Is there a way to modify the built-in constant of pi in Rhino? After watching the following playlist, I would like to see what happens if pi is adjusted to π = 4 / (square root of φ)

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No, it is a constant.

Good [from my perspective] and the easiest way to change the number of places pi [or any value I imagine] is calculated out is : Tools>Units>Display precision correct?

No, that is for display. We don’t have different values of pi based on display precision.