An Interview with Thomas Ludwig

Thomas Ludwig (aka lyc on deviantart and lycium everywhere else) is a very friendly, talented, well traveled and busy fellow. He was born in Germany, lived in Indonesia from age 5-8 and in South Africa until age 19, before starting a nomadic life roughly centered around New Zealand (where his family now lives) and Europe.

He will be speaking at SIGGRAPH in Vancouver, Canada this year. SIGGRAPH is the world’s largest, most influential annual conference and exhibition in computer graphics and interactive techniques: Five days of research results, demos, educational sessions, art, screenings, and hands-on interactivity featuring the community’s latest technical achievements, and three days of commercial exhibits displaying the industry’s current hardware, software, and services.

I learned a lot during our many chats leading up to this interview and have been pleasantly surprised at the detail Thomas has provided to my questions.


MM:  How did you meet Daniel White and Paul Nylander?

Thomas Ludwig:  Daniel White (twinbee) first posted on Fractal Forums (where I was an active user since its inception in 2006) with a paper sketch of what kind of 3D Mandelbrot he was hoping to find. Some interesting discussion of mathematical limits (particularly Liouville’s theorem) followed, which made me rather pessimistic about finding algebraic 3D fractals that are detailed everywhere, without the smooth “whipped cream” effect that is well-known from the quaternionic Julia sets.

Paul Nylander I’d run into before in my internet travels; he maintains a very informative and interesting website full of math and physics stuff.

MM: Both Daniel and Paul have praised you for your contributions to the discovery of the 3D Mandelbulb! In your own words, can you recall exactly what your role was and how that influenced the process?

Thomas Ludwig: I was lucky to be in the right place (active on Fractal Forums) at the right time, but as they say, fortune favors the prepared: I had been writing an experimental rendering engine at the time, with global illumination by Metropolis light transport (specifically the super elegant variant by Csaba Kelemen).

Daniel White’s description basically gives you an inside/outside test at a point, i.e. given a point (x,y,z) in 3D space you can determine if it is inside or outside his 3D fractal; this is in contrast to how 3D fractals are normally described, in terms of distance estimation functions (à la John C. Hart).

Having only an inside/outside test made the rendering extremely slow since you have to “march” along visibility rays using a fixed step distance, which cannot be too high else you will miss details, and in a fractal that is a very problematic matter. It also makes it very difficult to estimate the surface normal (orientation) once you have found the intersection point.

At the time CPUs were not so fast, and to be honest, MLT is not the best algorithm for simple lighting situations, so the first images had quite a lot of noise in them. 

I was quite happy with the slice of the power-2 Mandelbulb, which indeed looks like the outline of the 2D Mandelbrot, but begins to extend upwards in interesting ways; I’m quite bored of the Mandelbrot (having used FRACTINT in the MS-DOS days), and the Mandelbulb is to me much more visually interesting. 

MM: In your opinion, how has technology advanced the science of fractal and chaos theory as it pertains to rendering 3D images and fractals in particular? What tools do you believe have most improved the process of discovery for the field of Fractal Sciences as well as the artist?

Thomas Ludwig: A major advance over the previously described inside/outside-based rendering method is the use of distance estimators, which I believe was first derived for the Mandelbulb by Íñigo Quílez, after he did some great work simplifying the process from being in terms of spherical angles to simple algebra. Nothing less from a great demoscene wizard!

I can’t say I’ve noticed other big improvements on the technical front; most of the specialized fractal rendering software has quite dated rendering technology. Fortunately, the much more difficult problem of how to effectively expose all of this weird fractal stuff in a user-friendly way has been attacked by software such as Mandelbulb3D, Mandelbulber and Fragmentarium (the latter in a more indirect way, as a general shader editing tool).

MM: I know you are one of three employees in Glare Technologies Limited which was created in 2008 as the parent company for Indigo Renderer and Chaotica. Can you tell us a bit about Indigo Renderer and Chaotica and how they differ from Mandelbulb 3D?

Thomas Ludwig: Chaotica is the easier one to differentiate, because (currently) it’s a purely 2D renderer; there is no depth of any kind, and it uses the Iterated Function System (IFS) instead of ray tracing. IFS is a bit like astrophotography: you have a virtual “film” or sensor, which leave to expose, i.e. collect virtual photon hits.

Indigo Renderer is a 3D raytracer, like Mandelbulb 3D, but it places much more emphasis on physical accuracy and is a professional product aimed mainly at architectural and product visualization markets.

MM: Does Chaotica use GPU for rendering or just CPU?

Thomas Ludwig: It currently only runs on CPU, however, it’s important to note that the engine runs in double precision (which GPUs do at 1/32 normal speed), and is highly optimized to the point where it gives the GPU a good run for its money.

Of course, there will be cases where double precision is not needed and people have multiple GPUs / weak CPUs, and for those cases, Chaotica will have GPU rendering in the next version (which has been a very long time coming).

MM: Is Indigo Renderer designed for use in conjunction with other programs or as a standalone application? If so, what other programs can use it?

Thomas Ludwig: It can be used standalone to a degree, though it’s almost always used with other programs such as 3ds max, Cinema 4D, Blender, SketchUp, etc.

