Geometry

On definitions - Do definitions utilized in geometry "work" in any dimension? Is it OK if they don't? What do teachers, especially in the K-12 grade levels, need to know to create or use more universally applicable definitions?​​​

 

Here is an example: When students are initially taught about parallel lines, they are defined as lines which never touch. This definition works in two-dimensions but does not work for three or higher dimensions. Lines that are skew in three-dimensions do not touch but are not parallel.  A better definition would be parallel lines are always equidistant from each other. ​

 

This allows wonderful questions such as: 

  - Can curved lines be parallel?

  - Can shapes be parallel? If so, which ones?

  - What would parallel look like in the 4th or 5th dimension?

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What are other examples of definitions or "rules" which only work for one part of the math world?