Talk:Doyle spiral

Image development
Back home, so I can work on some images. Here's a simple draft of a (7,8) spiral as an SVG with the seven spirals enumerated (I have created the same figure for the eight spirals and single spiral). Of course the program works with any reasonable (p,q). Suggestions on colors, different indications, etc.? Ovinus (talk) 05:39, 28 June 2022 (UTC) Annoyingly, something seems to be removing or ignoring. Will fix. Ovinus (talk) 05:44, 28 June 2022 (UTC)
 * Looks pretty good. I like the idea of using numbers to clarify how the arms are counted. Maybe more contrasty colors for adjacent spirals, rather than trying to make them into a smooth gradient? Checking against an online color blindness simulator might also be a good idea. If you used a (6,8) spiral instead of (7,8) then it would be possible to make versions highlighting all three types of arms, and you would only need two colors to make all pairs of adjacent arms contrast, but with the one you chose, one of the arms contains all the circles making it difficult to highlight. I looked at recoloring your image so that most circles were neutrally colored and with overlapping bands of colors for a selected arm of each type (like the 1911 lead image, but in color and with multiple types of arms shown) but I didn't save and upload as I don't think it came out looking very good. —David Eppstein (talk) 07:21, 28 June 2022 (UTC)
 * Thoughts on these (could also be combined into a single GIF with links to the SVGs)?


 * According to these color combinations (#000 text on (#3a3 or #77c) on #fff background) are accessible. Ovinus (talk) 19:15, 28 June 2022 (UTC)
 * I tried adding these to the article, more or less as presented above but with captions. —David Eppstein (talk) 20:05, 28 June 2022 (UTC)


 * I was curious about the Mobius transformations and decided to see what they actually look like. I admit this is screen recorded from my web app, as the shit video quality suggests. I was too lazy to properly convert to video, although I will eventually. There is also float precision annoyance that I must iron out. Essentially it just takes the transformation $$z\mapsto \frac{z}{cz+1}$$ and lets $$c$$ go from 0 to 0.5 or so. It clarifies a couple points I didn't really intuit after reading the passage. First, the result of a Mobius transformation on a Doyle spiral for which the preimage of complex infinity is inside a disc maps the disc to the complement of a disc, and thus doesn't result a circle packing at all. I suppose it's trivial now that I see it, but I had no clue before. Second, it is possible to have a circle packing for which each circle has six tangencies and those tangencies form a ring, but that is not a Doyle spiral, if the ring of tangencies is allowed to not enclose the circle (see ~0:25 in the video). Anyway, thought you might enjoy it. Ovinus (talk) 05:21, 3 July 2022 (UTC)


 * Not to inundate the article with images, but here's one I came up with to illustrate how the Mobius transformation works. The "inverted disc" phenomenon might be OR, so I just rendered this one. Ovinus (talk) 22:42, 5 July 2022 (UTC)
 * You can find images generated by a Möbius transform of a Doyle spiral, sort of like this one, in Wright's "Searching for the cusp". A tighter crop might help better show the way it spirals in to two points rather than one (the second spiral center is where the Möbius transform takes the point at infinity). —David Eppstein (talk) 04:40, 6 July 2022 (UTC)
 * Oh good. Yeah, I tried zooming in on the image. I could also make a square-ish one if that feels more shapely. Ovinus (talk) 13:34, 6 July 2022 (UTC)
 * In the caption, more specifically, it is the three outer circles that are not surrounded. —David Eppstein (talk) 14:37, 6 July 2022 (UTC)
 * Ye. Alright, added. Ovinus (talk) 07:54, 23 July 2022 (UTC)