How do you prove this statement in geometry?

  • Thread starter phoenixthoth
  • Start date
  • Tags
    Geometry
In summary, a polygon, polyhedra, and hyperspace with nonnegative area, volume, and measure, respectively, cannot be formed with fewer than 3, 4, and n points. This is because even a single point can be considered a polygon with zero area, and in higher dimensions, the concept of volume or measure may not exist. Additionally, it is possible to prove this statement using geometry by constructing a small prism based on the polygon formed by n-1 points, which will have a volume approaching zero as the length of the edge approaches zero.
  • #1
phoenixthoth
1,605
2
A polygon with nonnegative area cannot be formed with fewer than 3 points.
A polyhedra with nonnegative volume cannot be formed with fewer than 4 points.
A hyperspace with nonnegative measure cannot be formed with fewer than n points.

What I mean by "3 points" is that the cardinality of the set of vertices is 3.
 
Mathematics news on Phys.org
  • #2
phoenixthoth said:
A polygon with nonnegative area cannot be formed with fewer than 3 points.

Actually a single point is a polygon of zero area which is nonnegative.

When you say "prove this statement in geometry" are you saying "prove this statement, which has to do with geometry" (because a proof using linear algebra is pretty easy) or are you asking "prove this statement using geometry" which I think would be significantly harder
 
  • #3
phoenixthoth said:
A polygon with nonnegative area cannot be formed with fewer than 3 points.
A polyhedra with nonnegative volume cannot be formed with fewer than 4 points.
A hyperspace with nonnegative measure cannot be formed with fewer than n points.

What I mean by "3 points" is that the cardinality of the set of vertices is 3.

Take any n-1 points and build a small prism based on the polygon they create in a n-1-dimensional hyperplane (the polygon has to be completely inside the prism). Prism's volume is not more that S*h, where S is the "area"(n-1-dimensional) of the polygon and h is the length of the edge not coplanar with the hyperplane. Now it's clear that S*h tends to 0 as h tends to 0. By definition it means that the set of points of the polygon has "zero Lebesgue measure".
I hope you mentioned the word "cardinality" for no reason, because in an infinite-dimensional space volume as we're used to it (Lebesgue measure) doesn't exist.
 
  • #4
Sorry, I meant positive measure.
 

1. How do you use theorems to prove a statement in geometry?

To prove a statement in geometry, you can use a combination of theorems, definitions, and postulates. First, identify the given information and what you are trying to prove. Then, use the given information to construct a diagram and label any known measurements or angles. Next, use the definitions and postulates to make logical deductions and establish relationships between the given information and what you are trying to prove. Finally, use the appropriate theorem or postulate to formally prove the statement.

2. What is the difference between a proof and an explanation in geometry?

A proof is a logical and deductive series of steps that demonstrate the validity of a statement. It must follow the rules of geometry and use definitions, theorems, and postulates to arrive at a conclusion. An explanation, on the other hand, is a more informal way of showing why a statement is true. It often involves providing examples or visual aids to support the statement, but does not necessarily follow a strict logical structure.

3. How do you know if a proof is valid?

A valid proof in geometry must follow the rules of logic and use deductive reasoning based on definitions, theorems, and postulates. In addition, all statements and steps in the proof must be justified and supported by evidence from the given information or previously proven statements. It is important to double-check your proof for any errors or logical fallacies before considering it valid.

4. Can a statement be proven in multiple ways in geometry?

Yes, there can be multiple ways to prove a statement in geometry. Different proofs may use different combinations of theorems and postulates, or may approach the problem from a different angle. It is important to choose the most efficient and logical method of proof, but as long as the proof is valid and follows the rules of geometry, it can be considered correct.

5. What do you do if you cannot prove a statement in geometry?

If you cannot prove a statement in geometry, it may be because the statement is actually false. In this case, you can try to find a counterexample or a situation where the given information does not lead to the desired conclusion. If no counterexample can be found, then the statement may actually be true, but just difficult to prove. In this case, you may need to approach the problem from a different angle or try using different theorems or postulates to arrive at a proof.

Similar threads

I A way to comprehend the geometry of a hypercube
    • General Math
Replies
2
Views
288
I Chiral symmetries in E[SUP]^n[/SUP]
    • General Math
Replies
1
Views
993
B A Simpler Way to Find the Shaded Area?
    • General Math
Replies
1
Views
1K
B Before understanding theorems in elementary Euclidean plane geometry
    • General Math
Replies
12
Views
1K
I Higher Dimensional Spheres viz. Cubes
    • General Math
Replies
1
Views
355
A Non-Euclidean geometry help
    • General Math
Replies
3
Views
1K
MHB Proving Triangle Area ≤ $\frac{1}{2}$ in a Square with $(n+1)^2$ Points
    • General Math
Replies
1
Views
759
I Questions about algebraic curves and homogeneous polynomial equations
    • Differential Geometry
Replies
4
Views
1K
I What is the role of geometry and dimensional operations?
    • General Math
Replies
3
Views
254
B Validating Darts Tournament Skill Levels: A Non-Intrusive Approach
    • General Math
Replies
4
Views
656

Hot Threads

Recent Insights

Back
Top