On Sat, Aug 8, 2020 at 2:07 AM Bill Gosper <billgosper@gmail.com> wrote:
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
Furthermore, In[163]:= Re[hilbert[EllipticTheta[3, 0, 1/4] - 1]] == Im[hilbert[EllipticTheta[3, 0, 1/4] - 1]] In[169]:= %163 /. Equal -> Inactive@Equal /.T : _[3, 0, q_] :> Round[T, 2^-691] // N[#, 99] & Out[169]= 0.564468413605938579334729274272475662306258269970439046444505596028480133179578406659130640162469148== 0.564468413605938579334729274272475662306258269970439046444505596028480133179578406659130640162469148 Furthermore, Out[165]= 2 Re[hilbert[EllipticTheta[3,0,1/4] -1]]==EllipticTheta[3,0,1/2] - 1 In[170]:= %165/.T:EllipticTheta[3,0,q_]:>Round[T,2^-691] Out[170]= 19291155418386032923976523898952230335209822947123972593482103762825322166697992580548005399101441/17087896287367280659160173649356416916821636178853222159576332862577757806245124400183696695492608== 176976953011670941809141181908675150324088333548023139766658020760156234157287878168592948466917627088529068875077124932719130590886008368402908435838005525299617304740133977367092117568827971628334841857/156764265941034957982331212844852467344711417043899710759469297619722251722129607859661177881884230709880082871203965476543290384119266386721367084105368877945996036265148061460008137163052639879920877568 In[171]:= N[%,99] Out[171]= 1.12893682721187715866945854854495132461251653994087809288901119205696026635915681331826128032493830== 1.12893682721187715866945854854495132461251653994087809288901119205696026635915681331826128032493830 —rwg