MM: What accomplishments can you attribute to these programs? Are they still in ongoing development?

Thomas Ludwig: Chaotica gets used in a bunch of Hollywood productions, most recently Annihilation. It’s been used by beeple a few times, which I’m personally very happy about because he’s one of my favorite artists.

Indigo Renderer is much better known than Chaotica I think, but in a more “indie” way; in the 3D world, Indigo is the underdog. We compete with much, much larger companies.

Both are still in active development, it’s what we do at Glare Technologies 🙂

MM: You have written various programs over the years. Can you expound on these and provide links and samples, if available?

Thomas Ludwig: I’ve been programming more or less nonstop since age 15 or so, so there will be literally hundreds of small programs, almost all without proper names. I’m mainly interested in topics involving numerical computing, such as ray tracing, fractals, gravitational simulations (N-body problem), … things like that.

I specialize in computer graphics, and the list of programs meaningful to others is pretty complete with Chaotica and Indigo Renderer.

MM: Do you have any other projects in the works at this time?

Thomas Ludwig: Far too many! I’ve recently started learning about Machine Learning (ML), particularly neural nets and Monte Carlo Tree Search (MCTS); I’d like to eventually be able to produce a general AI like DeepMind’s AlphaZero (not to be confused with AlphaGo or AlphaGo Zero). I’ve loved Go for a long time but have remained a very weak player, whereas my girlfriend is a strong and passionate daily player; there’s some dark humor in writing an AI to eventually beat her at it 🙂

Besides that, I’d like to start making some programming tutorial videos. The plan for many years was to write a book based off collected chat logs from the many people I’ve taught over the years, but I’ve had to give up on that since I’m just too disorganized and thinly spread for such a big undertaking.

MM: Considering your experience and such a storied history in the fractal community, what advice would you offer to a new artist just starting to explore fractals?

Thomas Ludwig: Hahaha, I don’t think it’s really that “storied”, I just lurk a lot online and do what I can 🙂 I think by the time someone is already interested in fractals, the course has been set, and it’s mostly a question of controlling one’s fractal addiction!

Interest is the driving force, as long as they have that, it’s just a matter of how much time they can afford to spend on it. Joining an online community such as Fractal Forums, or perhaps the Fractal Chat channel on Discord, is probably a good idea since there are many people from all walks of life, using all kinds of programs; there’s something for everyone at all skill levels.

MM: Are you still enthusiastic about the future of fractal research and the possibility of new discoveries?

Thomas Ludwig: There’s a lot of buzz around fractals at the moment, and they’ve been steadily increasing in mindshare over the years, so I have no doubt that there will be a lot of interesting developments in the coming years; probably the best is yet to come!

Ours is a very young field, which is very exciting because there’s lots for everyone to do.

MM: My most humble thanks to Thomas for making time over the past few weeks to be a part of this interview given his busy schedule. His love of the history of fractals has been very evident throughout this process.

This interview was conducted by Ricky Jarnagin, creator of the Facebook group Mandelbulb Maniacs  | May 8, 2018







An Interview with Paul Nylander

It has been an honor and a pleasure to interview Paul Nylander who played a significant role in the discovery of the 3D Mandelbulb. Please visit his website to learn much more about the process and coding that went into the discovery of not only the first 3D Mandelbulb but the dozens of other programs he used during the process:


MM: Please speak to your background, education, family life and whatever you want the public to know about you.

Paul Nylander: Hi, my name is Paul Nylander. My education is in mechanical engineering and physics. My engineering experiences have been somewhat random ranging from plutonium detectors to Disneyland rides. Now I am working on surgical devices, which I must say is the best job I’ve ever had. I have also worked as a C++ and C# software developer. On my spare time, I enjoy math, science, and art.

MM: How did you learn about fractals?

Paul Nylander: The first time I saw a picture of the Mandelbrot set fractal was at a bookstore in 1998. I was fascinated by it and I wanted to learn how to create it, but I couldn’t understand the math. So I went around my college math department asking professors for help, but many of them didn’t even know what a fractal was. It wasn’t until 2001 that I finally found a website that explained it in a way I could understand. I stayed up late and finally created my first complex fractal, but it was distorted and looked like cobwebs. I felt that I had discovered something great. I excitedly showed it to other people, but they didn’t seem very interested. But I continued learning new fractals and adding them to my website.

MM: You had the opportunity to engage with many of the pioneers during the early development of 3D fractals. Do you still communicate with any of them?

Paul Nylander: Not very often, but I do have much respect for them. I feel Daniel White deserves the most recognition for coming up with the best 3D Mandelbrot formula and advancing everyone’s appreciation of 3D fractals. Thomas Ludwig also played a critical role in creating the first high-quality rendering. Rudy Rucker beat everyone to the punch with his similar formula in 1990, but it didn’t attract much interest at the time because he did not have a high-quality rendering of his beast. I can only imagine what might have happened if Rudy had access to the same rendering capabilities back in 1990.

MM: What went through your mind when you first rendered the 8th order Mandelbulb set based on following the generalized variation of Daniel White’s original squaring formula? How long had you been working toward that discovery before it happened?

