= on filaments, often filament tips or branching poi= nts. These are easily seen to have sequences of mini Mandelbrots that

shrink toward them. The centers of their cardioids a= ccumulate to them, and the Misiurewicz points lie in the closure of

the set of component centers. That takes care of all p= oints with rational external angle on the boundary of M.

3. He= re we assume that M is locally connected, a conjecture with strong eviden= ce for it, but to the best of my knowledge

not ye= t proven (or disproven). If this is true, all points with irrational exte= rnal angle are accumulation points of points with

= rational external angle, and lie in the closure of the closure of the se= t of component centers. But the closure of a closure

&nb= sp; is the original closure, which puts the whole of the boundary of M in= the component centers' closure.

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------=_NextPart_001_0004_01C103CF.82D38B60-- Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Multiple Bogeys" Subject: Re: (fractint) Re: July's FOTD Date: 03 Jul 2001 14:55:36 -0400 ------=_NextPart_001_0005_01C103D0.3779E1E0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable > I meant a month without a *quadratic* midget. Someone here seems to be a prime candidate for public office... :-)

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> I meant a= month without a *quadratic* midget.

Someone here seems to= be a prime candidate for public office... :-)

Get Your Private, Free E-mail from MSN Hotmail at <= a href=3D"http://www.hotmail.com">http://www.hotmail.com.

------=_NextPart_001_0005_01C103D0.3779E1E0-- Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Multiple Bogeys" Subject: Re: (fractint) Re: July's FOTD Date: 03 Jul 2001 14:56:48 -0400 ------=_NextPart_001_0006_01C103D0.629FB840 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable > Now there's a thought... quadratic midgets occur in all sorts of formul= as, > but has anyone ever seen, say, a cubic Mandelbrot in any formula other = than > z^3+C? Sure. I constructed a quartic years ago to exhibit one. And I did it deli= berately. And it worked. The only requirement is that the critical point = you use be a zero of the derivative with multiplicity two.

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------=_NextPart_001_0006_01C103D0.629FB840 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable

> Now there= 's a thought... quadratic midgets occur in all sorts of formulas,
>= but has anyone ever seen, say, a cubic Mandelbrot in any formula other t= han
> z^3+C?

Sure. I constructed a quartic years ago= to exhibit one. And I did it deliberately. And it worked. The only requi= rement is that the critical point you use be a zero of the derivativ= e with multiplicity two.

= Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com.

