Quasi-Circles
In the previous pages, we have seen a number of times, that "circles" appeared, where only straight lines were drawn. Let's turn it around. Is it possible to draw (that is approximate) circles by straight lines? Yes, of course, it can! It means drawing polygons, with all angular points at – or close to – a circle. But the limiting factor is our checkered paper: the lines should begin and end on a raster-point. Now it becomes more interesting. Over the years, I experimented a lot with all sorts of circle approximating polygons on checkered paper. The resulting drawings were so many, that I feel I need to categorize them, especially so, now I have started the Gallery.
So, let's examine the possibilities to approximate circles with not too long, raster limited, straight lines. Or, which of these quasi-circles pass through or almost through (more than four) grid-points? That means, that the radius R of such a circle must satisfy the Pythagorean equation R² = x² + y², where x and y are integer numbers. In the following table, the value of R² is denoted for each pair of integer values x and y up to 10.

 R² x 0 1 2 3 4 5 6 7 8 9 10 x² 0 1 4 9 16 25 36 49 64 81 100 y y² 0 0 0 1 4 9 16 25 36 49 64 81 100 1 1 2 5 10 17 26 37 50 65 82 101 2 4 8 13 20 29 40 53 68 85 104 3 9 18 25 34 45 58 73 90 109 4 16 32 41 52 65 80 97 116 5 25 50 61 74 89 106 125 6 36 72 85 100 117 136 7 49 98 113 130 149 8 64 128 145 164 9 81 162 181 10 100 200

Now we have to find radii, that are equal or almost equal. It is easiest to put all values of R² in ascending order, so that equal and almost equal values become adjacent. Table 2 illustrates this. Following the R² column, the appropriate x and y values are denoted, followed by a "circle number" with the relevant characteristics (if applicable – "no suitable circle" means, that the sides of a possible polygon are too far off the circumference of a circle), and an example of how it looks like on checkered paper. Sometimes, an alternative is possible with the same characteristics.
Some of the quasi-circles are "exact" in the way, that the angular points lie exactly on a circle; these are displayed with a dotted circle. Mostly, values of R² are combined, that do not differ more than 2, but quasi-circles with a difference in R² of more than 2 also exist in the Gallery, as do polygons with more than 2 different radii – especially some greater ones. On checkered paper, it is not absolutely mandatory to stick to the raster-points. There are a number of quasi-circles, that use halfway values on the raster. Therefore, some polygons in table 2 have half-sized counterparts shown additionally.
In the Gallery, the various quasi-circles will be put together within their own "family".