Exploring Geometry Beyond the Physical World

In summary, euclidean geometry is a mathematical system that, like any other, does not perfectly describe the real world. In order to apply it to real world problems, one must select real world objects to correspond with the undefined terms of the system and show that the axioms still hold true. However, due to the imprecise nature of real world measurements, this can never be entirely accurate. Mathematicians often debate the meaning of "real world." It is easier to do geometry in lower dimensions because they are easier to visualize and work with compared to higher dimensions.
  • #1
roger
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I wanted to know if euclidean geometry is to do with the real world ?

generalisations of vector space to anything that satisfies the axioms for a vector space can be made, but how can geometry be studied without reference to the real world ?

roger
 
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  • #2
By doing what mathematics always does: Setting up axioms, "undefined" terms, and definitions and deriving whatever theorems can be proved from those.

In fact, Euclidean geometry, like any mathematical system, does NOT describe the "real world" perfectly. In order to apply any mathematical system to a "real world" problem, you have to select "real world" things to associate with the undefined terms. Then you have to show that the axioms are still true when referred to those "real world" things. But that's NEVER true. "Real world" things are subject to imprecise measurements. The best we can ever hope for is that the mathematical structure will approximately match the "real world".

By the way, the reason I keep putting "real world" in quotes is because I am not certain what you mean by it. In my "real world", I might well have to take a test on geometry next week!
 
  • #3
What do mathematicians mean by the phrase; '' has nothing to do with the real world'' ?

I don't know what is meant by real world either, but I expected most mathematicians to, since I've read about them debating such issues.

so for example why would it be easier for me to do geometry in 2d or 3d than 10d ?
 
  • #4
It would be easiest (and most trivial) to do geometry in 0d. I was about to say most pointless, but since 0d is a point, that wouldn't quite be accurate :tongue: It's probably easier to do geometry in 2d than 3d because 2d is a lot easier to draw, and it's easier to gain an intuition for either 2d or 3d than 10d, since our eyes can't see anything beyond 3d.
 

1. What is Euclidean geometry?

Euclidean geometry is a branch of mathematics that focuses on the study of shapes, sizes, and positions of objects in space. It is named after the ancient Greek mathematician Euclid, who is known as the "Father of Geometry". Euclidean geometry is based on a set of axioms and postulates, which are statements that are accepted as true without proof, and it is often referred to as the "geometry of flat space".

2. What are the fundamental concepts of Euclidean geometry?

The fundamental concepts of Euclidean geometry include points, lines, and planes. Points are considered to be infinitely small and have no dimensions. Lines are made up of an infinite number of points and have length but no width or depth. Planes are made up of an infinite number of lines and have length and width but no depth. These concepts form the basis of all geometric figures and relationships in Euclidean geometry.

3. What are the five postulates of Euclidean geometry?

The five postulates of Euclidean geometry are:

  1. Given any two points, a line can be drawn connecting them.
  2. Any finite line segment can be extended infinitely in both directions.
  3. Given any three non-collinear points, a unique circle can be drawn through them.
  4. All right angles are congruent.
  5. If a line intersects two other lines and forms interior angles on one side that add up to less than 180 degrees, then the two lines will eventually intersect on that side.

4. How is Euclidean geometry used in real life?

Euclidean geometry has many practical applications in daily life, particularly in the fields of architecture, engineering, and design. It is used to create and analyze geometric shapes and structures, such as buildings, bridges, and roads. Euclidean geometry is also important in navigation, as it helps us understand and calculate distances and angles. Additionally, it is used in computer graphics to create three-dimensional models and animations.

5. What are some key differences between Euclidean geometry and Non-Euclidean geometry?

One of the main differences between Euclidean geometry and Non-Euclidean geometry is that Euclidean geometry is based on the five postulates mentioned above, whereas Non-Euclidean geometry does not follow these postulates. Non-Euclidean geometry allows for the existence of curved or non-flat surfaces, whereas Euclidean geometry only deals with flat, two-dimensional surfaces. Non-Euclidean geometry also has different types of parallel lines and different ways of measuring angles and distances compared to Euclidean geometry.

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