From: owner-fractint-digest@lists.xmission.com (fractint-digest) To: fractint-digest@lists.xmission.com Subject: fractint-digest V1 #86 Reply-To: fractint-digest Sender: owner-fractint-digest@lists.xmission.com Errors-To: owner-fractint-digest@lists.xmission.com Precedence: bulk fractint-digest Wednesday, January 21 1998 Volume 01 : Number 086 ---------------------------------------------------------------------- Date: Wed, 21 Jan 1998 03:23:54 -0600 From: "Paul N. Lee" Subject: Re: (fractint) Lee's Truecolor PNG site Wizzle wrote: > > I have to stand with Rich on this one...... and say that > plugins are NOT the same as support for a format. I have > the required software to view PNG.... not the issue. I > think the issue is.... correct me here Rich...... we will > be hampered with web posting until Netscape takes that > next step and gives the same support to png as it does to > gif and jpg. I recall reading early last year about how > png was the coming thing..... but it wasn't incorporated > into Netscape version 4.0. Should we lobby Netscape?? > can we leverage the Great Browser Wars?? > Actually plugins are support, just a secondary or subordinate role to a leading performer. They aid the cause, policy, or interests of software; providing for or maintaining, by supplying necessities. (See dictionary under "support".) Netscape's latest browser release level handles PNG file formats, but how many individuals are currently using this level?? Until users of Microsoft or Netscape browsers get them up to speed, plugins will give them support for PNG files. And what about the rest of the world that don't use either one of the two "big boys" browsers??? Browser ________________ Opera-3.0 IBrowse Cyberdog IBM WebExplorer Lynx Echo OS2 Walker AmigaVoyager MacWeb Mercer OS NCSA Mosaic SPRY Mosaic ... (and about 100 more) - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: 21 Jan 1998 12:35:43 +0100 From: Brederlow Subject: Re: (fractint) Lee's Truecolor PNG site "Paul N. Lee" writes: > Wizzle wrote: > And what about the rest of the world that don't use either one of the > two "big boys" browsers??? > > Browser > ________________ > Opera-3.0 > IBrowse > Cyberdog > IBM WebExplorer > Lynx > Echo > OS2 Walker > AmigaVoyager > MacWeb > Mercer OS > NCSA Mosaic > SPRY Mosaic > ... > (and about 100 more) All amigabrowsers can (or should) handle png. The system provides Datatypes to load those. Every Programm able to handle the Picture datatype handles png, gif, iff, tiff, bmp, ppm,... all the same way. I think png is far better than gif, because its free, compresses better and has 24 Bit. May the Source be with you. Mrvn - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 05:17:22 -0800 From: kathy roth Subject: (fractint) Lee's Truecolor Fractals Gedeon Peteri wrote >>Exactly the same problem I ran into with Lee's image, and others. I >>corrected it by editing the par in Notepad. I found that the end of the >>line cited did not have a backslash. Also, make sure there are no double >>spaced lines. I sometimes get them too, especially with Lee's postings. >>When editing them out, one must take great care that nothing else is >>deleted. Perhaps this is your problem too. >>Gedeon That was it! They were double spaced. I corrected that and the color maps from Lee Skinner work fine, and at least some of the Paul Carlson formulas work too. I can't believe how many things I've fooled around with in the last couple of months, and was beginning to have some.... ..doubts....and the problem was they were double spaced! Jay Hill wrote >>Are you sure you used my copy of Lee's par file?? It has not >>the missing \ problems. Now you need to put the formula part >>in a .frm file, the rest in a par file. What I like to do as a quickie >>is stick frm: in front of the formula. Yes, I used your copy and I do understand the \ and saving the .par and .frm. It was the double space. The odd thing is a few of the double spaced ones worked. Maybe only certain things are sensitive to the double space issue. This was not actually wasted time for me because I learned a lot repeatedly taking things apart and putting them back together. Thanks again for putting them on your web site. I have been enjoying Fractal of the Night (like those potatoes.) Maybe I should do a Fractal of the Early AM. (I did work till 3AM- not spending every waking moment on this.) - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 10:17:36 -0800 From: Mark Christenson Subject: (fractint) Browsers / GIF vs PNG At 02:50 AM 1/21/98 -0600, Paul Lee wrote: > Support for PNG is there in the latest levels (which >I have and use), but I still prefer to use my Navigator 3.04 Gold. Okay, I think I'm ready to come out of the cave (still on Navigator 2.02). If Paul, or anyone else, could provide input on Nav 3.xx vs 4.xx (to *my* address, unless you think it would be of general interest) I would appreciate it. I'm not interested in company propoganda, but in real-life experience with issues like configurability, compatability with plug-ins, speed and ease of use for still images and motion video (the main reason I've eliminated Mircodespot Exploiter from consideration) and memory footprint (hard and volatile - this is I'm still on Win 3.1). One of the reasons I am reluctant to use Nav 4.x is this goofy "office suite" approach both of the majors are taking with regard to their browsers. "I don't want a memory hogging office suite, dammit!" What I *do* want is a fast, capable, robust browser. Period. So if 4.x can be stripped of the fat, that would be fine. As to the issue of .gif output, my concern isn't with fractal addicts like ourselves (who will put aside food, drink, sex, and fear of plug-in installation to pursue our habit), it's with losing the general audience. Most of the Web audience tend to throw up their hands and move on when confronted with such obstacles, because plug-ins aren't exactly plug-and-play. I get about 30 visitors a day, and guess about 20 of those would give up and move on. Since I'm getting long-winded, I will