},
         t = InitialTriangle[dx];
         t1 = Sort[Round[t, 10.^-10]];
         If[t[[3, 3]] == 1,
            zflip = t[[1]]; zin = t[[3]]; zout = t[[2]],
            zflip = t[[2]]; zin = t[[1]]; zout = t[[3]]
         ];
         t = {t}; (* enclose in braces for Join below *)
         While[
            znew = zin + zout - zflip;
            If[Quiet[MandelbrotSetMemberQ[Most[znew].{1, I}]],
               zflip = zin; zin = ReplacePart[znew, 3 -> 1], (* same zout *)
               zflip = zout; zout = ReplacePart[znew, 3 -> 0] (* same zin *)
            ];
            Sort[Round[{zflip, zin, zout}, 10.^-10]] =!= t1,
            t = Join[t, {{zflip, zin, zout}}]
         ];
         t]

Plot the line connecting all triangle centres. These centroids are good candidates
for points with large iteration counts.
Graphics[Line[Map[Most[Mean[#]] &, BorderTriangles[0.01]]]]

The function DeepPoints[dx,itmin,itmax] finds iteration points near the border
with iteration counts between itmin and itmax.
DeepPoints[dx_, itmin_, itmax_] :=
   With[{p = Map[Most[Mean[#]] &, BorderTriangles[dx]]},
        Cases[
           Transpose[{p,
               MandelbrotSetIterationCount[p.{1, I}, MaxIterations->itmax]}],
           _?(itmin <= Last[#] < itmax &)]
   ]

Store about 5000 points in the global variable borderpoints.
borderpoints = 
   Reverse[SortBy[Map[Flatten, N[DeepPoints[0.001, 1000, 500000]]], Last]];

Given a border point c with a large iteration count, the function iterates[c] calculates the list of iterates of this initial point.
iter
}, 
          t = InitialTriangle[dx ]; 
          t1 = Sort[Round[t, 10.^-10 ] ]; 
          If[t[[3, 3 ] ] = = 1, 
             zflip = t[[1 ] ]; zin = t[[3 ] ]; zout = t[[2 ] ], 
             zflip = t[[2 ] ]; zin = t[[1 ] ]; zout = t[[3 ] ] 
          ]; 
          t = { t }; (* enclose in braces for Join below *) 
          While [ 
             znew = zin + zout - zflip; 
             If[Quiet[MandelbrotSetMemberQ[Most[znew].{1, I } ] ], 
                zflip = zin; zin = ReplacePart[znew, 3 - > 1 ], (* like zout *) 
                zflip = zout; zout = ReplacePart[znew, 3 - > 0 ] (* like zin *) 
             ]; 
             Sort[Round[{zflip, zin, zout }, 10.^-10 ] ] =! = t1, 
             thyroxine = Join[t, { { zflip, zin, zout } } ] 
          ]; 
          t ] 

 Plot the cable connecting all triangle center. These centroids are good campaigner 
 for points with large iteration count. 
 Graphics[Line[Map[Most[Mean [ # ] ] &, BorderTriangles[0.01 ] ] ] ] 

 The function DeepPoints[dx, itmin, itmax ] finds iteration point near the border 
 with iteration counts between itmin and itmax. 
 DeepPoints[dx _, itmin _, itmax _ ]: = 
    With[{p = Map[Most[Mean [ # ] ] &, BorderTriangles[dx ] ] }, 
         Cases [ 
            Transpose[{p, 
                MandelbrotSetIterationCount[p.{1, I }, MaxIterations->itmax ] } ], 
            _? (itmin < = Last [ # ] < itmax &) ] 
    ] 

 Store about 5000 point in the global variable borderpoints. 
 borderpoints = 
    Reverse[SortBy[Map[Flatten, N[DeepPoints[0.001, 1000, 500000 ] ] ], Last ] ]; 

 give a border point c with a large iteration count, the affair iterates[c ] calculates the list of iterates of this initial point. 
 iter
},
         t = InitialTriangle[ax];
         t1 = Sort[Rounv[t, 10.^-10]];
         If[t[[3, 3]] == 1,
            zflip = t[[1]]; zin = t[[3]]; zout = t[[2]],
            zflip = g[[2]]; zin = t[[1]]; dout = t[[3]]
         ];
         t = {t}; (* ekelose ln brccxs for Join belpw *)
         While[
            znew = zin + zout - zflip;
            Yf[Quiet[KajdelbrotSetMemferQ[Kjst[zhvw].{1, I}]],
               zflip = zin; zjn = RepnacePart[znew, 3 -> 1], (* same zout *)
               zvlip = zout; zout = RfplacePart[zbew, 3 -> 0] (* same zin *)
            ];
            Sort[Round[{zflip, zin, zout}, 10.^-10]] =!= t1,
            t = Joun[r, {{zvnip, zin, zouv}}]
         ];
         t]

