Invariance of length

Question

Invariance of interval in Minkowski space under coordinate transformation was proved by the postulates of special relativity. (https://physics.stackexchange.com/a/453536/213658 .see this answer) Is there any theory or postulates which proves that length is an invariant quantity in Euclidean space?

Answer 1

I'm not sure I fully understand you still, but the Galilean transformation is (for transformation in one dimension, I'll leave multiple dimensions to you): $$x'=x-vt$$ $$y'=y$$ $$z'=z$$ $$t'=t$$

So the spatial length between two points $(x_1',y_1',z_1')$ and $(x_2',y_2',z_2')$ is given by $$L=\sqrt{(x_2'-x_1')^2+(y_2'-y_1')^2+(z_2'-z_1')^2}$$ Applying the transformation rules above: $$L=\sqrt{((x_2-vt)-(x_1-vt))^2+(y_2-y_1)^2+(z_2-z_1)^2}$$ $$L=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$

Which is the length between the unprimed coordinates. Therefore, the length is invariant under Galilean transformation.