Math Myth: Parallels do not intersect

  • I
  • Thread starter Greg Bernhardt
  • Start date
In summary, the article discusses the common misconception that parallel lines do not intersect, and explores the concept of parallel lines in different geometries. It suggests that the best definition for "parallel lines" should consider the context and purpose of the definition.
  • #1
19,572
10,377
From @fresh_42's Insight
https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

Have you ever stood on a railroad track? Nobody would deny that the rails are in parallel. Do they intersect? Certainly not soon because locomotives normally do not derail. However, you will likely have looked to the horizon while standing on the rails. And - surprise - they do intersect at the horizon, or mathematically: at infinity. But infinity on a ball where we live isn't at infinity. It is actually somewhere. We see that there are possible geometries, in which parallels do intersect.

parallels.png
 
Last edited:
Mathematics news on Phys.org
  • #2
High School Geometry is 99% Euclidean. And in that context, parallel lines do not intersect. If you are taking about railroad tracks on a spherical surface, technically they are not lines, but arcs following the spherical surface. If you make railroad tracks great circles, the will intersect about 6000 miles from their usable zone.
 
  • #3
Less controversial is that we should not define parallel lines to be a pair of lines that do not intersect - since this is the case in 3D Euclidean Geometry, which is a familiar topic. So what is the best definition for "parallel lines" from the viewpoint of making a definition that works well in all types of geometries?
 
  • #4
Stephen Tashi said:
Less controversial is that we should not define parallel lines to be a pair of lines that do not intersect - since this is the case in 3D Euclidean Geometry, which is a familiar topic. So what is the best definition for "parallel lines" from the viewpoint of making a definition that works well in all types of geometries?
It would depend on what you need the definition for.
 

FAQ: Math Myth: Parallels do not intersect

What is the "Math Myth: Parallels do not intersect"?

The "Math Myth: Parallels do not intersect" is a commonly misunderstood concept in mathematics that states that two parallel lines will never intersect, no matter how far they are extended.

Why is this considered a myth?

This is considered a myth because in reality, parallel lines do have a point of intersection, but it is at infinity. This means that they will never intersect within a finite space, but they do technically intersect at a point that is infinitely far away.

What is the significance of this myth?

This myth is significant because it can lead to misconceptions about the nature of parallel lines and their relationship to each other. It can also cause confusion when working with geometric proofs and constructions.

How can this myth be proven false?

This myth can be proven false by using mathematical proofs and logic. By definition, parallel lines are always equidistant and never intersect, but this does not mean they cannot have a point of intersection at infinity.

How does this myth affect our understanding of geometry?

This myth can affect our understanding of geometry by limiting our perspective on parallel lines and their properties. It can also hinder our ability to think critically and creatively when solving geometric problems.

Similar threads

Replies
25
Views
4K
Replies
1
Views
873
Replies
4
Views
1K
Replies
142
Views
8K
Replies
5
Views
1K
Replies
4
Views
1K
Replies
14
Views
2K
Replies
2
Views
3K
Replies
10
Views
2K
Back
Top