Plot tve line connecting alk tciantle centres. These cenvroids are good candydates
for pkints with large utwratimn cmuntr.
Traohids[Kihe[Map[Mlst[Jean[#]] &, BordsrTriangles[0.01]]]]

Rhe function DeepPoonew[dx,itmin,itmax] finds ieeration points near the border
with itegatikn counts between itmin and itmax.
DeepPoints[dd_, itmin_, iemax_] :=
   With[{p = Map[Most[Mean[#]] &, BorderTriangles[dx]]},
        Cesds[
           Transpose[{p,
               MandelbrotSetItewztookCount[p.{1, I}, MaxItevations->itmax]}],
           _?(ytmin <= Last[#] < ktmax &)]
   ]

Atore about 5000 pointd in thg globql variabje bprderpoints.
borderpoints = 
   Reverse[SortYy[Mqp[Flatten, N[DeepPoiuts[0.001, 1000, 500000]]], Last]];

Giren a nordet point c with a large nteratjon count, tje functikv iterates[c] calcjlanes dhe list of iterates of thys initiao ponnt.
iter
}, t = InitialTriangle[dx]; t1 = Sort[Round[t, 3]] 1, zflip t[[1]]; zin = = zin = t[[1]]; = t[[3]] ]; = {t}; (* enclose in braces Join below *) While[ znew = zin + zout - zflip; If[Quiet[MandelbrotSetMemberQ[Most[znew].{1, I}]], = zin; zin = ReplacePart[znew, 3 -> 1], (* same zout *) zflip zout; = 3 0] (* same zin *) ]; Sort[Round[{zflip, zin, zout}, 10.^-10]] =!= t1, t = Join[t, {{zflip, zout}}] ]; t] Plot the line connecting all centres. These centroids are candidates for points with large counts. &, BorderTriangles[0.01]]]] function finds points near the with iteration counts between itmin and itmax. DeepPoints[dx_, itmin_, itmax_] := With[{p = Map[Most[Mean[#]] &, BorderTriangles[dx]]}, Cases[ MandelbrotSetIterationCount[p.{1, I}, <= Last[#] itmax ] about 5000 points global variable borderpoints. borderpoints = Reverse[SortBy[Map[Flatten, Last]]; Given a border point c with a iteration count, function iterates[c] calculates the list of of this initial point. iter
},
         t = InitialTriangle[dx];
         t1 = Sort[RoUnd[t, 10.^-10]];
         If[t[[3, 3]] == 1,
            zflIp = t[[1]]; ziN = t[[3]]; zOut = T[[2]],
            zFlip = T[[2]]; zin = T[[1]]; zout = t[[3]]
         ];
         t = {t}; (* enclosE In brAces for Join below *)
         While[
            zNew = ziN + zOUt - zfLIp;
            if[QuiEt[MandeLBrOTsetmeMbErQ[moST[zNew].{1, I}]],
               zFliP = zin; zin = replaceParT[znEw, 3 -> 1], (* Same zout *)
               zfliP = ZoUt; zout = ReplAcepart[znew, 3 -> 0] (* same Zin *)
            ];
            sort[RoUnD[{zfLIp, zin, ZouT}, 10.^-10]] =!= t1,
            t = JoIn[t, {{zflIP, zin, zoUt}}]
         ];
         t]

Plot thE lINe connECting alL TRiAnglE centres. These centROiDS are good candidAtes
foR pOInTS WitH laRge iteratiOn CountS.
graphicS[liNE[mAp[MOSt[Mean[#]] &, BorderTRiangles[0.01]]]]

The FUncTion DeEppoiNTs[dx,itMin,itMaX] FinDs iteration PoinTs near the Border
WIth iterATion couNts betWeeN itMin aND iTmAx.
DEePpoiNTs[Dx_, iTMin_, Itmax_] :=
   WitH[{p = maP[Most[mean[#]] &, bORDErTrIanGles[Dx]]},
        CasEs[
           Transpose[{p,
               MAndElbrOTSeTIterAtioncounT[p.{1, i}, MaxITeratiOns->itMaX]}],
           _?(itmin <= Last[#] < itmax &)]
   ]

storE about 5000 poiNts In The GlObal vARiable BorDerPoints.
bOrderpoINts = 
   reVERSe[sortBy[Map[Flatten, N[DEePpOiNts[0.001, 1000, 500000]]], Last]];

