Point, line and plane are concepts that people can use to help describe what they see each day. With these concepts, as well as scale and volume, we can give meaning to what we see.
Points in design are different from points in geography. In the context of writing, each point is a letter character. However if you wanted to look as a single letter, it is also made up of lines. I find it really hard to see just one part of design at a time. This is because I like to see the picture that they make together. You can also justify this image as being a plane of text as it fills an large space. But to me it is best to look at this as points and then lines because this is how we will best understand the content write. Each point has a meaning that becomes clear when put into the context of a line.
Here I have a Tinkerbell on a card that I have made for a friend. Within this photo, you can see how many points put next to each other can give the appearance of being a line. In this case, glitter makes up most of the image and each piece of glitter can be a point or, if you look at the piece as a whole, the glitter can make up a plane with the black and gold each making up their own plane. If you back up the perspective to the entire card, you can even argue that the faerie is a point of interest on the card. There are so many possibilities for design and perspective to come into play. You really have to decide how you want to look at things within design.
When I am drawing, I focus heavily on these concepts. If you were to take away the colour making the image purely black lines, you would still be able to tell that there are trees in the foreground. However, there might be difficulty determining what is in the background. This image is made up of several planes and lines. This is where colour can give meaning to the planes.
No matter where you look, you will see these design elements. It is up to you how you want to interpret them.