FOTD -- September 27, 2004 (Rating 4) Fractal visionaries and enthusiasts: Today's fractal gives us a new view of a midget lying in the Seahorse Valley area of the Mandelbrot set. This valley, which separates the main bay from the largest bud, is perhaps the best-known feature of the Set. When its aspect in the M-set alone is considered, Seahorse Valley appears as two two-dimen- sional wedges that approach but never quite reach the X-axis. But the Mandelbrot set is but a single slice of a monstrous, four-dimensional abstraction known as the Julibrot, and Seahorse Valley itself is but a single slice of a much broader four-dimen- sional part of the Julibrot. Being a 4-D object, the totality of Seahorse Valley cannot be visualized, but it can be discussed. In the M-set, the valley terminates in two sharp points. The M-set lacks two dimensions of the Julibrot however. When one dimension is added, the valley may be visualized as terminating in two sharp edges that approach each other but never quite touch. This much can be easily visualized. But when still another dimension is added, the valley must be seen as terminating in two sharp surfaces, which could extend indefinitely in the plane of their two dimen- sions and still be as sharp as a razor at every point. The idea makes no sense. A flat surface is a two-dimensional wall that cannot be sharp. This is true enough in three-dimen- sional space, but in four-dimensional space, a 2-D surface with no extent in the remaining two dimensions is a razor edge that could slice cleanly through any soft 4-D object it touches. These words are easy enough to understand, but like so many other features of 4-D space, the thing they describe cannot be visualized. Curiously enough, a properly programmed computer would have no trouble moving and turning 4-D objects in 4-D space, but even when it did so, we would see only morphing and turning 3-D shapes on our flat 2-D screens. Is the inability to visualize four-dimensional objects a limitation of our 3-D visual apparatus, is it a limitation of our minds, or are we merely unable to imagine an abstraction that does not physically exist? Regardless, today's image is easily viewable. I named it "Seahorse Scene" because, even though it is sliced at an unimag- inable angle, the scene is still located in Seahorse Valley. I rated it at a 4 because I can see little in it that is worth more. When the render time of 10-1/4 minutes is considered, the overall value equals 39. All this rating stuff can be avoided by visiting the FOTD web site and downloading the finished image from there. The FOTD web site may be accessed at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Continuing near perfect weather here at Fractal Central on both Saturday and Sunday kept the fractal cats quite happy. They spent the better part of both days lounging in the yard, sleeping and remembering the days when they still had their kittenish enthusiasm. Today is starting acceptably well. Tomorrow promises rain, so the duo had best enjoy themselves as much as possible today. For me, things are nearly caught up. Unless something unexpected happens, I should be able to return the FOTD to its one-a-day schedule on October 1. And I might even have time for some deep philosophical discussion. The next FOTD fractal will appear on Sep 29. Until then, take care, and search for the entrance to the fourth dimension. Jim Muth jamth@mindspring.com jimmuth@aol.com START 20.0 PAR-FORMULA FILE================================ Seahorse_Scene { ; time=0:10:14.17--SF5 on a P200 reset=2003 type=formula formulafile=allinone.frm formulaname=multirot-XZ-YW-new passes=1 center-mag=+0.00000000000102891/+0.000000000000197\ 56/6.53521e+011/0.02319/-9.79555128726789448/-85.9\ 814495713471274 params=42/147/2/0/-0.7475607140431\ 548/0.1243880653191712/-0.7475607140431548/0.12438\ 80653191712 float=y maxiter=4200 inside=0 logmap=1146 periodicity=10 colors=000qeYqfXrgWrhVsiUtjUsiTshSrgRrgQqfPqeOpeNp\ dMocLocKnbJnaImaHm`Gl_Fl_EkZDkYCjYBjXAiW9iW8hV7hU6\ gU5gT4fS3fS2eR1eQ0eQ0fP4fO8fNCgMGgLKgKOhJShIWhH_iG\ ciFgiEkjDojCskAzjCwjEujGsjIqjKojLmjNkjPijRgiTeiVci\ WaiY_i_YiaVicTieRifPihNhjLhlJhnHhpFhqDhsBhu9hw7hy5\ hz3ix4iv4iu4is4jr4jp4jo4jm4kl4kj4ki4kg4le5ld5lb5la\ 5m_5mZ5mX5mW5nU5nT5nR5nQ5mP6lP7kP7jP8iP9iP9hPAgPBf\ PBePCePCdPDcPEbPEaPF`PG`PG_PHZPIYPIXPJXPJVOKUNLSML\ RLMPKNOJNMIOLHOJGPIFQGEQFDRDCSCBSAAT9ATADRAFQAHOBK\ NBMLBOKCRICTHCVFCXED_DDaBDcAEf8Eh7Ej5Fm4Fo2Fq1Fs0D\ l5BeAAZF8SK6LP5EU37Z21c96WFAPLFHRJAVO2XN3YM3ZL3_K4\ aJ4bI4cH5dG5eF5gE6hD6iC6jB7kA7m97n88o78p68q58p8BoB\ DnEFmHHlJJkMLjPNiSPhURgXUf_WebYde_cgabjcame`pg`ri_\ jk_cmZXnZQpUHsZJqcKphLomMnqNmpMloMlnLknLkmKjlKjlKi\ kJijJhiIhiIhhHggHggHffGfeGedFedFdcFdbEcbEcaDc`Db_C\ b_CaZCaYB`YB`XA_WA_b1dWA_ } frm:multirot-XZ-YW-new {; Jim Muth ; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect e=exp(flip(real(p1*.01745329251994))), f=exp(flip(imag(p1*.01745329251994))), z=f*real(pixel)+p3, c=e*imag(pixel)+p4: z=z^(p2)+c, |z| <= 36 } END 20.0 PAR-FORMULA FILE==================================