------=_NextPart_001_0006_01C103D0.629FB840-- Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Mark Christenson Subject: Re: (fractint) Re: July's FOTD Date: 03 Jul 2001 21:42:57 -0700 At 04:07 PM 7/3/01 -0000, Andrew wrote: >> >On June 15, 2001 Jim Muth wrote... >> > >> >"I have decided that the month of July will me a midgetless month." >> >>I meant a month without a *quadratic* midget. > >Now there's a thought... quadratic midgets occur in all sorts of formulas, >but has anyone ever seen, say, a cubic Mandelbrot in any formula other than >z^3+C? No cubics, but I have witnessed z^4 midgets in one of my gravibrots. Will advise which if you're interested. I could direct you to Bud's Fractal Pages, but they're temporarily offline... Aloha, Bud Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Jim Muth Subject: (fractint) C-FOTD 04-07-01 (Mand-Newt Eruption [8]) Date: 04 Jul 2001 10:04:54 -0400 (EDT) Classic FOTD -- July 04, 2001 (Rating 8) Fractal visionaries and enthusiasts: Today's FOTD image has no quadratic midget. In fact, it has no midget at all. It is simply a design created by number 13 in my series of 14 MandNewt formulae, which use variations of the famous Ikenaga function to create fractals. Formula numbers 13 and 14 are recent additions to the MandNewt series, which until now has consisted of only 12 formulae. Almost the entire image consists of bof60 inside fill. This fill changes an otherwise flat inside area into a series of overlapping bubbles, which can make interesting images. Today's scene is one of these images. It is a rather stately scene with its left-right symmetry. Some may see a series of hot-air balloons filling with a grainy green-and-pink gas; others may see a scene in some surrealistic outer space. I see a kind of eruption, and have therefore named the picture "Mand-Newt Eruption". I rather like the hazy effect created by the bright blue areas. In my opinion, this plus the overall appearance gives the image an exceptional rating of 8, which may be a bit liberal. The 4-minute render time of the parameter file at a resolution of 640x480 is bearable, but this is one of those images that looks best at the highest resolution possible. The download of the GIF image will soon be available on Paul's web site at: and on Scott's site at: The fractal weather today here at Fractal Central was uneventful, with partly cloudy skies and a temperature of 84F (29C). The cats must have approved, for they spent most of the afternoon lounging in the yard. Today however is Independence Day in the USA, and that means fireworks. The cats don't like noise, so I suppose they'll be skittish most of the day. As for myself, I'm simply going to do as little as possible. Until next time, take care, and fractals are fun; arguing politics is not. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ Mand-Newt_Eruption { ; time=0:04:22.60--SF5 on a P200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandNewt13 passes=1 center-mag=8.84911\ /-9.84213e-014/3.120633/1/-90 params=-4.159/-3.693\ /-4.144/-0.685/0.119/-0.783 float=y maxiter=90 inside=bof60 logmap=yes periodicity=0 colors=000I8\ UJAXKC_LEbMGeNIhOKkPMnQOqRPtQPwPOxONxNMyMLzLKyKJxJ\ IwIHuHGsGFqFEoEDmDCkCBiBAgA9e98c87a76_65Y07W06U05S\ 04Q03O02M01K00I00G00E0AC0KW3NX5PZ9R`CSaEWcGXeIZhL`\ jNclPemRgnShpWhuUjrUjpSlnSlmRllRmjPmhPmgNncNnaLp`L\ pZLpXKrWKrUIrSIsPGsNGuLEuKEuICvGCvE0w0CvCLuNUsWare\ jpnrnvymzzmzznzznzznzzpzzpzzpzyrz000<158>000 } frm:MandNewt13 {; Jim Muth z=g=pixel, a=real(p1), b=imag(p1), c=real(p2), d=imag(p2), e=real(p3), f=imag(p3): h=z^a+(g-b)*z-f j=c*z^d+g z=z-e*h/j, 0.000000000000000000000000000001 <= |h| } END 20.0 PAR-FORMULA FILE================================== Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Andrew