Regular embedding

Imagine you have a toy puzzle with different pieces that fit together perfectly. Each piece has a special shape that matches the others, and when you put them all together, they make a beautiful picture.

Now, imagine that you have a big piece of paper that you want to draw that same puzzle picture on. But you can't just stick the puzzle pieces onto the paper because they are 3-dimensional and would stick out. Instead, you have to draw each individual piece onto the paper so that they fit together perfectly like in the puzzle.

This is kind of like what regular embedding is all about. It's a way to take a complicated mathematical object, like a shape or a surface, and flatten it out onto a 2-dimensional surface, like a piece of paper or a computer screen. But you can't just squash it down like a pancake because that would ruin all the important details and relationships between the different parts of the object.

So instead, mathematicians use special formulas and techniques to carefully map each piece of the 3-dimensional object onto the 2-dimensional surface. They have to make sure that the shape stays intact and that all the different parts fit together properly, just like in the toy puzzle.

Regular embedding is really important for understanding and studying all sorts of complex mathematical concepts, from geometry to topology to algebra. It helps mathematicians visualize and work with these objects in new ways, allowing them to discover new patterns and connections that might not be immediately apparent in their original 3-dimensional form.