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Property of the plane, need clarification 
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#1
Mar2812, 06:50 PM

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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. 


#2
Mar2912, 01:01 AM

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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.



#3
Mar2912, 07:44 AM

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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.



#4
Mar2912, 11:09 AM

P: 3

Property of the plane, need clarification
Thanks so much guys. A few more questions
"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? 


#5
Mar2912, 11:12 AM

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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 


#6
Mar2912, 11:15 AM

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I assumed you were older than 14, by your language.



#7
Apr112, 06:20 PM

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#8
Apr112, 09:51 PM

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"Things like spheres and cylinders" do NOT satisfy this unless you specify that they have the same radius!



#9
Apr212, 12:40 PM

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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 xyplane) 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 


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