Coppin" Subject: Re: (fractint) C-FOTD 04-07-01 (Mand-Newt Eruption [8]) Date: 05 Jul 2001 12:44:03 -0000 >From: Jim Muth Reply-To: fractint@lists.xmission.com To: >fractint@lists.xmission.com CC: philofractal@lists.fractalus.com Subject: >(fractint) C-FOTD 04-07-01 (Mand-Newt Eruption [8]) Date: Wed, 4 Jul 2001 >10:04:54 -0400 (EDT) > > >Classic FOTD -- July 04, 2001 (Rating 8) > >Fractal visionaries and enthusiasts: > >Today's FOTD image has no quadratic midget >I rather like the hazy effect created by the bright blue areas. In my >opinion, this plus the overall appearance gives the image an exceptional >rating of 8, which may be a bit liberal. That is indeed a very cool image! Thanks. Andrew. _________________________________________________________________________ Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com. Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Jim Muth Subject: (fractint) C-FOTD 05-07-01 (Platform in Space [7]) Date: 05 Jul 2001 10:30:32 -0400 (EDT) Classic FOTD -- July 05, 2001 (Rating 7) Fractal visionaries and enthusiasts: Since the FOTD is already late, and it is a slow one, I'll make the discussion short. To begin, in keeping with the theme of a midgetless July, the image has no midget. The iterated formula that drew the image is Z^1.03+C. I named the picture "Platform in Space" because that's what it reminds me of. I rated it at a 7 because I feel it's above average. It's an unusually slow image to render, taking over 2 hours of computer time on a tired old Pentium 200mhz machine. Therefore the best way to view it is to give Paul and Scott a chance to render it and then visit their Web sites at: and: The fractal weather on the Independence Day holiday featured fireworks from nature. A series of heavy thunder-showers in the evening put a damper on the man-made variety. The temperature of 86F (30C) did little to ease the fractal cats' skittishness. And it's now time for an active day doing other things. Until next time, take care, and hope for the best. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ Platform_in_Space { ; time=2:01:41.52--SF5 on a P200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotBC1 function=recip passes=1 center-mag=-8.54499/-3.05261/4.14962/1/-29.999 params=1.03/0/-7.53/800 float=y maxiter=12000 inside=0 logmap=57 symmetry=none periodicity=10 colors=000ZN`YMaXLbWKcVJdUIfTIgSHhRGiQFjPEkODlODmC\ 5PE7NF9MGBKIDJJEHKGGMIENKDOLCNNENOFNQGNRINSJNUKNVM\ NXNNYONZQM`RMaSMcUMdVMeWMgYMhZMj_MkaMlbLncLoeLqfLr\ gLsiLujLvkLxmLynLzoQujVpeZl`cgWheTgcSfbRfaRe_QdZPd\ YPcXOcVObUNaTMaRM`QL`PL_OKZMJZLJYKIZIJYJIYJIYJIYJI\ YJHYJHXJHXJHXKGXKGXKGXKGWKFWKFWKFWKFWLEWLEWLEVLEVL\ DVLDVLDVLDVMCUMCUMCUMCUMBUMBUMBTJCUMBVOBWRAXTAYW9Z\ Y9_`8`b8ad8`bB_aE_`HZ_JYYMYXPXWRWVUWUXVSZVRaUQdTPf\ TOiSMlRLnRKqQJtSHuQIvOJvMKvLKvJLwHMwGMwENwCOwAPx9P\ x7Qx5Rx2Sz4Rz6Rz8QzAQzCPzEPzGPzIOzKOzMNzONzQMzSMzU\ MzWLzYLz_KzaKzcKzZMzUOzPQzKSzFUzBVzBUzCTzDRzZKzZLz\ ZLz_Lz_Lz`Mz`Mz`MzaMzaNzaNzbNzbNzbNz`PzZRzXTzVVzTW\ zRYzP_zNazLbzJdzHfzFhzDizEgzFfzGezHdzIczJbzKazL_zM\ ZzNYzOXzPWzQVzRUzSSzTRzUQzVPzWOzXNzYMzdMzkMzrMzqLz\ pLzoLznKzmKzlKzkJzjJziJzhIzgIzfIzeHzdHzcHzbHzaGz`G\ zeTzdTzcSzbRzbQzaPz`Oz_Nz } frm:MandelbrotBC1 { ; by several Fractint users e=p1, a=imag(p2)+100 p=real(p2)+PI q=2*PI*fn1(p/(2*PI)) r=real(p2)-q Z=C=Pixel: Z=log(Z) IF(imag(Z)>r) Z=Z+flip(2*PI) ENDIF Z=exp(e*(Z+flip(q)))+C |Z| Subject: Re: (fractint) C-FOTD 04-07-01 (Mand-Newt Eruption [8]) Date: 05 Jul 2001 20:03:01 -0400 ------=_NextPart_001_0001_01C1058D.7ED06A00 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable > Today however is Independence Day in the USA, and that means > fireworks. If you like fJuly 4 fireworks, then there's a particularly apt choice of = sci-fi movie for the night ...

