From: Larrie Lane
The reason for setting the repeats to 4.00 for a cylinder or tube is that it is in theory a cube with 6 sides.
I'm not sure where you're getting that, Larrie. Cylinders in SL do not use cubical topology. They use honest to goodness cylindrical topology, meaning one stitched side and two endcaps (24 facets each, just to be complete about it). While it's certainly possible, and in some cases even preferable, to make a sphere with cubical topology (six sides), that's not how SL does it.
That said, let's talk about why it actually is that the 4:1 ratio happens to work well. It has to do with simple geometric math, in combination with the power-of-two requirements for textures in OpenGL.
The math part is easy. We all know that the circumference of a circle (or cylinder) is Pi times its diameter. So, in order for a square texture to remain perfectly square as it wraps around a cylinder, it would need to repeat Pi times. But there's an obvious problem with that. Since Pi is not a whole number, the texture won't tile wholly at Pi repeats. In order to make it to tile perfectly, you'd want to round it down to 3 horizontal repeats. It is actually three repeats, not four, that is the optimal number for the least possible stretching of a square texture on a cylindrical side.
It's pretty easy to see this, just by putting an image of a circle in your texture. Repeat the texture 4 times, and you'll see the circle squish horizontally by about 33%, to become an oval. But repeat it just 3 times, and the circle will look almost perfect. It is stretched horizontally, just slightly, by the missing Pi decimals, if you want to get technical about it, but that's hardly noticeable. It's as close to perfect as you're going to get with whole numbers.
OK, so if three is optimal, how come textures with a canvas size ratio of 4:1 are so often used? Well, here's where we get into OpenGL's limitations. Textures must be measurable in powers of two. There's no such thing as 3:1 sized texture that can be measured in powers of two in both directions. The closest you can get is 4:1. If you want your texture to repeat just once as it wraps the cylinder, the best canvas size ratio you can use, even though it's far from ideal, is 4:1.
There is an even better solution. Mount a Pi:1 sized texture on a 4:1 sized canvas, and then repeat it 0.785 times horizontally. The unused portion of the canvas will be hidden; only the Pi:1 sized portion will show. You do waste a few pixels that way, but if you want a perfectly sized texture for a cylinder, that's the only way to do it.
For an example, let's say your canvas is 512x128. The subject area would be 402x128, and the remainder of the canvas could then just be left blank. When you apply the texture to your cylinder, set the horizontal repeats to 0.785, and just the subject area will show. The blank padding area will disappear, and the texture will appear to be perfect on the cylinder.
In that particular example, you'd have 41 kilobytes worth of wasted texture memory from the hidden pixels. You'd have to do it 25 times, with 25 separate textures, to waste a single megabyte. The waste is pretty minimal.
Bottom line, the 4:1 ratio has very little to do with the actual topology of the cylinder. I don't recommend using it. If you're repeating a square texture, 3:1, not 4:1, is what will keep it as square as possible. If you're using just a single repeat, then a Pi:1 sized image, mounted on a 4:1 sized canvas, is best.
From: Larrie Lane
As you start reducing the size of your prim you will need to reduce the repeats. So for example, if you make your cylinder or tube 5x5x5 then the repeats would have to be 2.000 horizontally and 0.500 vertically. Why, is because you have just halved your prim size and therefore you must half the texture size using the repeats. If you were to apply this to a cube then the repeats would be 0.500 both Horizontal and vertical.
Now you've lost me, Larrie. If you're talking about repeats per face, then the physical size of the surface(s) is irrelevant. A face is a face is a face.
If you're working with repeats per meter, then sure, the repeat values would change as the surface size changes. But why would you use repeats per meter on a cylinder side? RPM works best for planar surfaces. There's not generally much call to bother with it on a non-planar surface like the side of a cylinder.
Usually, all you'll be concerned with is how many times the texture repeats as it makes its way around the circumference. Sometimes, you'll also care about how many times it repeats height-wise. But almost never would you need to care how many of those repeats happen to fit into a meter.
From: Larrie Lane
If you are still having problems then try posting a picture of your texture on the forum with the prim type and size or try contacting me inworld.
While it's very generous of you to offer your in-world time, I must say, I really hate it when people go that direction. When a question is asked here, but not fully answered here, a great many people lose out.
First, every individual who is actively involved in the discussion deserves to be able to follow it through to its full conclusion. When you pull the discussion off the forum, those people are shut out. Second, for every one person that posts a question on the forum, there are always countless others with the same question who don't post. Some people, for whatever reason, just read and don't ever speak up. Just because those people are silent doesn't mean they don't still deserve the same benefit of learning. Answering a discussion in-world helps only one person, but answering it here helps lots of people simultaneously.
So For everyone's sake, it's always best to keep the discussion on the forum if that's where it started.