When will euclidean geometry become ?

In summary, the conversation discusses the application and limitations of Euclidean geometry in real-world scenarios. It is pointed out that Euclidean geometry works well on small scales, such as building a house, but breaks down on larger scales due to the curvature of the Earth. Spherical geometry is suggested as an alternative for constructing on a large scale. Examples of structures that use spherical geometry are mentioned, such as microwave towers. The conversation also touches on the importance of teaching geometry and the inclusion of proofs in education systems. It is noted that while proofs may not be commonly taught, they are still a valuable aspect of geometry and can be seen in various applications, such as the surface area of a sphere.
  • #1
The mentalist
5
0
Hello everybody,
Based of some information that I recently learnt(which I don't know if they are right or wrong), I start asking myself this question about the euclidean geometry.
Ok, this geometry is basically founded on straight lines, and what I have learned is there is no such a thing as a straight line in our planet.So.alot of human activities is based on this geometry,and it does really work after all but only for our eyes , so there must be some mistakes but very little one that we can't observe,because they very tiny .So ,my question is , when will the euclidean geometry become useless ? I mean if it is making some little mistakes some where ,then there must be cases which the mistakes can no longer be hidden
Or I am just talking randomly and all this question is based on mistaken information, please do enlighten me.
Thanks in advance.
 
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  • #2
Of course, we use Euclidean geometry where it's applicable. And that is: on small scales. So if we want to build a house, then our notions of parallel and perpendicular work, because the scales are so small.

However, if we start to go to larger scales (like: larger distances), then Euclidean geometry breaks down. A classical and historical example is for a ship to find the shortest distance between two points. Here, the curvature of the Earth comes into play and things are much more difficult than in Euclidean geometry.
 
  • #3
micromass said:
Of course, we use Euclidean geometry where it's applicable. And that is: on small scales. So if we want to build a house, then our notions of parallel and perpendicular work, because the scales are so small.

However, if we start to go to larger scales (like: larger distances), then Euclidean geometry breaks down. A classical and historical example is for a ship to find the shortest distance between two points. Here, the curvature of the Earth comes into play and things are much more difficult than in Euclidean geometry.

Ok, so let's assme that we want to build on very large scale ,what will we use in the place of euclidean geometry,since it will break down ?
 
  • #4
I guess we would use some kind of spherical geometry. Do you have any concrete example of such a structure?
 
  • #5
micromass said:
I guess we would use some kind of spherical geometry. Do you have any concrete example of such a structure?
No,I am just quoting your words .Ok ,so would you give some exemples on constructions work based on spherical geometry.
 
  • #6
The mentalist said:
No,I am just quoting your words .Ok ,so would you give some exemples on constructions work based on spherical geometry.

I don't know any. But I'm pretty ignorant of these things. You should ask in the engineering forums.
 
  • #7
An example is microwave towers. How do you know where to point the microwave dish? You can't just point it parallel to the ground, it won't find the other tower.
 
  • #8
verty said:
An example is microwave towers. How do you know where to point the microwave dish? You can't just point it parallel to the ground, it won't find the other tower.
Microwave towers are line of sight. Theoretically they are pointed parallel to the tangent of the midpoint curve between the two towers.
 
  • #9
Euclidean Geometry's usefulness is not its application to real world solutions, necessarily. It's beauty lies in the mathematical structure it created and the concept of proof. The rigor of the geometric proof, sadly, is not taught anymore in schools. Just because the politics in control of schools see no use for it does not mean it is not a worthwhile endeavor. It is a great foundation for all things mathematical.
 
  • #10
There is nothing wrong with the "math" in the examples in #2 an #7, but in my view they are both Euclidean geometry. Euclid's "Elements" is not just about "straight lines". The second book (out of 13) is mostly about circles, and it progresses to 3-D geometry.

Engineers use Euclidean geometry every day in situations that are much mode complicated than just the surface of a sphere (i.e. the earth, in the two examples).

IMO the only applications of non-Euclidean geometry in Physics would involve special or general relativity.

I think the OP (and others) may be confusing "Euclidean geometry" with "geometry of a two dimensional plane".
 
  • #11
The rigor of the geometric proof, sadly, is not taught anymore in schools. Just because the politics in control of schools see no use for it does not mean it is not a worthwhile endeavor.

That is not universally true everywhere. Some education secondary (and college) systems teach Geometry and with proofs included as a big part of instruction.
 
  • #12
symbolipoint said:
That is not universally true everywhere. Some education secondary (and college) systems teach Geometry and with proofs included as a big part of instruction.

I would love to know where. Everything I have seen uses either Algebra, or Cartesian coordinates as part of the curriculum.

Geometry was refined by Hilbert to those essential Postulates and Theorems necessary for geometric proofs. In all of the geometric proofs numbers never appear as the measure of anything. Only references to right angles or n times an object. as in surface of a sphere is 4 times that of a great circle.
 
  • #13
coolul007 said:
I would love to know where. Everything I have seen uses either Algebra, or Cartesian coordinates as part of the curriculum.

Geometry was refined by Hilbert to those essential Postulates and Theorems necessary for geometric proofs. In all of the geometric proofs numbers never appear as the measure of anything. Only references to right angles or n times an object. as in surface of a sphere is 4 times that of a great circle.

But... many proofs are easier with coordinates, especially those with sides in some ratio, those with midpoints, those relying on gradients. Some are just down-right infeasible in purely synthetic form, hence why you see geometry problems so often in math competitions.
 

1. When will euclidean geometry become obsolete?

Euclidean geometry is based on a set of axioms and principles that have been used for centuries and are still widely applicable in many areas of mathematics and science. While there have been advancements in non-Euclidean geometries, Euclidean geometry is still considered a fundamental and useful tool in many fields. Therefore, it is unlikely that it will become completely obsolete in the near future.

2. Will euclidean geometry ever be replaced by non-Euclidean geometries?

Non-Euclidean geometries, such as hyperbolic geometry and elliptic geometry, have been developed as alternatives to Euclidean geometry. However, each geometry has its own set of axioms and principles that can be applied to different situations. Euclidean geometry is still widely used and considered a valuable tool in many areas, so it is unlikely that it will be completely replaced by non-Euclidean geometries.

3. Can we prove that euclidean geometry is true?

Euclidean geometry is based on a set of axioms and principles that are accepted as true without proof. These axioms are used to build logical arguments and proofs in geometry. While we cannot prove that the axioms themselves are true, we can use them to prove theorems and results in Euclidean geometry.

4. Are there any limitations to euclidean geometry?

Euclidean geometry has been successfully used to describe and understand the physical world for centuries. However, it does have limitations, particularly when dealing with very large or very small scales, or in non-Euclidean situations. In these cases, other geometries may be more appropriate.

5. Why is euclidean geometry important?

Euclidean geometry is the foundation of many other mathematical and scientific fields, such as physics, engineering, and computer graphics. It provides a framework for understanding and describing the physical world and has practical applications in various areas of our lives. Additionally, studying Euclidean geometry can help develop critical thinking and problem-solving skills.

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