Get Your Private, Free = E-mail from MSN Hotmail at http://www.= hotmail.com.

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> Today how= ever is Independence Day in the USA, and that means
> fireworks.

If you like fJuly 4 fireworks, then there's a = particularly apt choice of sci-fi movie for the night ...

Get Your Private, Free E-mail from MSN H= otmail at http://www.hotmail.com.<= br>

Get Your Private, Free E-mail from MSN Hotmail= at http://www.hotmail.com.
------=_NextPart_001_0001_01C10E58.1C348740 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable

> I used Jo= hn Goering's formula to draw the image and do the
> rotating. = This formula can draw more orientations within the
> Julibrot than= any other I know of, though there are still some
> directions that= it cannot reach.

Here's some math that might be useful to devise = a formula to draw any 2D slice of a 4D fractal.

<= DIV>Any plane can be exactly specified by two non-collinear vectors. Two = 4-space vectors can be specified with 8 real values -- 7 if they are= normalized. Four of the formula parameters, say p1-p4, could be used to = give these vectors.

The cheap way to convert= the pixel coordinates into the plane of two vectors v1 and v2 is:

x*v1 + y*v2

Of course= , the resulting image is skewed, aspect mangled, or otherwise distor= ted if v1 and v2 aren't equal in magnitude and perpendicular. However, if= they are not collinear, we can preprocess v2 to be orthogonal to v1 by a= projection calculation:

v2 <- v2 - (v1*v= 2)(v1/|v1|)

where (v1*v2) is the dot product= of v1 and v2 and |v1| is the magnitude of v1, (the square root of v= 1*v1). (Observe that if v2 is already perpendicular to v1, the dot produc= t is zero and v2 is not changed. If v2 and v1 are collinear, the dot prod= uct is |v1||v2|, the term on the right ends up with the direction of v1 (= which is that of v2) and the magnitude of v2 and thus is v2, and v2 becom= es zero, so things go wrong.)

Making them eq= ual length is easy -- replace v1 by v1/|v1| and v2 by v2/|v2| so both hav= e length 1. A simple procedure is then:

v1 &= lt;- v1/|v1|

v2 <- v2 - (v1*v2)v1

v2 <- v2/|v= 2|

I leave the translation of this into a fr= actint formula using four complex variables to represent v1 and v2 as an = exercise for the reader.

At this point, you = get a rectilinear image -- no skew or aspect squashing unless the input c= oordinates (using the zoom box) are themselves subjected to such distorti= on. However, one can still rotate the image by playing with v1 and v2 as = well as by rotating the zoom box. These effects are additive.

Get Your Private, = Free E-mail from MSN Hotmail at http:/= /www.hotmail.com.

------=_NextPart_001_0001_01C10E58.1C348740-- Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Multiple Bogeys" Subject: Re: (fractint) Another use for fractint Date: 17 Jul 2001 00:53:57 -0400 ------=_NextPart_001_0003_01C10E5A.F5BE00C0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable > Some time ago, I discovered another practical use for fractint. > I created a batch file f.bat which simply called fractint.The f.bat fil= e was > under the /bat directory, which I included in the path. > Then, when some program under DOS lets the screen in an abnormal state,= for > example with different colors or letter size, I simply type f and when > fractint appears I press Esc. This easily returns the screen to its nor= mal > state. Blind-typing "mode co80" at the prompt will accomplish the same thing :-) (If you want a single-letter command for it, you can put "mode co80" in a= n "f.bat". :-))

Get Your Private, Free E-mail from MSN= Hotmail at http://www.hotmail.com= .

------=_NextPart_001_0003_01C10E5A.F5BE00C0 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable

> Some time= ago, I discovered another practical use for fractint.
> I created = a batch file f.bat which simply called fractint.The f.bat file was
>= ; under the /bat directory, which I included in the path.
> Then, w= hen some program under DOS lets the screen in an abnormal state, for
&= gt; example with different colors or letter size, I simply type f and whe= n
> fractint appears I press Esc. This easily returns the screen to= its normal
> state.

Blind-typing "mode co80" at the= prompt will accomplish the same thing :-)

(If you want a sing= le-letter command for it, you can put "mode co80" in an "f.bat". :-))

Get Your Private, Free E-mai= l from MSN Hotmail at http://www.hotma= il.com.