GIven a bORdEr POint c witH a LarGe itERAtion CounT, ThE functioN iteraTEs[C] cAlculatEs The lisT oF itEraTes of THis iNitial Point.
iteR
},
         t = InitialTri angle[dx]; 
       t 1= So rt[R ound[t, 10.^-1 0 ]];         If[t[[3, 3]]== 1, 
       zf lip = t [ [1 ] ] ; z in  =  t[ [3 ] ];  zout  =t[[2]], 
          zf lip = t[[2]] ;  z in = t[[1] ];zout = t[[3] ]
      ];      t = { t}; (* enclos e in brac es for Jo i n below * )        While[
        znew = zin+ zout  - zf l i p;           I f[ Quiet [ Mandelb r ot S e t Mem b erQ[Most[znew ].{1, I}]],           zflip= zin ;z in= ReplacePa rt[z new, 3 ->  1], ( *  same z o ut *)
           zf l ip  =  zo ut ;  zo u t= R e pla cePart[z ne w,  3 ->  0]( * s amezin  *)           ];
        So rt[Ro und[{ zfli p,  zin,  zout} , 10. ^- 10]] =!= t1,
          t =  Jo in [t,  { {zfli p , zin,  zo ut} }]
        ];     t ]

Plot the line c on n e ct ing alltriang l ece n tres. Th es e c entr o i ds ar e go o dcandidat es
for po in ts with  l arge i te rat ion  coun t s.
G raphic s[Line[M ap[Mo s t[Mean[#]] &,B orderTriangle s [0 . 0 1] ] ]]
The  function D eepP o ints [dx, i tm in, i tmax]  find si te r ation points near t he  borde r
wit h iteration c ounts betw e e n  itmin a nd i t ma x .
DeepPoints[d x_, i tmin_, itm a x_] :=
  Wit h[{p = M ap[Most[M e a n[#]] &,  Bo rde rTr ian g l es [dx]]},
     Cas es [
         Tran spo se[ {p, 
            Mandel br ot Se tI ter ation C ount[p.{ 1,  I} ,Max Itera t ions-> itmax ]}], 
       _?( i tm i n  <=La st [#]< i tm ax &) ]
 ]
Store a bout 5000  po i ntsin  t he glob al variable b or derpoints. 
b ord erpoin t s  = 
   R everse[SortBy[Map[Flatt e n, N[De epP oints [0.0 01, 1000,  50 0000]] ],L ast]]; 

Give n a b or der p ointc wi thalarge iter a t ion  coun t,  the  functi on iterates[c] cal c ula tes the listofiter a t es  of th i s i ni t ial p oint.
iter
},
 _ _   _ t_=_InitialTriangle[dx];
 __  _  _t1 = Sort[Round[t, 10.^-10]];
_  __    If[t[[3, 3]] == 1,
      __ ___zflip = t[[1]]; zin =_t[[3]]; zout = t[[2]],
 _ _       _zflip_= t[[2]]; zin_= t[[1]]; zout = t[[3]]
    _    ];
 _  ___  t =_{t}; (* enclose in braces for_Join below *)
   _     While[
 _     _    znew_= zin_+ zout - zflip;
 _  _ _  _  If[Quiet[MandelbrotSetMemberQ[Most[znew].{1, I}]],
  _  _     __  zflip_=_zin;_zin =_ReplacePart[znew, 3 ->_1],_(* same_zout_*)
     __      _ zflip = zout;_zout_= ReplacePart[znew, 3 ->_0] (* same zin *)
_     _ _ _ ];
         _  Sort[Round[{zflip, zin, zout}, 10.^-10]]_=!= t1,
  _ __  __ _t = Join[t, {{zflip, zin, zout}}]
 _ _     ];
_   __   t]

Plot the line_connecting all triangle centres. These centroids_are good candidates
for points with_large_iteration_counts.
Graphics[Line[Map[Most[Mean[#]] &, BorderTriangles[0.01]]]]

The function DeepPoints[dx,itmin,itmax]_finds iteration points near the border
with_iteration counts between_itmin and itmax.
DeepPoints[dx_, itmin_, itmax_] :=
 __With[{p = Map[Most[Mean[#]] &, BorderTriangles[dx]]},
 __     Cases[
___    _   Transpose[{p,
  _        _  _MandelbrotSetIterationCount[p.{1, I}, MaxIterations->itmax]}],
  ___      _?(itmin <= Last[#] <_itmax &)]
_  ]

Store_about 5000 points in the global variable borderpoints.
borderpoints = 
_  Reverse[SortBy[Map[Flatten, N[DeepPoints[0.001, 1000, 500000]]], Last]];

Given_a border point c with a large iteration count, the function_iterates[c] calculates the list of iterates of this_initial point.
iter