Define hilbert:[0,1] → [0,1] ⨉[0, i] be the Hilbert Curve function. E.g., In[58]:= #[[1]] & /@ hilbert /@ Range[0, 1, 1/6] Out[58]= {0, 1/2 + I/2, I, 1/2 + I/2, 1 + I, 1/2 + I/2, 1} (corner,middle,corner,middle,corner,middle,corner) (But the Hilbert function is dense with *quadruple* points! In[233]:= unbert[1/2 + I/4] Out[233]= {5/48, 7/48, 41/48, 43/48} In[234]:= hilbert /@ % Out[234]= {{1/2 + I/4}, {1/2 + I/4}, {1/2 + I/4}, {1/2 + I/4}} ) Then, empirically, Out[38]= hilbert[1/2 EllipticTheta[3,0,1/4]] == 2+I/2-1/2 EllipticTheta[3,0,1/2] (Hilbert meets Jacobi.) In[49]:= $RecursionLimit=2500;%38/.R:EllipticTheta[3,0,_]:>Round[R,2^-999] Out[49]= 46727909977607706913365982377862004313596030844679844841051801456313857598476681679840123330654943383021969653600824586099255880151054521703858175/49947976805055875702105555676690660891977570282639538413746511354005947821116249921924897649015871538557230897942505966327167610868612564900642816+I/2== 18234098153986631008405986432405557662078277564803117560901382754419566447242273596263185647274799844525865896201189350917120311482141620852228639269436148149700137184604665725070440348534903185396966774064837413388299772999715897332003677006757173294606789913563827485196972130849897054207/19490628022799998160706764775750376621752453715190015053735812914425897381532852204931230131764020518450609832462817336366918339730406188093155974592625306839062555399912946059741579310980107296705599186958436757747371195850789749891492727230937931225655477606208555094163657179983828221952+I/2 In[50]:= N[Subtract@@%,9] Out[50]= 5.69026240*10^-160 In[51]:= N[%49,160] Out[51]= 0.9355315863940614206652707257275243376937417300295609535554944039715198668204215933408693598375308515525975283883147938845745807397312533384005722661851704410260+0.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000I== 0.9355315863940614206652707257275243376937417300295609535554944039715198668204215933408693598375308515525975283883147938845745807397312533384005722661851704410254+0.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 I