finish with a bullet chart of issues: - - I could convert the PNG files to GIF, but 256 color renditions of 24 bit images are usually pretty ugly (an oxymoron?), and, if you've how most most image conversions organize the palette, completely uneditable. - - (256 color) GIF images are the only way to display images undithered on the Web (if Nav 4.x has 24 bit support, please set me straight on this). - - JPEG has the same problem as cited above. And to the Fractint developers, a question: how much overhead would really be required to retain GIF output capability and 256 color palette support? (I know, you probably think I'm a Ludite... you're not the first.) Thanks, Bud P.S. - I have nothing against 24 bit color or good file compression. - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 10:21:55 -0800 From: Mark Christenson Subject: (fractint) Browsers / GIF vs PNG / correction Fumble-fingers strikes again. What I *meant* so say was: >... > - I could convert the PNG files to GIF, but 256 color renditions of 24 bit > images are usually pretty ugly (an oxymoron?), and, if you've _seen_ how most > most image conversions organize the palette, completely uneditable. >... Sorry for ... lack ... words ... - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 12:25:34 -0700 From: Rich Thomson Subject: (fractint) list traffic Would people find it useful if we created a (moderated) list JUST for the posting of formula files and PARs and web URLs? I suggest it still be kosher to include a par/frm when you're trying to use it to illustrate a discussion point, or question, but perhaps if we had a separate "announce" list for people to announce, distributue, share, get pars/frms/URLs/etc., it might help segregate the traffic and bring the discussion list back down to a smaller number of messages. Just a thought... let's hear what you think about the idea :) - -- ``Between stimulus and response is the will to choose.'' -- Steven Covey =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 3D Paint: The Power to Create in 3D; Rich Thomson email me for more info rthomson@ptc.com - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 14:31:03 -0500 (EST) From: Ian Kaplan Subject: Re: (fractint) list traffic > > > Would people find it useful if we created a (moderated) list JUST for > the posting of formula files and PARs and web URLs? I suggest it > still be kosher to include a par/frm when you're trying to use it to > illustrate a discussion point, or question, but perhaps if we had a > separate "announce" list for people to announce, distributue, share, > get pars/frms/URLs/etc., it might help segregate the traffic and bring > the discussion list back down to a smaller number of messages. Just a > thought... let's hear what you think about the idea :) Yes. An alternative might be to create a weekly or monthly digest, but that's assuming we could find someone to donate several hours a week to the task... - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 13:12:39 -0700 From: Rich Thomson Subject: Re: (fractint) Lee's Truecolor PNG site In article <9801201852.AA06945@minuit.iuta.u-bordeaux.fr> , "J.P. Louvet" writes: > Can you explain me what is alpha channel ? Damien covered it pretty nicely, but I'll add a little more info. Think of an alpha channel as describing the opacity (or transparency, if we think of opacity going from 0 to 1, with 1 being completely opaque and 0 being completely transparent, then transparency = 1 - opacity) of each pixel. If alpha is stored just like any other pixel channel, we typically use 8 bits for the alpha channel. This gives a 32-bit pixel, 8 bits each for red, green, blue, and alpha. You can experiment with alpha in lots of paint programs like photoshop. They might call the alpha channel a "mask" or a "transparency mask" or an "opacity mask", etc. Alpha channels let you do very interesting things when combining images. When an image is combined with another through an alpha channel, the alpha channel is used as a blending factor: combination = image1 * alpha + image2 * (1 - alpha) Since alpha and 1-alpha represent two fractions that add up to one, this computation keeps the image pixels inside the valid range (they don't get smaller -- fading to black, or larger -- saturating to white). Fun things you can do with alpha and fractals: o color normally, but use the value of the last iterate to determine the transparency o color normally and extract one of the red/green/blue channels to use as an alpha channel on another image o color several views of the same fractal and use alpha channel to combine them. (Damien's example showed this, I think) The "whitesq" fractint trick used in formulas can be thought of as implementing a sort of transparency. It essentially allows you to interleave two fractals together, selecting one or the other based on the value of whitesq. Every other pixel then corresponds to one of the two images interleaved toether. Your eyeball averages together the adjacent pixels and they appear to be blended together if you look from a distance. An alpha channel of 0.5 assigned to each separate image and then both of them combined would achieve a similar effect. This kind of transparency is called "screen door" transparency because the two images being blended together appear as if enmeshed together through a screen. By varying the screen, you can get achieve different levesl of blending between the two images. (For instance, take a pixel pattern of ABAABABAAB... that repeats to cover the screen instead of the ABAB... repeating over the screen and you end up with more of the "A" image than the "B" image appearing on the screen and if things are done right, your eye will think you are viewing mostly the A image with a fainter copy of B superimposed. An alpha channel provides much better support (especially when you consider fractint as just one program in an artist's toolbox) for transparency. It isn't much of an issue now, given that fractint doesn't yet support a file format that handles alpha (GIF does not) and doesn't yet