一个关于 π 的怪异公式A crank formula for π
作者发现了一个基于物理常数来计算圆周率 π 的怪异公式，并对其进行了深入分析。该公式涉及到一个在微波区域选择的波长 λ，作者最初以为可以通过调整这个波长值来使方程成立。但正如评论所指出的那样，方括号内的部分在数学推导上存在缺陷。最终证明，这个公式仅仅是一种牵强的数字巧合，缺乏真正的物理和数学意义。
John
I ran across a cranky formula for π based on physical constants here
and decided to play around with it.
The source describes λ as “wavelength (chosen in the microwave region)” and I thought perhaps you could chose a value of λ to make the equation work. But as a comment pointed out, the bracketed expression is simply 2hc², independent of λ, due to Planck’s blackbody law. That means we can simplify the expression above to
Now the values of h and c are known. In fact, they’re now exactly known by definition: other SI units are defined in terms of h and c. The mass of an electron is known to 11 significant figures.
But E in the equation above is “Total energy of the universe.” I don’t even know what that means. Does it refer to the observable universe? Does it include dark energy? Does it include the energy equivalent of mass?
I asked a couple LLMs that the total energy of the universe might mean and what its value might be, and they said something like “Depends. It might be zero. It might be infinite. But if I had to say, I’d say around 1070 Joules.”
If we solve the equation above for E we get 2.8480347886530404 × 1019 Joules. I have no idea how to justify that.
The expression for π is not dimensionless. I suppose you could choose some nonstandard units that would make the equation work.
The source I linked to above cites Mathematical Cranks by Underwood Dudley, but I couldn’t find it in the book.
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