- #1

Omega0

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- TL;DR Summary
- A set of two symmetric non-linear algebraic equations has an asymmetric solution. What is the deeper reason behind this:

Hello,

please let me split me split my question into 3 blocks. The first: The problem and the solution. The second: The question. The third: Maybe weird thoughts about about a similar problem.

$$ \begin{align*}

6^x+6^y &= 42 \\

x+y &= 3

\end{align*}

$$

This is solved by ##x=1, y=2## or ##y=2, x=1##.

First, it is 100% clear that if a solution for x and y is found, they need to be exchangeable. The problem is absolutely symmetric.

And this is what puzzles me. What is the intrinsic reason that the solution isn't symmetric, meaning ##x = y##? What mathematics is behind this?

Please note that the first equation is nonlinear. If the set of equations would be symmetric and linear, there wouldn't be any solution or an infinite amount of solutions.

Let me tell you about my thoughts: It remembers me around the Karman Vortex Street. With the difference that we have here only 2 points in the solution space. The problem to get the Karman Vortex Street is nevertheless perfectly symmetric,, mathematically. The equations are non-linear. If you simulate them you'll find out that it takes some time to get the oscillation - but it will come!

In nature it is totally clear: We have chaotic movement etc., no problem - but for mathematics, I thought we wouldn't have a broken symmetry for such a simple problem. The problem above appears to me like a broken symmetry.

Thanks for your thoughts!

Jens

please let me split me split my question into 3 blocks. The first: The problem and the solution. The second: The question. The third: Maybe weird thoughts about about a similar problem.

**The problem and the solution**$$ \begin{align*}

6^x+6^y &= 42 \\

x+y &= 3

\end{align*}

$$

This is solved by ##x=1, y=2## or ##y=2, x=1##.

**The questions I have**First, it is 100% clear that if a solution for x and y is found, they need to be exchangeable. The problem is absolutely symmetric.

And this is what puzzles me. What is the intrinsic reason that the solution isn't symmetric, meaning ##x = y##? What mathematics is behind this?

**My thoughts about the problem**Please note that the first equation is nonlinear. If the set of equations would be symmetric and linear, there wouldn't be any solution or an infinite amount of solutions.

Let me tell you about my thoughts: It remembers me around the Karman Vortex Street. With the difference that we have here only 2 points in the solution space. The problem to get the Karman Vortex Street is nevertheless perfectly symmetric,, mathematically. The equations are non-linear. If you simulate them you'll find out that it takes some time to get the oscillation - but it will come!

In nature it is totally clear: We have chaotic movement etc., no problem - but for mathematics, I thought we wouldn't have a broken symmetry for such a simple problem. The problem above appears to me like a broken symmetry.

Thanks for your thoughts!

Jens