support 24-bit rendering modes. A palette-based rendering can take advantage of alpha support under PNG (the colormap entries can have an alpha associated with the color). Alpha support will be more important as fractint begins to encompass PNG and truecolor rendering modes. - -- ``Between stimulus and response is the will to choose.'' -- Steven Covey =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 3D Paint: The Power to Create in 3D; Rich Thomson email me for more info rthomson@ptc.com - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 13:13:09 -0800 From: Mark Christenson Subject: (fractint) Re: Error Messages *** Angela, sorry for the redundant post. I meant to send this to the list *** At 06:40 PM 1/20/98 -0800, Angela wrote: >Bud... >I had problems with error messages also....and got rid of them by... >1. putting all my .frm files in the same directory as fractint.exe > ... Thanks for the fix. I can't believe I was so stubborn as not to try it. (Then again, maybe I can.) I may have redundant files on my system, but they're relatively small. I still consider it a bug, but it's one I can live with. My apologies to Stone Soup. And I'm diggin' the new 19.6 capabilities. That said, I'm still having trouble with bug #1. Bud - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 15:51:21 -0600 From: "Damien M. Jones" Subject: Re: (fractint) Browsers / GIF vs PNG Bud, - - (256 color) GIF images are the only way to display images undithered on - the Web (if Nav 4.x has 24 bit support, please set me straight on this). Uh, I've got Navigator 3 and 4, and IE 3 and 4... all show 24-bit images undithered in 16-bit and 24-bit display modes. If you're still in an 8-bit (256-color) display mode, then it *must* dither the image... no choice. But dithering is not required (or used) for higher color modes with these browsers. (Note: careful scrutiny of IE4 will show that it does dither 24-bit images in a 16-bit display mode, but the dithering is to reduce the images to 16-bit color, not to 8-bit color.) Damien M. Jones \\ dmj@fractalus.com \\ http://www.icd.com/tsd/ (temporary sanity designs) \\ http://www.fractalus.com/ (fractals are my hobby) - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 22:15:05 -0000 From: Peter Moreland Subject: Re: (fractint) 3d_stalks - -----Original Message----- From: Wizzle To: fractint@lists.xmission.com Date: 17 January 1998 01:48 Subject: Re: (fractint) 3d_stalks come up with hearts in >time for Valentines???? See if you like these... Heart_One { ; (C)1998 Peter Moreland ; Jan 21, 1998 0:00:15.04 P200MMX 1024X768 ; Email: peter@getitonbangagong.demon.co.uk ; *** No Commercial Use Without Permission *** reset=1960 type=manfn+zsqrd function=log center-mag=-0.61235100000007710/-0.02938849999992099/0.420909/1/-87.5/0.\ 001 params=0/0 float=y maxiter=32000 inside=0 potential=255/300/150 decomp=265 biomorph=0 colors=000zzP<23>zz1zz0zy0yx0<48>PCEPBFOAFN9GM8G<5>G2JF0KF0K<13>I0NI0OI0\ OI0PI0P<34>R0YR0ZR0ZS0_S0_<4>T0bT0bU1bU2c<62>zzzzzzzzy<35>zzQ } Heart_Two { ; (C)1998 Peter Moreland ; Jan 21, 1998 0:00:14.11 P200MMX 1024X768 ; Email: peter@getitonbangagong.demon.co.uk ; *** No Commercial Use Without Permission *** reset=1960 type=tetrate center-mag=-1.43442799999999600/-0.00860950000000882/1.763211/1/90 params=0/0 float=y maxiter=60 inside=epsiloncross outside=mult potential=255/511/0 invert=2.94416999816895/0/0 decomp=256 colors=000dGU<11>zg0<13>nGDmDEmCDmBDmACl8B<2>r6C<15>IfC<21>zz9<60>2I10H0\ 0I1<8>4WH<4>MU`YUzUTh<6>cOheNgeOg<25>soltpluqlvrlwslxumyvmzwn<47>C2W<8>V\ 6aY7bZ9``BZbEW } - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 17:42:45 -0500 From: "Jason Hine" Subject: Re: (fractint) list traffic Rich remarked: >Would people find it useful if we created a (moderated) list JUST for >the posting of formula files and PARs and web URLs? I would like this very much. I really enjoy the pars, but would rather download the separate list's last month's archive and spend an hour once a month cutting and pasting, instead of every night. The digest idea suggested by Ian would work as well, if not better? Jason - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Wed, 21 Jan 1998 22:40:33 -0500 From: Gedeon Peteri Subject: Re: (fractint) Lee's Truecolor Fractals kathy roth wrote: > That was it! They were double spaced. I corrected that and the color > maps from Lee Skinner work fine, .......It was the double space. The > odd thing > is a few of the double spaced ones worked. That is odd indeed! My first impulse was to say that none worked for me, but then I remembered that after I had discovered this problem, I did not even try to run any subsequently received double spaced pars; I edited them immediately. I brought up the issue of double spaced lines some time ago. Jay had the correct answer to the question why they occur, and the recipe for prevention. On October 11, 1997 he wrote: > I think the whole problem is line wrap. Some of us must > use emailers that do line wrap on transmit. This can't > line wrap the color part. So an extra line feed gets sent. > Try this, since I know mine line wraps. I include a par > here with color lines cut manually by me before > sending. I have been following his advice to cut lines manually. Before I did so, my own pars always got back to me double spaced; not once since then. I would like to add that it is not always only the color part. I have seen double spaced lines in connection with long center-mag (or corners) specifications too. > I have been enjoying Fractal of the Night (like those > potatoes.) Me too! > Maybe I should do a Fractal of the Early > AM. I encourage you to do so, though admittedly I would not get to see them until much later in the AM! I, for one, do not think that this list should be limited to formulas and pars posted in order to illustrate some point of discussion. Here, at any rate, is a pointless par. Gedeon Hot_lava_brot { ; (c) Gedeon Peteri ; Jan 20, 1998 reset=1960 type=magnet1m passes=b center-mag=+1.78301513414121800/-1.75824140412015300\ /101.3055/1/4.998 params=0/0 float=y