Fractal visionaries and enthusiasts:

Fractals are filled with bottomless spirals.   They are probably the most common features in the world of fractals.   But how often have we seen a set of concentric fractal rings with a very definite bottom?   After viewing today's Seahorse Valley scene, we have seen such a set at least once.

But are these rings truly what they appear to be.   And for that matter, are the everyday fractal spirals truly what they appear to be.   The Mandelbrot set does not exist in isolation.   It is a particular slice down the center of a four-dimensional monstrosity known as the Z²+C Julibrot.   The elements in the Julibrot are 4-D objects that twist and turn in four-dimensions.   Those very attractive spirals are actually 2-D slices of 4-D things that cannot be visualized but most likely approximate nested hyperspheres.   Today's image is simply a slice in a different direction through one of these nested hyperspheres.

The disk at the center of today's image is the next inner shell of the nested hyperspheres, which has not been cut entirely through.   A bit of parameter tweaking will punch a hole in the center of the flat central disk, which will then become another ring.   I have not yet tried tweaking the parameters, though it should not be very hard to find the right ones.

 I added a bit of extra life to the scene by rendering it with the outside set to 'summ'.   The rating of a 7 is the lowest rating that brings satisfaction for a fractal well done.

The calculation time of 8-2/3 minutes is slightly slow.   A more efficient way of viewing the sliced nested hyperspheres is to surf out to the FOTD web site at:and view it at leisure there.

Solid clouds, steady light rain and a temperature of 61F 16C kept a strict damper on things here at Fractal Central on Friday.   The fractal cats spent the day catching up on their sleep, though they did have a brief moment of excitement when a bedraggled stray cat passed by their window.

My day was about average, as are most days.   Hopefully, tomorrow will be the same.   Unless we get involved in some kind of expedition on Saturday, the next FOTD will be posted in 24 hours.   Until next time, take care, and we know that an electron is a unit of energy, but is it a particle, a wave, a wave packet, or none of the above?


Jim Muth
jamth@mindspring.com
jimmuth@aol.com

START PARAMETER FILE=======================================

Seahorse_Valley-06  { ; time=0:08:42.28-SF5 on P4-2000
  reset=2004 type=formula formulafile=basic.frm
  formulaname=SliceJulibrot4 float=y inside=0 passes=1
  center-mag=+1.074127267031372/-0.1217313849462725/6\
  907.325/0.03441/-45.7342470354590063/-52.7861725848\
  204202 maxiter=2820 outside=summ periodicity=10
  params=90.1/46.3/90.9/91.4/-0.75/0/0/0/2/0
  colors=000bhgchecibdj`ejZekXflUflSgmQhnOhnMioJipHj\
  pFkqDkrAlr8ls6mt4nw0mt2lq3ko5kl6ji8ig9idBhbCg_DgXF\
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  KMIMLHNLHPLHQKGSKGTKGVJFWJFYJFZIE`IEaIEcHDdHDfHDgG\
  CiGCnF9jGCfHFbIHZJKVKNRLPNMSJNVFOXBP_8QaBSaEUaHVaJ\
  XaMYaP_aS`aUbaXca_eabexafaagD`fF_fGZeHYeJXdKWdLVcM\
  UcOTbPSbQRaSQaTP`UO`VN_XM_YLZZKZ_JYaIYbHXcGXeFWfEW\
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  TkOTkPTkQTkRTkTSkUSkVSkWSkXSkYSjZSj_Sj`SjaSjbSjcSj\
  dSjeSjfSjgSjfUkeVkdWkcXkbYkaZk`_l_`lZalYblXdlWelVf\
  lUgmThmSimRjmQkmPlmOmmNlnNlnMkoMkoMkoLjpLjpKipKiqK\
  iqJhqJhrIgrIgrIgsHfsHfsGetGetGetFduFduDbtFduGfuHhu\
  IiuKkuLmuMnvNpvPrvQsvRuvSwvTxvUuwVrwVoxWmxXjxXgyYd\
  yYbyZcv_dt_dr`ep`fmafkbgi }

frm:SliceJulibrot4   {; draws most slices of Julibrot
  pix=pixel, u=real(pix), v=imag(pix),
  a=pi*real(p1*0.0055555555555556),
  b=pi*imag(p1*0.0055555555555556),
  g=pi*real(p2*0.0055555555555556),
  d=pi*imag(p2*0.0055555555555556),
  ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
  sg=sin(g), cd=cos(d), sd=sin(d),
  p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
  q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd),
  r=u*sg+v*ca*sb*cg, s=v*sin(a),
  c=p+flip(q)+p3, z=r+flip(s)+p4:
  z=z^(p5)+c
  |z| <= 9 }

END PARAMETER FILE=========================================