-5 square is tackled first. Then two-squares-wide rings are travelled sequentially. It takes four trips to complete each ring. The trips may be made clockwise or counterclockwise. In my drawing I've alternated the directions. The tours are not limited to these possibilities as there are other variants. Just the central 5-by-5 square can be drawn in 64 different ways, but it will always end up in one of the four corners (16 ways per corner).
My second picture duplicates the green starting and red terminal square of the finite 25-by-25 approximation of our infinite chessboard.
But it uses a radically different construction method. Our central 5-by-5 starting square is joined by another to its right; and that by another above. If one follows the connections one will understand why this also will travel the entire board to infinity. I've created the 5-by-5 patterns haphazardly, as (again) there are plenty of alternatives from which to choose.
I drew my pictures using Mathematica code from Dr. Colin Rose.