Paul Nylander: I had been working on 3D Mandelbrots for about 2 months before I rendered higher power Mandelbulbs. I thought they were pretty neat, but I didn’t think they were a breakthrough. Everyone’s goal was to find a 3D formula that that could produce all the wild variety and extravagantly ornate details of the 2D Mandelbrot Set. The problem with the standard quadratic Mandelbulb is that it has some distorted regions that look like stretched out taffy. The higher order Mandelbulbs certainly are an improvement, but they also have regions that look stretched out.

MM: You mention the “Persistence of Vision Raytracer” (POV Raytracer) multiple times on your website. How critical was this software to the development of the first 3D Fractal images?

Paul Nylander: Not at all. C++ was my tool of choice. It wasn’t until later that I learned how to render hypercomplex fractals on POV-Ray, although I must say it is a very slow way to render. However, I do think POV-Ray is a convenient tool for visualizing other types of mathematical beauties.

MM: I know you are a Mechanical Engineer. Rapidly changing technology has introduced many new industries requiring the expansion of the scope of design tools available to Mechanical Engineers, designers, and drafters. Considering advances in robotics, rendering, 3D design and printing, texturing and 3D modeling: How have technological advances affected CAD/CAM software and the ongoing skills required for Mechanical Engineers today?

Paul Nylander: There is much I can say about this, but I can only speak from my own personal experience, so my advice may not be applicable to everyone. I learned a great deal during my most recent job transition. Some people say the half-life of an engineering degree is 5 years, others say 2 years. I can say from my own personal experience that I have used very little of what I learned in college throughout my career. The key is to learn how to learn. If you are becoming bored at your job, then that is a sign that you have probably stopped learning and it’s time to move on to a different job. If you are afraid to take the risk of changing jobs, you may be putting yourself at even greater risk because your marketability as an engineer will plummet. If you find an employer that truly cares about developing your skills, then you have much to be thankful for. Unfortunately, most companies do not care whether you have many diverse skills; they are only interested in candidates who have the specific experience they are looking for.

Good mechanical engineers should also be machinists. There’s nothing more frustrating to machinists than engineers who design parts that are impractical to produce. Join a local maker space and learn how to use the CNC mill. Build something amazing. Bringing show-and-tell items that you designed and built yourself is a great way to impress prospective employers during a job interview. Also, if you have ever spent much money on your own projects, then you quickly learn to appreciate the value of a dollar, and you are less likely to be wasteful of your employer’s money. Robotics is one of the best things an engineer can have on their resume. If you don’t know robotics, I highly recommend buying some kits or participating in robotics clubs. One of the best career decisions I ever made was to buy my own license of SolidWorks many years ago. I might have become obsolete if not for that.

Don’t be afraid to suggest unconventional ideas and disagree with everyone else. It’s better to be laughed at than to always be silent during business meetings. The best engineers I know have lots of bad ideas, but they also have lots of good ideas. The important thing is that they have lots of ideas in general. And the only way you’re going to have lots of ideas is if you know about lots of things. That being said, I should note that the best engineers are not necessarily the highest paid engineers, based on what I’ve seen.
Self-employment is another option, but cash flow can be sporadic (“feast or famine”, as they say). If you really want to be self-employed, I recommend Michael Gerber’s book “The E-Myth Revisited – Why Most Small Businesses Don’t Work and What to Do About It”. It is not enough to be a great technician, you must also be a manager and entrepreneur if you want to succeed on your own. Too many times, highly skilled people start their own businesses thinking they got what it takes, only to become burned out when the work piles up, or when they micromanage their employees.

MM: You told me in an email that you have moved somewhat out of “my fractal phase”. What are you currently doing?

Paul Nylander: Yeah, I got burned out rendering fractals when I saw that it was taking too much of my time and it wasn’t paying my bills. So what am I doing these days? Well, besides work and family life, I have been trying to develop my own product lines of lamps and toys. I am currently selling a few expensive products to people who are willing to pay for them, but my long-term goal is to create affordable products for everyone. I have had some fun with 3D printing but I am especially interested in plastic injection molding, as it is still the most economical means of mass production. I recently joined a local maker space where I learned how to do CNC milling, plastic injection molding, laser cutting, vacuum forming, welding, etc. I have also been getting more into electronics, designing custom circuit boards with Arduino’s, LED’s, stepper motors, etc. One of my bigger side projects has been building a pick-and-place machine for assembling surface mount components on circuit boards.

I have also been reading lots of books lately, about topics like consciousness, causality, evolution, and religion. These things are interesting to think about, but also very serious to me. There are many things that I take on faith (everyone does), but I insist that faith must be reasonable. I feel it is important to be well educated on different views and encourage open discussions with other people.

MM: What is your impression of the fractal images, animated videos, virtual reality and even movies today? Movie executives from Guardians of the Galaxy 2 commissioned digital and fractal artist, Hal Tenny to provide images used to conceptualize some of the fractal work produced in the film. Others have developed ways to inject fractals into their latest projects. Are you following the progress and growth of fractal sciences as they become more mainstream?