maxiter=65535 bailout=64 fillcolor=222 inside=222 colors=0V0GKiGIhFGg<2>CEeBDdABc<10>00W`1F<6>z80<7>\ `GF000<6>dFH<7>0W0WZz<6>kmu<7>WXz0TW<13>M0mLTl<6>B0d\ <7>0TWWW0<6>gn0<7>uW1s01<6>h`1<7>W00G0GH0I\ <13>fml<15>G0G00G<14>deH<15>00GW00<14>zGW<15>W000GW\ <14>Eng<6>LZlMXmLVl<3>HMi } - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Sun, 18 Jan 1998 19:59:07 EST From: Nature102 Subject: Re: (fractint) On topic please In a message dated 98-01-18 19:51:31 EST, pfjakub@earthlink.net writes: << > And for the token fractal reference of this message, has anyone else ever >found a Nazi symbol in the Mandelbrot set, or can my name go down in the math >history books for that? :-) > It's hard to pick up nuances in people's posts, but I really can't fathom what a smiley face has to do with the Nazi symbol >:-( Sorry about going off-topic, but this particular bit of humor, if that is what it is, disturbs me. >> No! I really found a fractal that looks like a Nazi symbol! - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Sun, 18 Jan 1998 22:45:27 EST From: Nature102 Subject: Re: (fractint) Next version of DEEPER now avail. In a message dated 98-01-18 21:21:47 EST, tumnus@together.net writes: << P.s., I don't think that finding a fractal which resembles a Nazi symbol necessarily means that the finder practices or subscribes to Nazi-oriented acts or philosophies. Same goes for users of Microsoft products, I hope...! :) >> At least you THINK. ::Evil grin:: No, really, I'm not a Nazi. It's just weird to see stuff like that in fractals. Who knows, maybe eventually I'll find a fractal that looks exactly like the script for Romeo & Juliet. :-) In case anyone cares, here's the parameter file for the Nazi fractal: Nazi { ; It's a Nazi plot! Mr. Mandelbrot was brainwashed!! reset=1960 type=mandel center-mag=+0.39960241016028810/+0.19543117373730450/1.833849e+009 params=0/0 float=y maxiter=10000 colors=000L3S<53>KSK<2>KSQKSSKQSKOSKMS<40>xyyzzzyyy<36>000<36>x0Uz0Vz0G<\ 66>E0SL0S<5>L2S } Or you could just check out the image on my Web page at http://www.geocities.com/Area51/Cavern/8881/index.htm in the fractal section. (Just for the one or two people who might end up trying to figure out why they can't find it in the Wheel of Time section. :-P) - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ Date: Thu, 22 Jan 1998 00:20:36 -0500 (EST) From: ao950@freenet.carleton.ca (Paul Derbyshire) Subject: (fractint) The elusive Herman ring: Caught at last! Anyone who's seriously read through some of the stuff in The Beauty of Fractals or The Science of Fractal Images may have noticed that while great attention is paid to Siegel disks, attractors and parabolic cycles and points, there are also offhand mentions of something called Herman rings, but with no details. One fact I gleaned was that a Herman ring did have some characteristics of Siegel disks including an irrational rotation of a point around an invariant curve; I had images of a sort of "Siegel annulus" with no cycle at the center, or perhaps a curve within an attracting basin. A few attempts I made at constructing a function that would rotate the unit circle by a parameter alpha while displaying other dynamics elsewhere failed. On a hunch, I went into the university library the other day and ran a computer search for "Herman ring"; no luck. I tried the more generic subject "Complex mapping" which pointed me to an obscure Springer-Verlag book a single copy of which was buried in the basement. When I went down there, I eventually managed to find the book. (If you thought the Dewey decimal system was bad, the crazy system Carleton uses is its square; not only digits, but letters, digits, and more decimal digits. This one was QA337.5 C37.) Its front cover was adorned with a rather plain image of a simple quadratic Julia set. Inside the back cover was a well-fed index, with 'Herman ring' figuring prominently under the H. The reference on page 46 turned out to state the Sullivan classification TBOF and TSOFI also state, but a bit more mathematically. It pointed the reader interested in Herman rings to the other Herman ring page, 106. There I found what I was looking for: a whole wack of info, and an example function to iterate. The rational function is alpha*z^2*(z-c)/(1-cz), produces some of the most beautiful Julia sets I've ever seen, has attractors at 0 and infinity always, two critical points of conjugate fate at 3+c^2 plus or minus square root (9-10*c^2+c^4) all over 4c (plus zero and infinity), generates very peculiar Mandelbrot images in the 4-dimensional space of c and alpha (slices of fixed c and of fixed alpha look interesting), and lastly, generates Herman rings. Attached below is hring.frm, with two versions of the Julia sets, and the two slicings of the M-set. Below that is a PAR file chock full of two days worth of exploration. Six distinct Herman ring Julias are included, some with zooms, as well as Julias that are close to being Herman rings and have cyclic attractors, others that have only A(0) and A(oo) but a complex dendrite or Cantor dust approaching filling in the ring, and all exotically gorgeous. And it turns out a Herman ring is like a "Siegel annulus" or a Siegel disk with ahole punched out. Generate one of the Herman rings in the PAR, then use the orbits window to take a peek. Note how invariant curves near either the inner or outer skin of the ring are very crinkled and fractal, while those near the center are smoothish and near-circular. The "interesting" areas of the Julia sets tend to be quite large, so sometimes you want to zoom out from the default range after punching in new parameters. - --8<-- hring.frm -->8-- HRing_J { ; This thing is capable of generating Herman rings for alpha ; equal to exp(2*pi*i*a), a irrational. ; p1: alpha. p2: c. p3: Orbit trap radius about 0, reciprocal is ; used for infinity. ; Use outside=real, logmap=0, periodicity=0. z=pixel, a=p1, c=p2, r=real(p3), rr=1/r, iter=0, done=0: z2=sqr(z) z=a*z2*(z-c)/(1-c*z) iter=iter+1 IF(lastsqrrr) color=iter IF(color<1) color=1 ELSEIF(color>127) color=127 