Closest approximated rational triangle

In summary, there is no efficient way to determine the closest approximation to a triangle where all sides are rationally related. This is because there will always be another closer approximation between any two numbers. However, if the problem is redefined to only include natural numbers and a specific unit, then it may be possible to find a closest approximation using the formula a = x * unit, b = y * unit, c = z * unit.
  • #1
ktoz
171
12
Given 2 points on a plane, if you arbitrarily place a third, is there any way to determine the closest approximation to this triangle where all sides of the approximation are rationally related?

The only thing I can think of would be to draw a small circle around the third point that represents the acceptable search area for the third point of the approximation, but after that, I'm completely stuck.

Anyone know if there is a fairly efficient way to do this?

Thanks
 
Last edited:
Mathematics news on Phys.org
  • #2
No, because however close your approximation, there is another one which is closer. Between any two numbers (either real or rational) there are an infinite number of other rational numbers.
 
  • #3
AlephZero said:
No, because however close your approximation, there is another one which is closer. Between any two numbers (either real or rational) there are an infinite number of other rational numbers.

Hmmm. What if the problem is redefined to:

The closest approximation whose sides are related by:

x, y, z, n = element of natural numbers
x, y, z < max
unit = 1/n

a = x * unit
b = y * unit
c = z * unit
 
Last edited:

What is a closest approximated rational triangle?

A closest approximated rational triangle is a geometric shape with three sides where the lengths of the sides are represented by rational numbers, meaning they can be expressed as a ratio of two integers. It is the triangle that comes closest to a given shape or angle while still maintaining rational side lengths.

What is the importance of calculating the closest approximated rational triangle?

The closest approximated rational triangle is important in various fields such as architecture, engineering, and computer graphics. It allows for precise and accurate measurements and constructions, and can also be used for approximating and simplifying complex shapes and angles.

How is the closest approximated rational triangle calculated?

The closest approximated rational triangle is calculated by using the concept of continued fractions. The first step is to convert the given angle or shape into a continued fraction, then using a specific algorithm to find the closest rational approximation. This process can be done manually or with the help of computer programs.

What is the difference between a rational and an irrational triangle?

A rational triangle has sides that can be expressed as a ratio of two integers, while an irrational triangle has side lengths that cannot be expressed as a ratio of two integers. In other words, the side lengths of a rational triangle can be written as fractions, while the side lengths of an irrational triangle cannot.

Can the closest approximated rational triangle be used to calculate other geometric shapes?

Yes, the concept of the closest approximated rational triangle can be extended to other geometric shapes such as squares, rectangles, and polygons. This allows for precise and rational approximations of these shapes, which can be useful in various mathematical and practical applications.

Similar threads

MHB Prove Area of Equilateral Triangle
    • General Math
Replies
9
Views
2K
MHB Calculating Coordinates of Spherical Triangles
    • General Math
Replies
7
Views
3K
I Putting a point inside a triangle in 3D space
    • General Math
Replies
5
Views
2K
I Optimization problem with multiple outputs: impossible?
    • General Math
Replies
6
Views
1K
B Invariant under rotation: Banal, obvious, or noteworthy?
    • General Math
Replies
2
Views
1K
I Three inertial particles: Separating the twins from the paradox
    • Special and General Relativity
2
Replies
40
Views
2K
I with understanding Kepler's second law
    • Thermodynamics
Replies
9
Views
1K
Minimizing the friction of the end of a lever against the top of a moving piston
    • Mechanical Engineering
Replies
21
Views
1K
Exploring the Benefits of Ionocrafts and their Equilateral Triangle Shapes
    • Aerospace Engineering
Replies
2
Views
1K
Egg tossing crime -- From what location was the egg thrown?
    • Introductory Physics Homework Help
Replies
14
Views
2K

Hot Threads

Recent Insights

Back
Top