Paul Nylander: I have not been following it. Thanks for the heads up. I was quite excited when I saw the Mandelbulb in Big Hero 6, and also in the Annihilation trailer. Nobody told me about these cameo appearances; I just accidentally discovered them on my own, so I can only presume that the Mandelbulb and related fractals have continued to grow increasingly popular. If there are other major appearances of the Mandelbulb, I would be interested to know about it.

MM: Do you have any fractal art hanging in your home and if so, who created it? What advice do you have for those who are developing software and tools, studying and producing fractal works in this rapidly expanding time in history?

Paul Nylander: I have no fractal art in my home. I do have a cast bronze loxodrome sconce on my wall, although technically that’s not a fractal.

My advice to those who want to create fractal art is to concentrate more effort on 3D Kleinian quasi Fuchsian Limit Sets. I think the most jaw-dropping sights remain hidden in there. But this is just my personal opinion. After all, beauty is in the eye of the beholder. Honestly, I am a little surprised that people have taken more interest in the Mandelbulb than 3D quasi Fuchsians. I guess it is because they are so mathematically difficult to render. I understand Jos Leys developed an efficient algorithm to render them in 2016. This interview inspired me to take another look at quasi Fuchsian. I have some ideas that I would like try, so perhaps if time permits I will give it another shot.



MM: Paul has been so gracious throughout the process of this interview. I owe him a debt of gratitude for his willingness to participate allowing me to step into his world and pick his brain!

This interview was conducted by Ricky Jarnagin, creator of the Facebook group Mandelbulb Maniacs  | April 9, 2018

Links: David Makin Interview | Jorge Abalo Interview | The Mandelbulb 3D Master Training Resource Guide | Daniel White – The man that coined the name Mandelbulb | Interview with Thomas Ludwig



An Interview with David Makin

David Makin was one of the winners in the Mandelbrot Art contest in 2006, 2007 and 2009. He was on the judging panel for the 2010/2011 contest. Today we get the rare opportunity to pull back the curtains a bit and discover what it was like to be a mover and shaker in the early days when fractals were as alien to most people as nanotechnology is today.

Another thing David did was to render the first deep zooms of the “true 3D Mandy” in 2009, proving it to be a fractal at higher iterations, although he claims the lighting algorithm on the renders was awful !!

He is also responsible for creating the following formulas used in Mandelbulb 3D:

  • Makin3D-1
  • Makin3D-2
  • Makin3D-3-4
  • Makin-NSR-tri
  • Makin4D p2a
  • Makin4D p2b
  • Makin4D p4a
  • Makin4D p4b
  • Makin4D p8a
  • Makin4D p8b

It is with great pleasure that I introduce you to Mr. David Makin!


MM: We would like to start by asking the basics. Where you are from? What is your educational background? What is your marital status and what type of work do you do for a living? 


David Makin: My name is David Makin and I’m originally from Ashton-in-Makerfield near Wigan, which was in Lancashire back then (1962) but is now in Greater Manchester.  We moved to Colwyn Bay on the North Wales coast in 1987 and then round the corner to Rhos-on-Sea in Colwyn Bay (where we are now) about 20 years ago. By “we” I mean myself and my parents, though my mother passed last year. So now it’s just myself and my dad – I’m his registered “carer”, he’s 79 and I’m 55. I have a younger brother Stephen, who’s in his second marriage and has 3 children from the first and one from the second. So I have 2 nephews and 2 nieces but so far I haven’t found a woman who’ll have me !!

As for education I went to old-style Ashton Grammar School and then Winstanley College and my highest academic qualifications are 5 A levels (BBCDE in math chemistry, physics, further math and general studies). Basically because I messed up my further education at Uni by starting a Chemistry degree but not finishing it due to discovering computers and programming on a subsidiary course at which point I went through a manual on BASIC in 45 minutes and realized that’s what I ought to be doing.

Unfortunately this was around 1982 (just after Sinclair) and not being an AAA A level student I couldn’t get on a computer science degree due to their popularity so I ended up having to settle for an HND course in Maths, Stats and Computing at Manchester Polytechnic. While on the course at home I borrowed a Commodore Pet to program on and realized very quickly that I’d need to learn machine code to do what I had envisaged when first encountering computers/programming (because I wrote a duck-shoot game on the Pet in interpreted basic and it ran like a strategy game).

On the course as well as the history of computing and the hardware stuff we were learning Algol (effectively structured basic) and a language that I even then felt should be assigned to the bin – COBOL. At home I then bought a Dragon32 (think Tandy CoCo in the US) and started learning 6809 assembler.

Soon, because I was progressing faster at home I dropped the HND course and later released some software for the Dragon through John Penn Software over the next couple of years. From then until moving in 1987 I had various computers including an MSX, a C64, a C128 and a Sinclair QL but I never got on with them in programming terms. Then came 16-bit with the Atari and the Amiga and I was really getting into programming on my Atari MegaST when we moved to Wales. Once we’d moved I rang up what I thought was a game suppliers to buy some software but the company I rang (Mr. Chip software) said they made games but didn’t sell them, so I said have you got room for another programmer and ended up being vetted by sout (Shaun Southern) writer of Trailblazer and Kick-Start on the C64. It just so happened that I’d recently figured out the insane memory to pixel mapping on the ST and Shaun was trying to do the same so he told Doug (the boss) to sign me up.