ENDIF IF(lastsqrrr) color=(iter/maxit)*127 IF(color<1) color=1 ELSEIF(color>127) color=127 ENDIF IF(lastsqr=0) flag=1-flag ENDIF IF(imag(c)<0) flag=1-flag ENDIF IF(flag==1) z=(3+sqr(c)-sqrt(9-10*sqr(c)+sqr(sqr(c))))/(4*c) ENDIF : z2=sqr(z) z=a*z2*(z-c)/(1-c*z) iter=iter+1 IF(lastsqrrr) color=(iter/maxit)*127 IF(color<1) color=1 ELSEIF(color>127) color=127 ENDIF IF(lastsqr=0) flag=1-flag ENDIF IF(imag(c)<0) flag=1-flag ENDIF IF(flag==1) z=(3+sqr(c)-sqrt(9-10*sqr(c)+sqr(sqr(c))))/(4*c) ENDIF : z2=sqr(z) z=a*z2*(z-c)/(1-c*z) iter=iter+1 IF(lastsqrrr) color=(iter/maxit)*127 IF(color<1) color=1 ELSEIF(color>127) color=127 ENDIF IF(lastsqr8--- FixedPoint { reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=1.72043/0.391134/0.1937984 params=1/0/3/1.2/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000G08<29>xKKzLLzMM<29>zxxzzzzzz<29>rizqhzqhzphz<21>Vbz<6>04Gz0Gz\ 00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zVV<3>zzV<3>VzV<3>Vzz<2>Vbzhhz<3>z\ hz<3>zhh<3>zzh<3>hzh<3>hzz<2>hlz<62>zzz } InfAndZero { reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0/0/0.6666667 params=1/0/0.134/0.618/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000G08<29>xKKzLLzMM<29>zxxzzzzzz<29>rizqhzqhzphz<21>Vbz<6>04Gz0Gz\ 00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zVV<3>zzV<3>VzV<3>Vzz<2>Vbzhhz<3>z\ hz<3>zhh<3>zzh<3>hzh<3>hzz<2>hlz<62>zzz } InfAndZero_2 { ; Beautiful! reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.812377/1.06798/0.3208747 params=1/0/1.3/2/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000G08<29>xKKzLLzMM<29>zxxzzzzzz<29>rizqhzqhzphz<21>Vbz<6>04GA0U<\ 5>hennlrtsvzzzpuz<4>0Vz0GzVVz<3>zVz<3>zVV<3>zzV<3>VzV<3>Vzz<2>Vbzhhz<3>z\ hz<3>zhh<3>zzh<3>hzh<3>hzz<2>hlz<62>zzz } InfAndZero_3 { ; Beautiful! reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.812377/1.06798/0.3208747 params=1/0/1.5/2/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000G08<29>xKKzLLzMM<29>zxxzzzzzz<29>rizqhzqhzphz<21>Vbz<6>04GA0U<\ 5>hennlrtsvzzzpuz<4>0Vz0GzVVz<3>zVz<3>zVV<3>zzV<3>VzV<3>Vzz<2>Vbzhhz<3>z\ hz<3>zhh<3>zzh<3>hzh<3>hzz<2>hlz<62>zzz } InfAndZero_4 { ; Bubbles of beautiful color reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.812377/1.06798/0.3208747 params=1/0/1.6/2/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000G08<29>xKKzLLzMM<29>zxxzzzzzz<29>rizqhzqhzphz<21>Vbz<6>04G0AU<\ 5>ehnlnrstvzzzupz<4>V0zG0z<13>205000111<29>fxohzqhzl<62>zzz } InfAndZero_5 { ; Chaos reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.812377/1.06798/0.3208747 params=1/0/1.8/2/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000kgz<35>H2AG08I18<14>zLL<15>yxz<2>xuzxuzwtzwtzwsz<17>rizqhzqhzp\ hz<21>Vbz<6>04GU0A<5>nehrlnvstzzzupz<4>V0zG0z<13>205000111<29>xxfzzhzzh<\ 62>zzz } InfAndZero_5b { ; Galaxies! reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.812377/1.06798/0.3208747 params=1/0/1.8/2/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000<29>M8IN9JM8H<5>G08<14>wJKzLLzNN<14>yxz<2>xuzxuzwtzwtzwsz<17>r\ izqhzqhzphz<21>Vbz<6>04G000<45>LLUMMVMMVNNV<77>zzz } InfAndZero_6 { ; MegaGalaxies reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.812377/1.06798/0.3208747 params=1/0/1.85/2/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000<29>M8IN9JM8H<5>G08<14>wJKzLLzNN<14>yxz<2>xuzxuzwtzwtzwsz<17>r\ izqhzqhzphz<21>Vbz<6>04G000<45>LLUMMVMMVNNV<77>zzz } InfAndZero_7 { ; Flowerlace reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.812377/1.06798/0.3208747 params=1/0/1.9/2/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=@funky2.map } Siegel1 { ; A Siegel disk case reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J passes=1 center-mag=0.812377/1.06798/0.3208747 params=1/0/2/2/0.001/0 float=y maxiter=200 inside=0 outside=real colors=000zzz<28>rhdqgcrid<5>zrj<14>gf3ee0cc0<8>HG0<26>G16F07G08G08H08<2\ 4>O0W<6>vjzzzz<30>fllfllekkdjj<42>DEECEEBDDBDDBDD<21>DD6ED5GF6<21>okL } ParabolicFixed { ; reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J passes=t center-mag=0.812377/0.0440538/0.4506667 params=1/0/2/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000A0K<46>DGh<4>JKwLLzNNz<8>ijz<26>jytkzsjzrjzrizr<24>bzV<6>4G0K0\ 0<29>KDDKDDKEELFF<43>mllnllommommomm<21>mmtlmujkt<21>BFe } InfAndZero_8 { ; Weeeeeird reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J passes=t center-mag=0.812377/0.0440538/0.4506667 params=1/0/1.2/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000bdq<17>C3MA0KA0L<44>DGh<4>JKwLLzNNz<8>ijz<26>jytkzsjzrjzrizr<2\ 4>bzV<6>4G0<13>120000100<13>ICCKDDKDD<45>mllnllommommomm<21>mmtlmujkt<3>\ deq } InfAndZero_9 { ; Weeeeeird reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J passes=t center-mag=0.812377/0.0440538/0.4506667 params=1/0/1.01/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000bdq<17>C3MA0KA0L<44>DGh<4>JKwLLzNNz<8>ijz<26>jytkzsjzrjzrizr<2\ 4>bzV<6>4G0<13>120000100<13>ICCKDDKDD<45>mllnllommommomm<21>mmtlmujkt<3>\ deq } InfAndZero_9b { ; Weeeeeird reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J passes=t center-mag=1.00325/-0.0195923/5.24031 params=1/0/1.01/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000bdq<17>C3MA0KA0L<44>DGh<4>JKwLLzNNz<8>ijz<26>jytkzsjzrjzrizr<2\ 4>bzV<6>4G0<13>120000100<13>ICCKDDKDD<45>mllnllommommomm<21>mmtlmujkt<3>\ deq } InfAndZero_10 { ; Boiling Basins reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=1.00325/-0.0195923/0.3534904 params=1/0/1.1/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000bdq<17>C3MA0KA0L<44>DGh<4>JKwLLzNNz<8>ijz<26>jytkzsjzrjzrizr<2\ 4>bzV<6>4G0<13>120000100<13>ICCKDDKDD<45>mllnllommommomm<21>mmtlmujkt<3>\ deq } InfAndZero_11 { ; A quadratic dendrite mini Julia can be seen. reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.973756/0.290225/5.523287 params=1/0/1.1/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000bdq<17>C3MA0KA0L<44>DGh<4>JKwLLzNNz<8>ijz<26>jytkzsjzrjzrizr<2\ 4>bzV<6>4G0<13>120000100<13>ICCKDDKDD<45>mllnllommommomm<21>mmtlmujkt<3>\ deq } InfAndZero_12 { ; A quadratic Cantor mini Julia can be seen. reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.973756/0.290225/5.523287 