Rendering Kleinian group fractals
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I'm a C# programmer who loves fractals! I've spent days trying to figure out the math behind Kleinian group fractals with no success. There are a lot of Mandelbulb3D parameter sets available (as well as UltraFractal, Fragmentarium and other fractal applications) But I really want to understand the math itself, and the parameter sets really don't help Here are a couple of beautiful examples: https://www.deviantart.com/sabine62/art/Tentacle-653134425[^] https://www.deviantart.com/schmiegel/art/Kleine-Abschnitte-659324989[^] I've found a lot of scientific articles on Mobius transformations, spherical reflection, and elaborate sounding stuff like "Quasi-Fuchsian", but it's bits and pieces here and there. I can find and have figured out how to calculate the curvature of a new circle tangent to 3 other circles, but not how to calculate the center-point of the new circle. I've about worn google out I think.... does anyone know of a good resource for how to create these fractals in language for a programmer who's not a mathematician? I know I can do the math with the right resources to learn it, I just can't read symbolic math (i.e. Greek symbols) Thanks in advance! Don [Irrelevant link deleted]
The problem with your question is that to understand the math, you have to understand the language of math. Anyone explaining those concepts to you is going to use the "Greek symbols" you don't know. So, go learn the "Greek symbols" and everything will make a lot more sense.
Asking questions is a skill CodeProject Forum Guidelines Google: C# How to debug code Seriously, go read these articles.
Dave Kreskowiak -
I'm a C# programmer who loves fractals! I've spent days trying to figure out the math behind Kleinian group fractals with no success. There are a lot of Mandelbulb3D parameter sets available (as well as UltraFractal, Fragmentarium and other fractal applications) But I really want to understand the math itself, and the parameter sets really don't help Here are a couple of beautiful examples: https://www.deviantart.com/sabine62/art/Tentacle-653134425[^] https://www.deviantart.com/schmiegel/art/Kleine-Abschnitte-659324989[^] I've found a lot of scientific articles on Mobius transformations, spherical reflection, and elaborate sounding stuff like "Quasi-Fuchsian", but it's bits and pieces here and there. I can find and have figured out how to calculate the curvature of a new circle tangent to 3 other circles, but not how to calculate the center-point of the new circle. I've about worn google out I think.... does anyone know of a good resource for how to create these fractals in language for a programmer who's not a mathematician? I know I can do the math with the right resources to learn it, I just can't read symbolic math (i.e. Greek symbols) Thanks in advance! Don [Irrelevant link deleted]
Not sure Mathematica code would help, but, if you haven't seen this: [^], and, posting a message here: [^] would, I think connect you with a large network of people programming fractals.
«Where is the Life we have lost in living? Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information?» T. S. Elliot
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The problem with your question is that to understand the math, you have to understand the language of math. Anyone explaining those concepts to you is going to use the "Greek symbols" you don't know. So, go learn the "Greek symbols" and everything will make a lot more sense.
Asking questions is a skill CodeProject Forum Guidelines Google: C# How to debug code Seriously, go read these articles.
Dave KreskowiakSo, you can't help the OP, but tell him, in essence: "get a brain." So typical of far too many of your zero-content posts. Do you really need rep-points that bad ? Down-voted.
«Where is the Life we have lost in living? Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information?» T. S. Elliot
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So, you can't help the OP, but tell him, in essence: "get a brain." So typical of far too many of your zero-content posts. Do you really need rep-points that bad ? Down-voted.
«Where is the Life we have lost in living? Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information?» T. S. Elliot
W T F? My message wasn't "get a brain". It's more "if you want to understand this stuff and open the door to more of it, then you 're going to have to learn the language of math". First, how are you going to explain the math to him if he can't understand the language of math? Second, after, what, 15 years around here, I don't give a rats ass about rep points. Never have. Never will.
Asking questions is a skill CodeProject Forum Guidelines Google: C# How to debug code Seriously, go read these articles.
Dave Kreskowiak -
W T F? My message wasn't "get a brain". It's more "if you want to understand this stuff and open the door to more of it, then you 're going to have to learn the language of math". First, how are you going to explain the math to him if he can't understand the language of math? Second, after, what, 15 years around here, I don't give a rats ass about rep points. Never have. Never will.
Asking questions is a skill CodeProject Forum Guidelines Google: C# How to debug code Seriously, go read these articles.
Dave KreskowiakDave Kreskowiak wrote:
after, what, 15 years around here, I don't give a rats ass about rep points. Never have. Never will.
