I did some more exploring in the Bristling Energy fractal. The name Two Left Turns came from the process I used to find the image. I made two left turns when I chose the path to zoom into. The two left turns added the two octagons to the fractal.

The parameter file used to make the top image is below:

Two_Left_Turns { ; Exported from Fracton.
 ; time = 5 min 42 sec at 640 x 480 with no aa
 ; on a Mac Pro 2 x 2.8 Ghz Quad-Core Xeon
 reset=2004 type=mandel passes=1 float=y
 center-mag=-0.126245138931413858672422416646500180\
 6066151164938915275145862382678838012749e1/0.40833\
 58177658692924030543625981883443850512333421119460\
 012321903238149605257/7.58518528e+65/1/0/0
 params=0/0/0/0/0/0/0/0/0/0 maxiter=12000
 inside=135 logmap=6951 proximity=0
 colors=80G80G80G80G80H80H80I81I71I71J71J72J72J72K6\
 2K63K63L63L63L63M54M54M54M54N55N55N45O45O46O46P46P\
 46P37P37Q37Q37Q37R38R28R28S28S29S29S29T19T1AT1AU1A\
 U0BV1CV2DW4EW5FX6GX7HY9IYAJZBKZCL_EM_FN`GO`HPaJQaK\
 SbLTbNUcOVcPWdQXdSYeTZeU_fV`fXagYbgZch_dhaeibfibfi\
 cgjdhkfikgjlhkljlmkmmlnnnpnoqoprorspqropqnopmnnllm\
 kkljjkiiighhffgeefdddcccbaba`a`__ZZZYYYXWXWVVVUUUT\
 TTRRRSSSUUUVVVXXXYYY___aaabbbdddeeeggghhhjjjkkkmmm\
 <9>zzx<9>wrg<2>ujS<4>q`9<5>lT0<5>fM1<4>`F2<3>U83<1\
 >O13ujS<1>q_9<2>lT0<2>fM1<2>_F2<2>U83O13ujSq_9<1>k\
 T0<1>fM1<1>_F2<1>U83O13ujS<1>q_9<1>lT0<1>fM1<1>_F2\
 <1>U83<1>O13<3>ujS<6>q_9<3>kT0<3>fM1<3>_F2<3>U83<3\
 >O13<2>zzz }