I don't see how those results directly translate into the previously mentioned functions, it seems Trappmann already defined and graphed half-iterates of the exponential function, there should then be a way to plot half-iterates of heights of tetration. I suppose we should start with the half-exponential itself.
This value L, uncoincidentally is a reflection of the first fixed point of e^x at lambertw(-1) or otherwise log(ssrt(e^(-1))), http://www.wolframalpha.com/input/?i=lambertw(-1), L~=0.318131505 + 1.33723570I It also looks awfully close to the (maximum curvature) vertex of the e^x function itself. The star curves you presented also look very similar to the -1 branch of the real W function, though that is possibly a coincidence.
But more importantly, what does the half-exponential function actually look like in terms of functions? Trappman suggests something that looks like a basic composite of logs and exponentials, like you just exponentiate an iterated logarithm summed with a constant, yet I have not found a simplification that proves f(x) for f(f(x))=e^x.
This value L, uncoincidentally is a reflection of the first fixed point of e^x at lambertw(-1) or otherwise log(ssrt(e^(-1))), http://www.wolframalpha.com/input/?i=lambertw(-1), L~=0.318131505 + 1.33723570I It also looks awfully close to the (maximum curvature) vertex of the e^x function itself. The star curves you presented also look very similar to the -1 branch of the real W function, though that is possibly a coincidence.
But more importantly, what does the half-exponential function actually look like in terms of functions? Trappman suggests something that looks like a basic composite of logs and exponentials, like you just exponentiate an iterated logarithm summed with a constant, yet I have not found a simplification that proves f(x) for f(f(x))=e^x.