The first game I wrote was Super Scramble Simulator for ST and Amiga – created by “Magnetic Fields” (the 16-bit name for Mr. Chip) – while Shaun did Lotus 1. I also worked on “Wrangler”, “Supercars” (ST) and did all the coding for Crystal Dragon (Dungeon Master style game on the Amiga). Release of Crystal Dragon was 1995 and unfortunately I’d overworked on it whilst also having too much special tobacco 😉 I ended up having a “psychotic episode” in the form of schizophrenia and was basically out of it for 2 years and told I’d always need to take risperidone – which I weaned myself off by 1998.

Getting back to computing in 1999 I discovered “Fractint” for the first time and never looked back with respect to fractals. But I still wanted a career in computer games to which end I joined Parys Technografx, formed by Stephen Parys formerly of Magnetic Fields and worked with them on “Tower of Souls” then doing work for Weight Watchers before moving on to the Parys Tech. Games for mobile devices.

At the start of this period I wrote MMFrac but stopped development after about a year in 2000 when I discovered Ultra Fractal (2) which basically already did everything I’d thought of at the time – nice one Frederik !!

I still occasionally added more code to MMFrac mainly for testing anything new, e.g. rendering 3D IFS, and the source is now a bit of a mess so for the last 10 years or so I’ve stuck to using UF instead, even for 3D. I always want to write code first and develop images later, that way my comprehension of what’s going on is much better, especially when I end up using other people’s code as then I really want to have worked out my own version first.

Now and since starting at Magnetic Fields I’ve always been “self-employed” and I used to say “programmer” but now I say “artist/programmer” and I’m trying/hoping to make more from my artwork than from programming – at 55 it’s not as easy to remember 100,000 lines of source code anymore 😀 (and I’m baaaadd at commenting). In addition to the fractal art I now also do digital photography and manipulation.


MM: What would you like to say to people using MB3D and creating fractals today? 

David Makin: To newbies: You don’t really need to know the math but you do need to know how to use the software, so learning to do fractals is not really any different from learning to use Photoshop. As with a car a little bit of specialist knowledge helps (how to change a sparkplug<>what bailout is for, where the petrol goes<>what iterations/maxiterations are) but you don’t need to know the inner workings of formulas.

To those just creating art: Keep playing and testing the limits, fractals can produce amazing surprises and literally infinite worlds.

To all: Have fun !! But be ready for membership of fractaholics anonymous 😉

For MB3D/4D users: Don’t forget that 2D can be just as artistic !! And do try other fractal software too.


MM: What might you suggest for them to research? 

David Makin: Depends who you mean – for physicists/mathematicians I’d say seriously investigate the idea that existence itself is one or many dynamic strange attractors (immediately explains all spooky effects at distance).

For fractal programmers who haven’t done so already study how the rendering methods work – both old and new, mathematically “correct” or brute-force because there’s always some formula that won’t conform. One of the most important things is how to get smoothed iteration values or distances so learn that if you haven’t already – there are many methods.

As for new research, to me this involves the interface from user to formula more than anything at the moment, this is where even the latest programs fall down a little. It’s possible for instance to combine 3D IFS and normal Mandelbrot/Julia fractals (or any other kind) so for instance you could have a Sierpinski triangle of Mandelbrots – the interface here would need to allow the user to control an IFS tree being able to put whatever fractal function they prefer at any node in the tree, to do this completely for n levels requires entries proportional to exp(n) so this could only work explicitly for smaller depths,  but greater depths could be achieved using algorithms to decide the choices at the nodes in the IFS tree – an extension to using cellular automatic control in LRIFS (language restricted IFS).

In addition, it strikes me that it should be possible to combine such PIFS (Programmed IFS) with CSG trees.


MM: What would you want people to know about yourself, your work and contributions to the field of fractals?

David Makin: Apart from being a fractaholic, I also do photography, mostly landscape/nature and I’m now a big fan of the Raspberry Pi – I even process most of my photos on a Pi 3 using The Gimp.

One day a week (usually Saturday) I voluntarily man the Rhyl Create Gallery in the White Rose Centre, Rhyl. Rhyl Create is a group of about 40 artists on the North Wales coast from Prestatyn to Bangor.

With respect to fractal art my work was one of the winners in the Mandelbrot Art contest in 2006, 2007 and 2009 and I was on the judging panel for the 2010/2011 contest.

I believe my biggest contribution to fractal programming was the eureka moment when I was half-asleep contemplating the problem with fractals where standard iteration smoothing won’t work because the divergence varies across the fractal and I suddenly realized that calculating the actual divergence at bailout would maybe fix the problem, which it does (usually). This led to the auto-power mode in fractal coloring.

At one point (for my formulas using recurrence relations such that the recurrence was spread over the iterations rather than using a fixed polynomial on each) I did get an email from a mathematician saying “do you know what you’ve done” and I confess I still don’t !! (if anyone knows why the significance please let me know).

The only other thing is I did render the first deep zooms of the “true 3D Mandy” in 2009, proving it to be fractal at higher iterations, although the lighting algorithm on the renders was awful !!