------=_NextPart_001_0003_01C10E5A.F5BE00C0-- Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Jim Muth Subject: (fractint) C-FOTD 17-07-01 (Fractal Limeade [6]) Date: 17 Jul 2001 11:06:12 -0400 (EDT) Classic FOTD -- July 17, 2001 (Rating 6) Fractal visionaries and enthusiasts: I have no real excuse for naming today's image "Fractal Limeade". True, the weather has been warm here lately, and limeade is one of my favorite summer drinks, but the only resemblance of today's image to a glass of limeade that I can find is the vaguely lime-green color of the prominent ring of features surrounding the non-quadratic midget. The image is a scene in the Z^1.75+C fractal. It was found with the aid of the MandelbrotBC1 formula, which reveals areas in the remote parts of the infinite logarithmic spirals of fractals created with fractional exponents. I rated the image a 6, which is a bit above the average of my standard offering. With the parameter file's render time of almost 9 minutes, running it is not the most efficient way of viewing the scene. A more efficient way is to download the pre- rendered GIF image from: or from: The fractal weather was hot and lazy today, with a temperature of 90F 32C, which kept the fractal cats stretched out on the porch most of the day. Unfortunately, with all the work before me, I can't afford to be lazy. So until next fractal, take care, and think about those 4-D rotations. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ Fractal_Limeade { ; time=0:08:49.54--SF5 on a P200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotBC1 function=floor passes=b center-mag=+0.74278632205274600/-0.079548597538634\ 95/1.830877e+007/1/105 params=1.75/0/-36/0 float=y maxiter=1450 inside=255 logmap=368 periodicity=10 colors=000_6WuVSsTSpRSnPSkMSiKSfIRdGRaDR_BRX9RV7RV\ 5TX6U_6Wb7Yf7_i8al8dp9fs9hvAjyAlwGhuMesRaqXZobVmgS\ kmOjrLhqOfpRdoUbnX`mZZlaXkdVjgXjkUiiRhhPggMfeJedHd\ cEcbBb`9a_Hc7G`6GZ6FW5ET5DQ4CN4CL3BI3AF29C2891871C\ C4FG7IL9LPCOUERYHVbKYfM`kPcoRftUhwWYuNNrECo6Gf8KYA\ OQCSHEV9GYCJ`ELbHNeJPhLRjOTmQVpSXrVZuX`wZbu_`s`Zqa\ XobVmcTldSjeQhfOfgMdhKbhIahH_fIYcIX`IVYIUVISSIRPJP\ MJOJJMGJLDJJAJI8JJCNJFQKITKLWKOZLRbLUeLXhM_kMbnMeq\ ShkYkfcnabn`bn`geekXikXjkXjkXjkXkkXkkXkhZje_jcai`b\ iYdiWehTghQhhOjgLkgGmhJlgLkgOkfQjfSjfOihLiiIikEhlB\ hn8hoCgpGgpKgqOfqRdrVbrZ`sbZsf_ti`tmauqbuucvxdvvZi\ tUXmDTfEP_FMTGIMHEFIB8J7mbKk`KhZLeXLcVL_XKWYJSWIO`\ HKfGGqFCjE8kD4lC0mC2XP4cmKXmZ4kYLOXa1Vb5Uc8TdBSeER\ fHQgLPhONiRMjULkXKl`JmcInfHoiGolImdJkXLiPMgHNfAac8\ bZ8cV8cQ8cM8dH8eD8fF7gG6iI6kJ5mK4oM4qN3sO3uP3wQ6yR\ AzSDzSFzTHzTJzUMzUOzVQ000 } frm:MandelbrotBC1 { ; by several Fractint users e=p1, a=imag(p2)+100 p=real(p2)+PI q=2*PI*fn1(p/(2*PI)) r=real(p2)-q Z=C=Pixel: Z=log(Z) IF(imag(Z)>r) Z=Z+flip(2*PI) ENDIF Z=exp(e*(Z+flip(q)))+C |Z| Subject: (fractint) Fractint version 20.1.13 Date: 17 Jul 2001 20:28:11 -0500 Fractint version 20.1.13 is now available at www.fractint.org. New features include: Added parameters p4 and p5 to the evolver. This required splitting the tweak central screen into two pages. Fixed an evolver bug that was causing the evolver to not exit cleanly. Changed the compile options on evolve.c to eliminate aliasing, which started to cause problems with this patch. I haven't updated the Xfractint sources yet. Jonathan Thanks for using Fractint, The Fractals and Frac