|
Fractal of the Day
by Jim Muth
|
|
Fractal visionaries and enthusiasts: The four-dimensional hypersphere -- the locus of a point in four-dimensional hyperspace which is at a constant distance from a fixed point -- in other words, a four-dimensional ball -- perhaps the simplest hyperfigure. All two-dimensional slices of a hypersphere of any number of dimensions appear as circles; all three-dimensional slices are spheres. The surface of the 4-D hypersphere, as well as the surface of all 4-D figures is three-dimensional -- in the hypersphere, it is a space of constant curvature. Though we cannot visualize a curved space, we can visualize limited portions of it, which, if the hypersphere is large enough, will seem like ordinary 'flat' 3-D space, in the same manner as a limited portion of the earth's surface appears flat. On the 3-D surface of an imaginary 4-D hyperplanet, there are six cardinal directions -- north, east, south, west, up, and down. The up and down directions do not lead up into the sky or down into the ground -- they are additional directions which lead around the surface of the planet and meet at the antipode. A 4-D hyperplanet has an equator -- an imaginary great circle line around its center. Like on the earth, this equator marks the center of the tropical zone, where the climate is constantly warm. But, unlike the earth with its two point-poles, a 4-D hyperplanet also has a polar circle -- a great circle completely around the planet, which marks the center of the area of frigid climatic conditions. All points of the polar circle are 90 degrees from all points of the equatorial circle. On a 4-D planet, it is possible to circumnavigate the hyperglobe in a straight line and never leave the coldness of the polar region. To define a point in the surface-volume of a hyperplanet, three sets of coordinates are necessary. In the next FOTD I'll go into several ways this might be done. Right now, it's time to get to today's image. I named the image "Something Different". Resembling a scene in a tropical rain forest, the image is not really all that different, but, considering that the features surrounding the midget were produced by the basic equal-iteration coloring method, the image is rather unusual. Whether the image deserves its rating of 7 is up to the individual viewer to decide. The render time of 6 minutes is only slightly annoying, and the download, which may be found at: The fractal weather today was perfect. The warm sun, light winds, and temperature of 79F 26C kept the fractal cats both happy and outdoors. But I won't be happy until the day's work is done. And the way to get it done is to get it started, so until next time, take care, and take it easy. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
SomethingDifferent { ; time=0:06:15.53--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=-0.12380630434162650/+0.095427904465020\
41/15137.63/1/-82.4999999998920828/4.9807873103713\
2627e-011 params=1.2/-0.98/-0.8/1.02/0.178/-50
float=y maxiter=1000 inside=0 logmap=90
colors=0000Q0zgBziRzs4dz0iz0nz0sz0yz0zz0zz0zz0zz0z\
z0zz0zz1zz4zz9zzCywEnnGdgHXJJMRKCKM2CO06P09R2CR6GR\
9JRCMRHPRKTROXRR_RVbRYRRVRRTBRRRRPRRORRMVPK_HJ_9HY\
1GY0C_0Ba19a47b76bB4dE2dH0fM0gP0gT0iX0i_0kb0kf0gb0\
fa0d_0bY0aX0_V0YT0XR0VP0TO0RM0TK2VK4VK7XJ9XJCYJGYJ\
H_HK_HMaHPaHTbGVbGYdG_dGagCak9bm7bp4br2du0dw0dz0fz\
0fz0fz0gz0gz0gz0bz0Yr0Vg0Pg0Kg0Hg0JgCJgYJssJzzGzzC\
zuBzm7zb4zV2zM0zE0z40z00z00z00y00u00r00n00i00g00g0\
0g00g00g01g01g22g42g64g76g96gB7gC7gE9gG9gHBgJBgKEg\
HHgGKgEMgCPgBTg9Xg7Yg6af4dd1fb0ia0m_0pb0rg0uk0yp0z\
s0yk0yg0yg0yg1yg4yg7yg9ygBygCygEygGygHygKygMygOzgP\
zgRzgTzgVzgYzg_zgazgbzidzmfzngzpizngzngzmgzmfwmfsk\
fpkfmkdiidfidbibzf0zd0za0z_0zY0yg0zd0zn2uzCwzGyzJz\
zKzzOzzPzzTzzXzzYzzazzbzzfzzgXzB_zGbzKfzOizTmzXpza\
szfwzizznzzszzryzpszpmzngzmbzmYzkRziMziHzgCzf6zf1z\
d0zb0zb0zV7zMGzGOz7zzzzzz
}
frm:MandelbrotMix4 {; Jim Muth
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j,
k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < l
}
END 20.0 PAR-FORMULA FILE==================================
|
times.
