1

I know that a regular pentagonal tiling does not work in Euclidian space, but does work on a sphere. But this got me wondering something that I hope people can help with here, because I can't find any way to do it myself.

All projection software I can find requires you to give the projection and it reverses it to apply it to the sphere, instead of vice versa. And looking it up I can't find any papers on the projection of a sphere tessellated with regular polygons that cannot tesselate Euclidian space.

So what would projections of a Sphere tiled with 12 regular pentagons look like? And around how much of the pentagonal shape can you keep by projecting. It would be helpful if you can find a way to show images, if that is not too big of a bother.

Edit: For those that might misread this, I am perfectly aware that pentagons can tile the sphere, as I have already stated. My question is about projections of said spherical tiling onto the plane, where I know regular pentagonal tiling doesn't work. And how much is preserved upon such projections. I hope this clarification helps.

Edit: I particularly want to see a graph that displays the distortions in size, angle, and distance between points from this. To see how much is actually preserved by the projection.

Paul Frost
  • 76,394
  • 12
  • 43
  • 125
Zoey
  • 208

1 Answers1

1

Perhaps the reason that there are no papers on this topic is because they are very classical topics that are explained in (sometimes very old) mathematical books or taught in mathematical courses.

For example, Euclid's Elements contains the theorem which, in modern language, produces all possible regular tilings of a sphere, including the regular dodecahedron tiling which can be represented as a spherical tiling.

Here is another wikipedia link depicting lots of spherical tilings. A key sentence you can read at that link:

All regular polyhedra, semiregular polyhedra, and their duals can be projected onto the sphere as tilings...

Lee Mosher
  • 120,280
  • While this helps, it doesn't fully help with how I was thinking of like, having, a graph that also covers it in order to show the distortions in angle, area, and distance between points. I should clarify that in my post. Furthermore, I know they work on the sphere, I said that in my post, I am asking what happens when projected onto the plane, where they don't work. – Zoey Jan 09 '23 at 17:05
  • Please reread my question and answer again. As this does not answer my question, but a question which was not asked. – Zoey Jan 09 '23 at 17:23
  • I read and reread your question, and all I have to say are these remarks which are too long for a comment, so I put them here as a possibly helpful partial answer. And in case you don't get any further answers, I'll add a suggestion to read up on stereographic projection which is the standard method of projection from a sphere (with one point removed) to the plane. In that link provided by @JeanMarie you'll find a discussion of stereographic projection. – Lee Mosher Jan 09 '23 at 17:33
  • Okay, and sorry for getting annoyed. I have read a large amount on all sorts of different projections, but all projections I know of work with separating the sphere into 4-gons distorted in various ways. Such as Mercator working my turning the sphere into equal sized quadrilaterals, and projecting them onto a cylinder of equal radius to the sphere. My confusion is just on how different projection styles would work when there is the shape that the sphere is tiled with just can't tile the plane. I will look at the page, though, thank you, and sorry again. – Zoey Jan 09 '23 at 17:39
  • No problem. Let me add a suggestion regarding your question itself: specific mathematical questions regarding one or two specific projections (e.g Mercator; stereographic) might get better responses. – Lee Mosher Jan 09 '23 at 17:41
  • True, being more clear on what kinds of projections would be better. I am just wondering what projections would retain the pentagonal shape the most. It has to do with a book where there are 12 pentagonal seas on the planet, but they still had maps that showed them as roughly pentagonal, which just leaves me curious about what kind of projection this would likely be. It does at least say that there is some distortion, as it only says they are "roughly correct", Which was the source of my question. – Zoey Jan 09 '23 at 17:46
  • I thought of one more morsel of food for thought. There are multitudes of people on this site who have the tools for starting a broad investigation into your questions about projections, those tools being (as far as I can see) not much more than 2-dimensional vector calculus and differential geometry. However, I doubt there is anyone who has *already* set out on such an investigation. This site is not well adapted to questions about broad investigations, but it would be a good place for specific thorny questions that might arise once one has set out. – Lee Mosher Jan 09 '23 at 21:51