I believe you. The issue is whether you use these forums to post snide, content-free, responses that may be perceived as mocking by new students. Also, once OriginalGriff has posted the obligatory "we don't do your home-work" message, are further me-too responses needed ? I don't question your ability to contribute quality content; in fact, I have frequently up-voted your responses.
«Where is the Life we have lost in living? Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information?» T. S. Elliot
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The problem with your question is that to understand the math, you have to understand the language of math. Anyone explaining those concepts to you is going to use the "Greek symbols" you don't know. So, go learn the "Greek symbols" and everything will make a lot more sense.
Asking questions is a skill CodeProject Forum Guidelines Google: C# How to debug code Seriously, go read these articles.
Dave KreskowiakThanks Dave, Maybe I should have been more clear, I can decipher the Greek however it's very painful, I read code much better than symbolic. If as you say symbolic is the only way to get there I can do that. I have had some success generating Kleinian fractals in 3D, but the Googleable information always seems to be incomplete. I guess I'm going to have to break down and buy a copy of "Indra's Pearls" it's just hard to justify the $60 price tag.
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Thanks Dave, Maybe I should have been more clear, I can decipher the Greek however it's very painful, I read code much better than symbolic. If as you say symbolic is the only way to get there I can do that. I have had some success generating Kleinian fractals in 3D, but the Googleable information always seems to be incomplete. I guess I'm going to have to break down and buy a copy of "Indra's Pearls" it's just hard to justify the $60 price tag.
I would highly recommend working through the pain. The symbology of math is no different from any other programming language. It's a specialized language developed over time to describe mathematical modeling concepts. The languages we use to write code are no different, though they are not specialized for the same purposes. Computer languages, to varying degrees, are more generalized, or more specialized, to describe logical, control, and data modeling. That's how we get stuff like FORTRAN, COBOL, C, the PLC languages. The language of math is just another "programming" language, most commonly used in writing the papers we all love to hate :) but are indispensable to conveying the concepts we use every day. If you learn it, you open yourself up to many more sources of information. After 40 years of doing this, I still hate it, can't remember a lot of it, and have to go back and teach myself parts of the language all over again.
Asking questions is a skill CodeProject Forum Guidelines Google: C# How to debug code Seriously, go read these articles.
Dave Kreskowiak -
Not sure Mathematica code would help, but, if you haven't seen this: [^], and, posting a message here: [^] would, I think connect you with a large network of people programming fractals.
«Where is the Life we have lost in living? Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information?» T. S. Elliot
Bill, Thank you for your posts! I did find the "learn the language" response frustrating, mainly because if math can be coded to run on a computer (and in the case of fractals MUST be run on a computer) how and why is the Greek even helpful? That being said, I can work through the Greek and figure out how to get it into code form. In my reply to that post I admitted my best hope is to spend the $60 and just buy the book. I truly appreciate the links you provided and have been using those resources as much as possible. I used to spend a lot of time on fractal forums back when I was wrote a raytracer for 3D flame fractals.
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I would highly recommend working through the pain. The symbology of math is no different from any other programming language. It's a specialized language developed over time to describe mathematical modeling concepts. The languages we use to write code are no different, though they are not specialized for the same purposes. Computer languages, to varying degrees, are more generalized, or more specialized, to describe logical, control, and data modeling. That's how we get stuff like FORTRAN, COBOL, C, the PLC languages. The language of math is just another "programming" language, most commonly used in writing the papers we all love to hate :) but are indispensable to conveying the concepts we use every day. If you learn it, you open yourself up to many more sources of information. After 40 years of doing this, I still hate it, can't remember a lot of it, and have to go back and teach myself parts of the language all over again.
Asking questions is a skill CodeProject Forum Guidelines Google: C# How to debug code Seriously, go read these articles.
Dave KreskowiakThank you Dave, While frustrating, your advice is not at all unexpected and is much appreciated.
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Bill, Thank you for your posts! I did find the "learn the language" response frustrating, mainly because if math can be coded to run on a computer (and in the case of fractals MUST be run on a computer) how and why is the Greek even helpful? That being said, I can work through the Greek and figure out how to get it into code form. In my reply to that post I admitted my best hope is to spend the $60 and just buy the book. I truly appreciate the links you provided and have been using those resources as much as possible. I used to spend a lot of time on fractal forums back when I was wrote a raytracer for 3D flame fractals.
I'm very glad you found the links helpful. Fractals are fascinating ! cheers, Bill
«Where is the Life we have lost in living? Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information?» T. S. Elliot