MM: We know you invented the MMFrac program, but it is unavailable for download these days. Then you went to UltraFractal and wrote some formulas for that program. You also wrote some formulas for MB3D, but we would like to know how that came about. 

David Makin: The 3D formulas I wrote for Ultra Fractal were initially done because the only 3D available at the time were very slow (in UF and Fractint) and I first used a method based on the iteration density of the fractal, stepping along the rays faster when the density was low and slower when high. This produced a much faster algorithm but still pretty slow so I was always trying to find better methods and later switched to the true distance estimation methods, including the one that was essentially created by myself and Krzysztof Marczak (Mandelbulber).

As for the different formulas themselves I was always trying to find the true 3D Mandy but Daniel White (and Paul Nylander) beat me to it. It wasn’t me who transferred my UF formulas to M3D, that was one or more of the M3D authors.


MM: We would like to ask what are you doing currently.  Could you prepare some interesting information or images about your program, your formulas and your contributions to MB3D?

Note: See resources included in response to this question throughout the remainder. 

David Makin: Recently I’ve been concentrating more on photography but I’ve realized that the latest OOP language in UF probably makes some things possible that weren’t previously, including formula authors being able to make other uses of the switch option so 3D objects can be viewed and manipulated in 4 window modes, this requires some work to say zoom one window and leave the other 3 unchanged but I think it’s possible, there’s now also a better way to do IFS I think.

I can prepare some information and images relating to my formulas for sure, but it’ll take longer if you want them rendered in MB3D coz I confess to having hardly used it, I normally stick to UF !

I’m on a Mac and hate booting it to Windoze.


MM: What do you think might add value to the programs being developed today?

David Makin: Print direct from fractal program (2D or 3D).

Have a general buffer that formula authors can read/write anything to and save out in their choice of format.


MM: With MB4D in the development stage, what advice, wisdom would you pass on to the development team? 

David Makin: As I said above, remember the importance of  the user interface.


MM: Do you have any plans on developing new formulas for MB4D?


David Makin: UF, yes. MB4D if on a Mac version 😉

This was a leaflet to go with a talk on fractals that David Makin did to a group of math and art students.

Introduction to fractals and fractal art

1. “Fractal” is a term coined in 1975 by a mathematician called Benoit Mandelbrot, a Polish born French and American Mathematician.

Fractals are relatively simple mathematical constructs that can produce very complex results when used to create images, usually showing the classic features of fractals – self-similarity at different scales with “infinite” detail at least in places.

Although these constructs (often called formulas) can be relatively simple a very large amount of repetitive calculation is involved in producing an image and computer programs are far better suited to doing this than pen and paper.

However, there are some simple ones that demonstrate the self-similarity of fractals very well. The simplest of these is the Cantor Set first described by Georg Cantor in 1876 and this can be created easily. Draw a straight line at the top of your page. Now draw a copy of it underneath it but with the middle third missing. Now draw a copy of the two shorter lines underneath themselves but with their middle thirds missing.Now repeat several times such that the remaining lines get shorter and shorter as you go down the page.

The result of this process if you were to continue forever is the “limit” Cantor Set, and what you have drawn is a finite approximation of the limit Set.

A more interesting fractal that can be easily produced is the Sierpinski Triangle described by Wadaw Sierpinski in 1915.

Draw a large equilateral triangle, now mark the center point of each side and then join them to make a central triangle and imagine this is removed.
Now repeat the process for the 3 remaining smaller triangles.
And again repeating the process to the infinite limit produces the Sierpinski Triangle and what you can draw is a finite approximation to this.

2. One of Mandelbrot’s tutor’s was a mathematician named Gaston Julia and he investigated an algebraic method that produced fractal objects in the 1920s – without the aid of modern technology. This introduction to “Julia Sets” led Mandelbrot to choose a job at IBM in the 70’s so he could use their computing power to study further.

Julia Sets and the now more famous Mandelbrot Set are fractals created algebraically using complex numbers which are a two-dimensional number form that shares all the characteristics of normal real numbers plus an extra one that means there’s a solution to the problem of obtaining square roots of negative numbers.

Imagine a normal graph such that the horizontal or x-axis is the real number line and the vertical or y axis is another number line – call this the imaginary number line.

Because these are at 90 degrees a change in the real (x) value could occur without the imaginary  value being affected and vice-versa.

Now define a multiplication of real 1 by imaginary 1 as imaginary 1 – this means that if we start with +1 up the horizontal axis and multiply by +1 up the vertical axis then the result is +1 up the vertical axis because multiplying by +1 real produces no change in the other value – another way of looking at this is multiplying +1 (real) by +1(imaginary) results in the +1(real) being rotated through 90 degrees to become +1 imaginary.

Now imagine multiplying +1(imaginary) by +1(imaginary) – simply the +1(imaginary) causes the other +1(imaginary) also to be rotated through 90 degrees in the same way, however we already started with a value at 90 degrees to the real number line so the result must be at 180 degrees to that i.e. at -1 (real).

Therefore, we have the result that the square root of -1 (real) is +/-1 (imaginary) also written as 1i or just i. Now I hope fairly obviously that for example the square of +/-2i is -4, the root of -9 is +/-3i etc.

