Why bases 00), and converges to c, were c is the solution of $\vspace{15}{a^{a^c}}\,=\,c$, which seems to have 2 roots, c₁ and c₂, with $\vspace{15}{a^{c_1}\,=\,c_2}$ $\vspace{15}{a^{c_2}\,=\,c_1}$ $\vspace{15}{{c_1}^{c_1}\,=\,{c_2^{c_2}\,=\,{a^{c_1.c_2}}$ $\vspace{15}{a\,=\,{c_1}^{\frac{1}{c_2}}\,={c_2}^{\frac{1}{c_1}}}$ I suspect that this relation is the key to solve tetration equations: $\vspace{15}{a^{c_1}\,=\,a^{a^{c_2}}\,=\,a^{a^{a^{c_1}}}\,=\,...$ Here is tetration base a=0.01: c₁ = 0,941488369 c₂ = 0,013092521 The negative axis probably converges to a real function akin to a cosine. I need something better than excel. RE: Why bases 0