params=1/0/1.1008/0.0002/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000bdqbap<16>3CM0AK0AL<44>GDh<4>KJwLLzNNz<8>jiz<26>yjtzkszjrzjrzi\ r<11>zfgzefzeezdezce<8>zVb<6>G04<13>201000001<13>CCIDDKDDK<45>llmllnmmom\ mommo<21>mtmmulktj<3>eqd } 2-cycles_1 { ; A pair of two-cycles reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.399329/0/0.6391454 params=1/0/1.3/0/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000bdqbap<16>3CM0AK0AL<44>GDh<4>KJwLLzNNz<8>jiz<26>yjtzkszjrzjrzi\ r<11>zfgzefzeezdezce<8>zVb<6>G04<13>201000001<13>CCIDDKDDK<45>llmllnmmom\ mommo<21>mtmmulktj<3>eqd } InfAndZero_13 { ; Spinblobs reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.399329/-0.0995335/0.554198 params=1/0/1.1/0.03/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000bdqbap<16>3CM0AK0AL<44>GDh<4>KJwLLzNNz<8>jiz<26>yjtzkszjrzjrzi\ r<11>zfgzefzeezdezce<8>zVb<6>G04<13>201000001<13>CCIDDKDDK<45>llmllnmmom\ mommo<21>mtmmulktj<3>eqd } 2-cycle_2 { ; Spiralicious reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J passes=1 center-mag=0.399329/0/0.4544423 params=1/0/1.95/0.1/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000A4Q<77>dbqdbqdcqecqedq<43>zzz9EU<78>dezdezdezefzefz<43>zzz } 4-cycle_1 { ; A four-cycle reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J passes=t center-mag=0.680396/0.569371/0.3903175 params=1/0/1.9/1/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=0009AQ4AQ<76>bdqbdqcdqceqdeq<43>zzzAQJ<47>SmVSmVSmVTmWTmW<74>zzz } InfAndZero_14 { ; Beautiful whirlpools. reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.680396/0.925056/0.3622146 params=1/0/1.95/1.6/0.001/0 float=y maxiter=16384 inside=0 outside=real colors=0009AQ4AQ<76>bdqbdqcdqceqdeq<43>zzzJAQ<47>VSmVSmVSmWTmWTm<74>zzz } FixedPoint_2 { ; Twistyphant reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=1.16401/3.08474/0.1293624 params=1/0/2.04/6/0.001/0 float=y maxiter=256 inside=0 outside=real periodicity=0 colors=0009AQ4AQ<76>bdqbdqcdqceqdeq<43>zzzJAQ<47>VSmVSmVSmWTmWTm<74>zzz } InfAndZero_15 { ; Weeeeird reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.660246/-0.140387/0.3334082 params=0/1/1.7/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=0009AQ4AQ<76>bdqbdqcdqceqdeq<43>zzzJAQ<47>VSmVSmVSmWTmWTm<74>zzz } InfAndZero_16 { ; Blobbery reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.660246/0.485287/0.3334082 params=0/1/1.6/1/0.001/0 float=y maxiter=256 inside=0 outside=real colors=0009AQ4AQ<76>bdqbdqcdqceqdeq<43>zzzJAQ<47>VSmVSmVSmWTmWTm<74>zzz } 2-cycle_3 { ; Weirdz reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.660246/0.485287/0.3334082 params=-1/0/1.7/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=0009AQ4AQ<76>bdqbdqcdqceqdeq<43>zzzJAQ<47>VSmVSmVSmWTmWTm<74>zzz } CubeRoots { ; Looks a bit like Newton to find cube roots! reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.660246/0.485287/0.3334082 params=0.5/0.85/1.7/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=0009AQ4AQ<76>bdqbdqcdqceqdeq<43>zzzJAQ<47>VSmVSmVSmWTmWTm<74>zzz } FixedPoint_3 { ; Weeeeeird reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.660246/0.485287/0.3334082 params=0.5/0.9/2/0.2/0.001/0 float=y maxiter=256 inside=0 outside=real colors=0009AQ4AQ<76>bdqbdqcdqceqdeq<43>zzzJAQ<47>VSmVSmVSmWTmWTm<74>zzz } InfAndZero_17 { ; Spirella reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.754067/0.062957/0.3334082 params=0.5/1/2/0.2/0.001/0 float=y maxiter=256 inside=0 outside=real colors=0009AQ4AQ<76>bdqbdqcdqceqdeq<43>zzzJAQ<47>VSmVSmVSmWTmWTm<74>zzz } InfAndZero_18 { ; Spiroa reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.438809/-0.0158831/0.5906087 params=-0.7373688780783199/-0.6754902942615236/1.492/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000A9QA4Q<76>dbqdbqdcqecqedq<43>zzzJAQ<47>VSmVSmVSmWTmWTm<74>zzz } 8-cycle_1 { ; Weird rings reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=1.32231/-0.0158831/0.3304479 params=-0.7373688780783199/-0.6754902942615236/3/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000A9QA4Q<76>dbqdbqdcqecqedq<43>zzzQAJ<47>mSVmSVmSVmTWmTW<74>zzz } HERMAN_01 { ; Herman Ring! reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=1.57026/-0.0158831/0.2523142 params=-0.7373688780783199/-0.6754902942615236/3.4/0/0.001/0 float=y maxiter=256 inside=0 outside=real periodicity=0 colors=0009QA4QA<76>bqdbqdcqdcqedqe<43>zzzQAJ<47>mSVmSVmSVmTWmTW<74>zzz } InfAndZero_19 { ; Forest and Lakes reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=1.57026/-0.0158831/0.2523142 params=-0.7373688780783199/-0.6754902942615236/3.4/0.31/0.001/0 float=y maxiter=256 inside=0 outside=real periodicity=0 colors=0009QA4QA<76>bqdbqdcqdcqedqe<43>zzzAJQ<47>SVmSVmSVmTWmTWm<74>zzz } InfAndZero_19b { ; Forest and Lakes zoom reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.45449/-0.0158831/1.855251 params=-0.7373688780783199/-0.6754902942615236/3.4/0.31/0.001/0 float=y maxiter=16384 inside=0 outside=real periodicity=0 colors=0009QA4QA<76>bqdbqdcqdcqedqe<28>rwsrwsswsswstws<10>zzvAJQ<47>SVmS\ VmSVmTWmTWm<74>zzz } InfAndZero_20 { ; Weird spirals reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.45449/-0.0158831/0.6502943 params=0.09801751303322832/-0.9951846899640191/1.5/0/0.001/0 float=y maxiter=16384 inside=0 outside=real periodicity=0 colors=0009QA4QA<76>bqdbqdcqdcqedqe<28>rwsrwsswsswstws<10>zzvAJQ<47>SVmS\ VmSVmTWmTWm<74>zzz } InfAndZero_20b { ; Conch-iousness reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.59078/-0.667078/13.00589 params=0.09801751303322832/-0.9951846899640191/1.5/0/0.001/0 float=y maxiter=16384 inside=0 outside=real periodicity=0 colors=000DJQ<44>ORdORdPSePTf<13>VYkWZkWZkW_kX_k<43>pypqzqqzqrzr<12>zzzJ\ DQ<44>XTkYUlYUlZVmZVm<28>hfqhfrifrifrjfr<28>yrrzsszsszss<12>zzz } InfAndZero_21 { ; Bliral reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.344086/-4.44089e-016/0.5226667 params=0.09801751303322832/-0.9951846899640191/1.7/0.2/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000DJQ<44>ORdORdPSePTf<13>VYkWZkWZkW_kX_k<43>pypqzqqzqrzr<12>zzzJ\ DQ<44>XTkYUlYUlZVmZVm<28>hfqhfrifrifrjfr<28>yrrzsszsszss<12>zzz } InfAndZero_22 { ; Bliral_2 reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.344086/-4.44089e-016/0.5226667 params=0.09801751303322832/-0.9951846899640191/1.7/0.02/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000DJQ<44>ORdORdPSePTf<13>VYkWZkWZkW_kX_k<43>pypqzqqzqrzr<12>zzzJ\ DQ<44>XTkYUlYUlZVmZVm<28>hfqhfrifrifrjfr<28>yrrzsszsszss<12>zzz } InfAndZero_23 { ; Goldenthreads reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.842819/-8.88178e-016/0.4285867 params=0.09801751303322832/-0.9951846899640191/2.2/0.02/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000K00<125>zz0U00<126>zz0 } InfAndZero_24 { ; Jewels, Emeralds, and Rubies reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=1.27159/-1.77636e-015/0.3404555 params=0.09801751303322832/-0.9951846899640191/2.7/0.02/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=0000K0<125>zz0U00<126>zz0 } InfAndZero_25 { ; Jewels, Red Opal, and Lapis Lazuli reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=-1.48477/0.251341/0.2669171 params=0.09801751303322832/-0.9951846899640191/-3.2/0.5/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=00000K<125>0zzU00<126>zz0 } FixedPoint_4 { ; Blues reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.295015/1.62085/0.2284811 params=0.09801751303322832/-0.9951846899640191/1/3/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000AKU<125>Nfz5AK<126>kpz } 4-cycle_2 { ; Dragons clutching Rubies reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.397337/1.92792/0.2038051 params=0.09801751303322832/-0.9951846899640191/1.4/3.7/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000KAU<125>fNzK5A<7>TEJ<6>tej<111>zkp } 4-cycle_3 { ; Lakes behind walls reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.397337/2.30239/0.1671202 params=0.09801751303322832/-0.9951846899640191/0.5/3.8/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=0000K0<35>MxdNzfNzf<87>zzz5AK<7>EJT<6>ejt<111>kpz } 4-cycle_4 { ; Fire and Ice reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=0.397337/2.30239/0.1671202 params=0.09801751303322832/-0.9951846899640191/-0.5/3.9/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000K00<35>xdMzfNzfN<87>zzz5AK<7>EJT<6>ejt<111>kpz } InfAndZero_26 { ; Firestorm reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J passes=1 center-mag=0.0541845/-0.0848576/0.2347194 params=0.09801751303322832/-0.9951846899640191/2/7/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000K00<35>xdMzfNzfN<87>zzz5AK<7>EJT<6>ejt<111>kpz } InfAndZero_27 { ; JackFrost reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=-1.47555/-0.756048/0.2347194 params=0.09801751303322832/-0.9951846899640191/-3/-1.2/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=00000K<35>dMxfNzfNz<87>zzzA5K<7>JET<6>jet<111>pkz } HERMAN_02 { ; Liouville number Herman ring reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=-1.47555/-0.133929/0.2347194 params=0.09801751303322832/-0.9951846899640191/-3.3/0/0.001/0 float=y maxiter=100000 inside=0 outside=real colors=000zJc<76>11B00A00A00A<45>000zz0<77>L10K00K00K00<45>000 } HERMAN_02b { ; Liouville number Herman ring zoom reset=1960 type=formula formulafile=hring.frm formulaname=HRing_J center-mag=-2.50837/0.310442/2.347194 params=0.09801751303322832/-0.9951846899640191/-3.3/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000zJc<76>11B00A00A00A<45>000zz0<77>L10K00K00K00<45>000 } C-Mandel_1 { ; The M-set in the c-plane, alpha is 1. reset=1960 type=formula formulafile=hring.frm formulaname=HRing_Mc center-mag=0/0/0.6666667 params=1/0/0/0/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000QJD<44>dROdROeSPfTP<13>kYVkZWkZWk_Wk_X<43>pypqzqqzqrzr<12>zzzQ\ DJ<44>kTXlUYlUYmVZmVZ<28>qfhrfhrfirfirfj<28>rrysszsszssz<12>zzz } C-Mandel_1z1 { ; The M-set in the c-plane, alpha is 1. reset=1960 type=formula formulafile=hring.frm formulaname=HRing_Mc center-mag=-1.19453/0.400443/1.626016 params=1/0/0/0/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000JDQ<44>ROdROdSPeTPf<13>YVkZWkZWk_Wk_Xk<43>yppzqqzqqzrr<12>zzzQ\ DJ<44>kTXlUYlUYmVZmVZ<28>qfhrfhrfirfirfj<28>rrysszsszssz<12>zzz } C-Mandel_1z1a { ; The M-set in the c-plane, alpha is 1. ; Scepter Valley reset=1960 type=formula formulafile=hring.frm formulaname=HRing_Mc center-mag=+1.41142913000977500/+0.02727379400260770/239.12 params=1/0/0/0/0.001/0 float=y maxiter=16384 inside=0 outside=real colors=000DJQ<125>mzmQDJ<126>zzc } C-Mandel_i { ; The M-set in the c-plane, alpha is i. reset=1960 type=formula formulafile=hring.frm formulaname=hring_mc center-mag=0.239075/0.298587/0.1755667 params=0/1/0/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000QJD<44>dROdROeSPfTP<12>kYVkYVkZWkZWk_W<44>pypqzqqzqrzr<12>zzzQ\ DJ<42>jSWkTXkTXlUYlUY<29>qfhqfhrfhrfirfi<29>rrysszsszssz<12>zzz } C-Mandel_-1 { ; The M-set in the c-plane, alpha is -1 reset=1960 type=formula formulafile=hring.frm formulaname=hring_mc center-mag=0.239075/0.298587/0.1755667 params=-1/0/0/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000QJD<44>dROdROeSPfTP<12>kYVkYVkZWkZWk_W<44>pypqzqqzqrzr<12>zzzQ\ DJ<42>jSWkTXkTXlUYlUY<29>qfhqfhrfhrfirfi<29>rrysszsszssz<12>zzz } C-Mandel_0.5 { ; The M-set in the c-plane, alpha is 0.5 reset=1960 type=formula formulafile=hring.frm formulaname=hring_mc center-mag=0.239075/0.0015422/0.3539652 params=0.5/0/0/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000QJD<44>dROdROeSPfTP<12>kYVkYVkZWkZWk_W<44>pypqzqqzqrzr<12>zzzQ\ DJ<42>jSWkTXkTXlUYlUY<29>qfhqfhrfhrfirfi<29>rrysszsszssz<12>zzz } C-Mandel_0.5+0.8i { ; The M-set in the c-plane, alpha is 0.5+0.8i. reset=1960 type=formula formulafile=hring.frm formulaname=hring_mc center-mag=0/4.9738e-014/0.009469697 params=0.5/0.8/0/0/1e-006/0 float=y maxiter=256 inside=0 outside=real periodicity=0 colors=000QJD<44>dROdROeSPfTP<12>kYVkYVkZWkZWk_W<44>pypqzqqzqrzr<12>zzzQ\ DJ<42>jSWkTXkTXlUYlUY<29>qfhqfhrfhrfirfi<29>rrysszsszssz<12>zzz } C-Mandel_iz1 { ; Seahorse valley? reset=1960 type=formula formulafile=hring.frm formulaname=hring_mc center-mag=6.6038/-0.00288984/2.033325 params=0/1/0/0/0.001/0 