If we have a number that is mixed real and complex, such as 3 + 4i then multiplying by that number will in fact produce a rotation defined by the numbers location relative to the real (x) axis and the size/scale factor is given by the real value they would have if rotated to be on the real (x) axis – so in fact the magnitude of 3+4i is 5 (Pythagoras) – this means that it can also be defined as length 5 but at the appropriate angle to the real (x) axis.

Now because complex number multiplication involves rotation as well as scaling if we also introduce addition into a formula thus producing translation as well as rotation and scaling then very interesting things happen to repetitive calculation and this is what Julia investigated.

3. Julia Sets are produced by taking a simple formula and applying it repeatedly feeding the new result back into the original calculation at each stage – this process is known as iteration, confusingly each stage of the process is also known as an iteration.

Julia investigated what is essentially the simplest formula for this, repeating z^2+c where z and c are both complex numbers, z being the feedback value and c being an arbitrary fixed constant or seed.

If this is considered in purely real number terms and the seed c is zero then the result is very straight forward – if we start with z as a real number <-1 or >1 then repeated squaring will result in divergence to infinity. If we start with z such that -1<z<1 then repeated squaring will result in convergence to zero. And finally starting with -1 or +1 then repeated squaring will tend to +1 in both cases. In the first case (z<-1 or z>+1) it’s said that the attractor is infinity, in the second case (-1<z<1) the attractor is zero and in the third case (z=-1 or z=1) then the attractor is +1. In fact with real numbers even when c is non-zero things never get more complicated than this case where the real number line is split effectively into 3 parts.

Doing the same thing with complex numbers and a constant/seed of zero is not much more interesting – in fact it just results in a circle of radius one where starting points outside the circle go to infinity, points inside go to zero and points at *exactly* 1 unit from the origin stay exactly 1 unit from the origin (though most rotate as the calculations are performed).

What excited Julia and later Mandelbrot was what happened when this constant was non-zero when using complex numbers, the results can be literally infinitely complicated. Now there are still essentially 3 possibilities, divergence, convergence and constancy but the boundary (where we have constancy) between the divergent and convergent areas can be convoluted to an infinite degree and it’s basically the shape of this that produces the Art of fractals.

4. Now Julia investigated these mathematical wonders by trying many starting values for each single constant value – such that each different constant value gives a different Julia Set. Mandelbrot decided to try something slightly different, he decided to use a fixed starting value of zero and examine what happened in a 2D plane for all the different constant values and this is what produced the classic Mandelbrot Set for z^2+c.

It turns out that Mandelbrot’s Set in fact gives us a nice “map” of all the Julia Sets such that each point on the Mandelbrot image corresponds to a particular Julia Set and not only that but the nearby geometry of the form of the Mandelbrot for the chosen point is matched closely in the corresponding Julia Set. This is great for artists as it means that once you’ve found shapes/detail you like on the Mandelbrot you can produce similar shapes and forms by switching to a Julia Set in that area.

5. Producing images from these fractals on a computer requires the same sort of approximation as the drawn ones i.e. it’s impossible to create the infinite limit version, therefore the methods used always involve some form of termination to give a decent finite approximation to the true fractal. Sometimes in Fractal Art this approximation is made deliberately poor for artistic reasons.


1. Fractal Software (commercial, all types) (commercial, 3D IFS and other 3D) (freeware, all types somewhat Ultra Fractal compatible) (freeware, strange attractors) (freeware, flame fractals)

2. Fractal Information

3. Other – including sources of other freeware etc. (links to galleries of some great fractal artists IMO) (*THE* fractal place)

The following images are copyright protected by David Makin.

We are so excited and thankful to Mr. David Makin for allowing us to peek into his world and provide us this rare interview! David has been a Maniac since June 8th 2014, which means you have been in the presence of a Fractal Legend and didn’t even know it!

This interview was conducted by Ricky Jarnagin, creator of the Facebook group Mandelbulb Maniacs   July 29, 2017

Daniel White – The man that coined the name Mandelbulb

It is an honor to present to you an interview with Daniel White. Daniel is one of the true pioneers of the 3D fractal movement.  He not only coined the name Mandelbulb according to Mathematician and sci-fi author Rudy Rucker, but he also played a significant role in the discovery of the 3D Fractal. Most noteworthy, he collaborated with other members of, which resulted in the discovery of the first 3D fractal images.


I was reading the New Scientist article entitled “The Mandelbulb: first ‘true’ 3D image of famous fractal” the other day that was written in 2009. The article mentioned the 2D Mandelbrot set, named after the mathematician Benoît Mandelbrot , who also coined the term “fractals”. But Daniel and others were not satisfied with the 2D Mandelbrot set. They wanted more. This story has been largely forgotten, until now.


MM: I wonder if you could get us started with some background information. Specifically, information about your education, current hobbies, career path and family life.

Daniel White: Sure, education-wise, my degree is in Music and Computing (taken at the University of East Anglia), so like the Mandelbulb, that combined degree sort of captures my interest in the intersection between the sciences and the arts. It’s not just the fractal world either; way before university, I’ve been interested in finding out the basis behind the Western musical scale and why exactly there are twelve notes (hint: it’s a lot more involved than one might expect, and can’t just be explained by simple ratios!).

