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Can someone explain how pi holds up when base 8 math is applied?

Pi is a quantity, a value that is independent of the way you represent it in symbols. Base 8 is simply a different way of representing a given value, in the same way that 4/6 and 6/9 are two different ways of representing the same value.

In the case of 4/6 and 6/9 there is an agreed canonical way of designating that value, namely 2/3. In the case of pi there is similarly an agreed canonical way of designating the value, namely the symbol π.

pi becomes 3.110375524210264302151423063050560067016321122011160210514763072002... in base 8 or, really, still pi

Pi is just our name for the ratio of a circle's circumference to its diameter regardless of the base.

This is why it doesn't make sense for an intelligent species not to know what 'pi' is. It's an important concept that pops up immediately in basic geometry and just keeps popping up over and over all the way to general relativity and beyond.

Yes, an intelligent species may not know pi but they would not have progressed even to the point of basic geometry in the maths and sciences.

Their approach to geometry could be quite different though. Eg. they could use non-Euclidean geometry by default, 2D only being a special/edge case, and Pi, while encountered in calculations once in a while, wouldn't have a status of a well-known constant: which I understand would be the minimum to meet the requirement behind the question. If not, the question would be akin to whether they "know" 1, hard to say what it means

Pi shows up either in Einstein's field equations, or in Newton's gravitational equations. It's probably possible to formulate them both without reference to pi, but off the top of my head I don't see how. (As mathematicians go, I'm not at all applied, though.)

Pi only shows up in the field equations multiplied with an arbitrary constant. Therefore they are trivial to formulate without.

The arbitrary constant is related to G, which is Newton's gravitational constant, and to the speed of light. If I've understood correctly, if you don't want pi to appear in the field equations, then it will appear in Newton's equations as part of G.

Well, G is an arbitrary constant but Pi is not. So while you can _formulate_ using a new arbitrary constant, it is HIGHLY improbable you reached tbe point of making that formulation without discovering Pi along the way.

It's really hard to do engineering without Euclidean geometry.

pi pops up in equations without using any numeric symbols. It might not necessarily be 3.14159... but the concept is still there

For example, pi is the smallest strictly-positive real such that sin(pi) = 0, where sin is defined by its power series. That is a base-independent definition.

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