Classic F.O.T.D., January 01, 1998 (Projective Plane) Fractal visionaries: The world of higher dimensions is filled with objects which are impossible in our three-space, and can be represented here in only a distorted way. The Klein Bottle is one such object. In three dimensions it appears as a closed figure which intersects itself and joins itself in such a manner that despite having no breaks it has only one side. The inside is also the outside. But this is a distortion of the true object, which can exist only in spaces of four or more dimensions. In four dimensions, the Klein Bottle is constructed by taking a rubber sheet, curling it and connecting one pair of edges so that a tube results, then bending the tube and joining the open ends into a doughnut shaped object. But before joining the edges, and with no cutting, the tube is given a half-twist and turned inside- out, so that without self-intersection, the resulting doughnut- shaped object has only one side. Its inside is also its outside. The Klein Bottle is difficult enough to visualize, but the Projective Plane is even more difficult. In fact it is difficult to even describe. In this case, the sheet of rubber is given a half-twist into a kind of Moebius Strip tube before being curled and given a second twist before the open ends are joined to each other, forming the Projective Plane. In this case, even a distorted model is nearly impossible in three-space. Well, if an accurate model of a Projective Plane is impossible in three dimensions, one could never hope to illustrate the monster on a two-dimensional screen. I named today's fractal "Projective Plane" only because that's what I thought of when I saw the image. Actually, it is a picture of a curious feature that appears at Z=0.00019,0.07388 C=-1.7435,0.0 in the Z^2.003 Julibrot figure. This object is extremely thin and exists only very near the Julia orientation, where it appears as a near- perfect rectangle. To create today's image, I gave the object a 2-degree double rotation from the Julia orientation, which distorted the rectangle into the curved shape in the picture. A little playing with the colors produced the effect of a flying sheet of rubber. The flying plane has landed at Paul's web site at: http://home.att.net/~Paul.N.Lee/FotD/FotD.html Tomorrow, I'll have another interesting FOTD. At this time I have no idea what it will be, but something will turn up -- as it always does. Until then, take care, and keep finding those fractal gems. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER-FORMULA FILE================================ Projective_Plane { ; 3-1/2 min on a P200 at SF5 reset=1960 type=formula formulafile=multirot.frm formulaname=multi20031 function=flip/ident/ident\ /flip passes=1 center-mag=-0.00037327503160875/\ +0.00003799627771399/514.8005/1/25 params=88/88/0.00022/0.0755/-1.74308/0 float=y maxiter=1800 bailout=25 inside=253 logmap=yes symmetry=none periodicity=10 colors=000QVZ<2>PXZ\ PYZP_Z<7>PlZPnZRt_<4>PmZPkZOhZ<6>MTZMRZMQZ<12>I9\ ZI8ZJ6a<19>I8OI8NH9KH9IH9HH9IH9IH9HH9FH9FH9H<11>\ HA9HA9JCB<2>OIGQJHRMJ<7>dgVejWgkY<3>nncooeqnf<3>\ wshxuixwjyylyzm<4>zwizvhzugzugztgzthzshzsh<10>pl\ `ol_mkZ<7>YfTWfSVdR<3>RZOQXNPYN<6>IUHHTGGUG<3>I_\ IJ`IJ`I<38>asSXrT<19>stPttPvvJvuM<3>oqWmpZmqZ<3>\ moZnoZopZskZwzZzwZzwZ } frm:multi20031 {; Jim Muth, best=ifif, fiif, fifi, iffi a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994, z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p2, c=cos(b)*fn3(real(pixel))+cos(a)*fn4(imag(pixel))+p3: z=z^2.003+c, |z| <= 100 } END PARAMETER-FORMULA FILE==================================