float=y maxiter=16384 inside=0 outside=real colors=000DJQ<60>oxopyppyppyp<61>zzzQDJ<59>povqpwqqxrryrry<62>zzz } C-Mandel_Golden { ; The M-set in the c-plane, alpha is exp(2*pi*i*a) with ; a the Golden mean 0.6180339... reset=1960 type=formula formulafile=hring.frm formulaname=hring_mc center-mag=1.70479/0.453716/0.08591065 params=-0.7373688780783199/-0.6754902942615236/0/0/1e-006/0 float=y maxiter=32768 inside=0 outside=real periodicity=0 colors=000bdq<17>C3MA0KA0L<44>DGh<4>JKwLLzNNz<8>ijz<26>jytkzsjzrjzrizr<2\ 4>bzV<6>4G0<13>120000100<13>ICCKDDKDD<45>mllnllommommomm<21>mmtlmujkt<3>\ deq } C-Mandel_Golden_z1 { ; The M-set in the c-plane, alpha is exp(2*pi*i*a) with ; a the Golden mean 0.6180339... reset=1960 type=formula formulafile=hring.frm formulaname=hring_mc center-mag=0.0663148/-0.0622686/0.3068238 params=-0.7373688780783199/-0.6754902942615236/0/0/1e-006/0 float=y maxiter=512 inside=0 outside=real periodicity=0 colors=000bdq<17>C3MA0KA0L<44>DGh<4>JKwLLzNNz<8>ijz<11>ipwiqwiqwiqw<29>p\ ezzde<14>U9EN59G04<10>502104103102000<14>ICCKDDKDD<43>ljjmkkmllmll<30>qq\ d } C-Mandel_Liouville { ; The M-set in the c-plane, alpha is exp(2*pi*i*a) with ; a the Liouville number equal to the sum over natural n ; of 2 to the power minus n factorial. reset=1960 type=formula formulafile=hring.frm formulaname=hring_mc center-mag=0.236231/1.77636e-015/0.07236409 params=0.09801751303322832/-0.9951846899640192/0/0/1e-006/0 float=y maxiter=3500 inside=0 outside=real periodicity=0 colors=000bdq<17>C3MA0KA0L<44>DGh<4>JKwLLzNNz<8>ijz<11>ipwiqwiqwiqw<29>p\ ezzde<14>U9EN59G04<10>502104103102000<14>ICCKDDKDD<43>ljjmkkmllmll<30>qq\ d } HERMAN_03 { ; Herman ring elephants! reset=1960 type=formula formulafile=hring.frm formulaname=hring_j center-mag=1.77553/-0.0389762/0.2198578 params=0.8/0.6/3.75/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000KFP<125>puzQAJ<46>mSVmSVmSVmSVmTW<75>zzz } HERMAN_03z1 { ; Herman ring elephants! reset=1960 type=formula formulafile=hring.frm formulaname=hring_j center-mag=3.35243/-0.928493/4.397157 params=0.8/0.6/3.75/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000AJQ<46>SVmSVmSVmSVmTWm<74>zzzPFK<12>SIMSIMTJNTKN<110>zup } HERMAN_03z2 { ; Herman ring elephants! reset=1960 type=formula formulafile=hring.frm formulaname=hring_j center-mag=3.35065/-0.364243/4.397157 params=0.8/0.6/3.75/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000A0c<21>J9yKAzKAzKBz<100>zzzA00<126>zup } HERMAN_04 { ; Gorgeous greens. reset=1960 type=formula formulafile=hring.frm formulaname=hring_j center-mag=1.80092/-8.88178e-016/0.2590888 params=-0.6/0.8/3.75/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000tyo<125>0A0Ac0<22>Jy9KzAKzAKzB<19>TzLTzLTzLTyLSyL<76>0A0 } HERMAN_04z1 { reset=1960 type=formula formulafile=hring.frm formulaname=hring_j passes=t center-mag=-1.37634e-005/-0.0603861/1.061839 params=-0.6/0.8/3.75/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000oty<125>00Ac0A<22>y9JzAKzAKzBK<19>zLTzLTzLTyLTyLS<76>A00 } A-Mandel_1 { ; The M-set in the alpha plane, c is 3.5 reset=1960 type=formula formulafile=hring.frm formulaname=HRing_Ma center-mag=0.808717/-4.44089e-016/0.5466667 params=0/0/3.5/0/0.0001/0 float=y maxiter=2000 inside=0 outside=real colors=000bCC<10>hDG<5>zLL<8>zfgzijzij<25>tjyskzrjzrjzriz<13>eezezd<9>bz\ V<6>4G0<13>120000100<3>644447<8>CCIDDKDDK<45>llmllnmmommommo<21>tmmumltk\ j<3>qeddbq<17>3CM0AK0AL<32>BCa } InfAndZero_28 { ; Gorgeous orange fireswirls. reset=1960 type=formula formulafile=hring.frm formulaname=hring_j center-mag=1.94166/-0.0603861/0.2046131 params=-0.95/-0.2/4/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000oty<125>00AcA0<22>yJ9zKAzKAzKB<19>zTLzTLzTLyTLySL<76>A00 } InfAndZero_29 { ; Gorgeous! reset=1960 type=formula formulafile=hring.frm formulaname=hring_j center-mag=1.94166/-0.0603861/0.2046131 params=-0.97/-0.2/4/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000cA0<22>yJ9zKAzKAzKB<19>zTLzTLzTLyTLySL<75>B11oyt<11>lvpkuokuoj\ tnisn<110>0A0 } HERMAN_05 { ; Gorgeous! reset=1960 type=formula formulafile=hring.frm formulaname=hring_j passes=t center-mag=1.94166/-0.0603861/0.2046131 params=-0.98/-0.2/4/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000oyt<125>11BKAc<31>KczKczKcy<92>B11 } InfAndZero_30 { ; BlueFuzzRings reset=1960 type=formula formulafile=hring.frm formulaname=hring_j center-mag=1.94166/-0.0603861/0.2046131 params=-0.65/0.75/4/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000yot<125>11BKAc<31>KczKczKcy<92>B11 } HERMAN_06 { ; Starfish! reset=1960 type=formula formulafile=hring.frm formulaname=hring_j passes=t center-mag=3.78026/0.0127637/1.199655 params=-0.8/0.6/4/0/0.001/0 float=y maxiter=256 inside=0 outside=real colors=000tqz<29>f_peZoeZoeZn<92>B11cAK<29>yaKybKzcKzcKycK<92>11B } InfAndZero_31 { ; Elephoids reset=1960 type=formula formulafile=hring.frm formulaname=hring_j2 center-mag=1.37766/0.053528/0.2059799 params=0.82/0.57/3.6/0/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=Ksvqtz<29>_fpZeoZeoZen<92>1B1KKc<30>KxyKzzKzz<93>BB1 } InfAndZero_31b { ; Conchoids reset=1960 type=formula formulafile=hring.frm formulaname=hring_j2 center-mag=3.44154/2.55217/3.742367 params=0.82/0.57/3.6/0/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000UAK<29>prxqtzqtz<75>C9IB8HA7GA7F97E<13>000KAU<30>xrpztqztq<75>\ IC9HB8GA7FA7E97<13>000 } InfAndZero_31c { ; Skin of A(infinity) closeup reset=1960 type=formula formulafile=hring.frm formulaname=hring_j2 center-mag=3.44781/2.61558/43.5159 params=0.82/0.57/3.6/0/0.001/0 float=y maxiter=2000 inside=0 outside=real colors=000Xfp<29>A28906906<75>nhqoirpjspksqls<13>zzzpfX<30>82A609609<75>\ nhqoirpjspksqls<13>zzz } - -- .*. Friendship, companionship, love, and having fun are the reasons for -() < life. All else; sex, money, fame, etc.; are just to get/express these. `*' Send any and all mail with attachments to the hotmail address please. Paul Derbyshire ao950@freenet.carleton.ca pgd73@hotmail.com - - - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint" ------------------------------ End of fractint-digest V1 #86 *****************************