I am reading through Kiselev's Geometry: Book I. It is a plane geometry textbook and in the introduction it says the following "One can superimpose a plane on itself or any other plane in a way that takes one given point to any other given point, and this can also be done after flipping the plane upside down" Can anyone clarify what the part in bold means, or is saying? For some background, here are some definitions given in the text The part of space occupied by a physical object is called a geometric solid. A geometric solid is seperated from the surrounding space by a surface. The flat surface is the plane.
this means the group of isomorphisms of the plane is transitive, but do not get bogged down in the introduction to any book. keep reading.
Take two sheets of paper. Draw a dot on each one. You can always place one on top of the other so that the dots line up. It also works if you flip one or both of them over.
Thanks so much guys. A few more questions Do you have a more intuitive definition, maybe along the lines of what Tinyboss gave? After all, it is an elementary Geometry book meant for those in grades 7-9. That is what I was thinking, but then one of the questions at the end of the section gave me pause. It says "Give an example of a surface other than the plane which, like the plane, can be superimposed on itself in a way that takes any one point to any other given point." By that definition, wouldn't any surface qualify?
Things like spheres and infinitely long cylinders satisfy this property. Things that do not satisfy this property: cubes: you cannot superimpose a cube on itself such that a corner is matched up with the middle of one of the sides finite cylinders: you cannot superimpose a finite cylinder onto itself such that the center of one of the ends is matched up with a point on the side of the cylinder
I am having trouble understanding why this is the case. At first I thought that we could not do this for a cube because the corner is a point and thus it has no length, width, or thickness, and as a result the middle of the cube could not balance on the corner. However, doesn't the same problem arise with the finite plane? Haha, I am, but I am not that much smarter than a fourteen year old. Plus, I am teaching my thirteen year old stepbrother so I wanted it in language he could understand.
"Things like spheres and cylinders" do NOT satisfy this unless you specify that they have the same radius!
If you take a flat bounded surface, like a rectangle, it does not have the super-imposing property: you can't move the corner into the middle of the rectangle such that it all lines up. Only taking the whole plane allows you to do this always. Note that "finite planes" don't exist based on the definition of a plane in your OP - it has to be a surface which separates two 3 dimensional volumes. You can do this with an infinite plane (for example the xy-plane) but if you only allow finite surface area you can't cut all of space into two distinct parts without being able to go around your plane