Labeled isogonal coverings of the line

Just as a tiling of the plane using a vertex star made up of four coplanar squares is an isogonal covering of the two-dimensional plane with congruent two-dimensional vertex stars, we can also consider an isogonal covering of the one-dimensional line with one-dimensional vertex stars. For this, our vertex stars are merely a central point with two line segments of equal length extending out in opposite directions. In fact, we can simplify this to be merely a single line segment of a fixed length. To construct the covering, first pick an arbitrary point on the line. Then mark off points at equal distances in both directions. In doing this, we see that there can only be a single such covering, up to isomorphism. (That is, the specific location of the central point is not relevant.)

However, this only treats line segments as having a single color. When we allow colored line segments, even with just two colors, we can then have an infinite number of coverings. For example, first start at the central point. For simplicity, we extend the same pattern in both directions. Represent the two colors as '0' and '1'. Then, consider the integers in base 2, or binary, with infinite strings of '0's filling them out:
and so on. If we color our line segments according to these numeric patterns we see that, just as there are an infinite number of integers, there will be an infinite number of color patterns. Of course, these patterns are not isogonal because they are not vertex-transitive. We mention these colored patterns here merely to point out that covering the line is not quite as boring as it may at first seem to be!

So, returning to our original, single covering of the line, we realize that this only represents an unmarked, or unlabeled, covering. Going back to our original idea of a vertex star as three points, or vertices, representing two line segments connected to a central point, we see that we can label the endpoint vertices. In doing so, we find 5 possible vertex star labelings:
      a a
      a+ a+
      a+ a-
      a b
      a+ b+
Now we can match them up with adjacency symbols to create incidence symbols that form one-dimensional isogonal line coverings. There are 11 such possible labeled "polyhedra." We show the complete list below.

But, rather than just consider these as abstract symbols, we can consider our line segments to be curved instead of straight, and this allows the symbols to be represented graphically. First we need to determine graphical representations for the five vertex stars:
      a a  :    —
      a+ a+ :    ∼
      a+ a- :    ^
      a b  :    —•
      a+ b+ :    ∼•

But when we do this, we find that some of the symbols no longer represent coverings of the infinite line, but instead produce finite closed curves! We show these below, as well as those that cover the infinite line, along with approximate graphical representations for each of their isogonal forms.

Labeled isogonal one-dimensional "polyhedra"

  1.    [a a;  a]     ————————   (straight line)
  2.    [a+ a+;  a+]     ∼∼∼∼∼∼∼∼   (sine wave)
  3.    [a+ a+;  a-]     ∞   (finite figure-eight)
  4.    [a+ a-;  a+]     ^v^v^v^v^v^v^v^   (sawtooth wave)
  5.    [a+ a-;  a-]     ◊   (finite diamond)
  6.    [a b;  a b]     —:——:——:——:——:—
  7.    [a b;  b a]     —•—•—•—•—•—•—•—•
  8.    [a+ b+;  a+ b+]     ∼:∼∼:∼∼:∼∼:∼∼:∼∼:∼
  9.    [a+ b+;  a- b-]     ∞:   (finite figure-eight with double-dots at one end)
  10.    [a+ b+;  b+ a+]     ∼•∼•∼•∼•∼•∼•∼•∼•∼•
  11.    [a+ b+;  b- a-]     •∞•   (finite figure-eight with single dots at each end)

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Last updated: April 18, 2019