After teaching piano back in Bedfordshire, I recently moved to the West mids to be closer to the (extended) family, but lately what has taken up so much of my time is trying to get my software business up and running smoothly. I rarely make any money from fractal-related projects these days, but then, of course, we know people who have – more power to them. I create programs such as (SunsetScreen – takes off the ‘blue glow’ on your PC screen in the evening), (SonicPhoto – converts pictures into sound), OpalCalc – multi-line Notepad style calculator with live answer updating as-you-type), WildGem (better than Regex!), MIDI Transform (edit MIDI files), and a few others. That’s the fun stuff – most of the real money is made coding up relatively dull business-type applications (such as converting JSON/vCard/CSV file formats to each other). Such is life!

Hobby-wise, I enjoy climbing, video games, Tesla (the car company), playing piano/keyboard, and am unfortunately more interested in politics than I used to be (haha). I think things like freedom of speech is paramount to the evolution of fruitful ideas, even if we may disagree strongly with those ideas (but in my opinion, both the Left and Right have good and bad ideas). I’ve also recently been more interested in DIY, and so-called “crypto-currencies” (not just Bitcoin, but superior technologies like Ethereum and the newer super-fast, scaleable, feeless, ‘Nano’ currency). It’s such an elegant concept that a single address (protected and derived by a single approx 50-character length err, ‘password’, or “private key” as they like to call it) can theoretically hold billions of pounds!

MM: Can you take us back to the beginning and walk us through your role and the roles of others in the development of the first 3D Mandelbulb images?

Daniel White: A lot of this history can be seen in the intro to my Mandelbulb article here:

It all started back in 2007/2008 when I created the Mandelbrot page on Skytopia. At the time, I was fascinated by the 2D Mandelbrot and wondered what a 3D version might look like. Imagine all that crystal-like detail in 3D I thought. Others had attempted 3D/4D versions such as the Quaternion fractals, but they usually turned out ‘smeared’ or lacking true fractal 3D detail (apart from the incredible and unique Quasi-Fuchsian fractal I found at the time.) (see below right)

The idea behind creating the 2D Mandelbrot is actually relatively simple. I tried to break down visually how the mathematics behind complex numbers created the shape, and it essentially boils down to millions or even billions of rotations and scaling functions. You can see the basic idea under the heading “My attempt to open Pandora’s box…” at the aforementioned URL (I think, given enough time, even an average 8 year old could understand the basic concept). So I thought, let’s use those same basic math operations but in 3D space! Instead of rotating around a circle, let’s try rotating around a sphere on two axis – the X and the Y. That was the original basis behind the Mandelbulb.

Needless to say, around the time of the seminal FractalForums thread, the atmosphere was electric, almost as if we were on a mission exploring uncharted seas for hidden treasure. Apart from myself, people such as David Makin, Thomas Ludwig (“Lycium”), Paul Nylander (“Bugman”) were instrumental, to begin with, and later Jérémie Brunet (“Bib”), Krzysztof Marczak (“Buddhi”) Lead Developer of the popular Mandelbulber progam, Johan Andersson (“Mandelwerk”), and I’m sure quite a few others. I later found out how Rudy Rucker independently envisioned and created a “pseudo-Mandelbulb” years earlier around 1987. Tom Lowe also came in with the incredible Mandelbox later in 2010 of course.

Anyway, it was Paul Nylander and David Makin who were the first to render images of the Power ~8 bulb, but in fact, before that, it was Thomas Ludwig who first rendered the Power 2 version which can be seen here:

From that image, there’s little in the way of that magical Mandelbulb-like fractal detail, but in fact if we look round the back, and then zoom into that bottom right section with the small columns, we get this image  shown to the right (which I rendered a year later when the Mandelbulb story blew up worldwide): 

– not quite as magical as the Power 8 images, but already we can start to see some very interesting fractal detail appear! At the time, back in early 2008, no one quite knew what he found, and I remember almost begging him to render some close-ups of that Power 2 cross-section. Had he done so, history may have played out quite differently!!

MM: Who was the person that introduced you to fractals?

Daniel White: I honestly can’t remember. I don’t think it was any single person – just hearing and reading about fractals (perhaps on Slashdot, Amiga Shopper/Format, CU Amiga, or in culture generally). I remember playing about with Chaos Pro back in the Amiga days before the year 2000. Pictures took a long time to render in those days!

MM: Back in 2007 when you were exploring the Mandelbrot set and trying to develop a 3D image like the one above, what was the predominate software being used at the time?

Daniel White: I coded it myself. My first images were simply greyscaled top-down layered slices (with the bottom slice being black, and the top being white). Images like these:  That’s about the easiest thing to start with if you have no idea about how 3D images are created. If I recall correctly, I tried a higher power Mandelbulb then and saw the slices build up. Since the quality was so terrible, I couldn’t really appreciate what I unintentionally discovered!

MM: Do you have an educational background in Math or Science, and what influenced your decision to get involved in fractal research?

Daniel White: Beyond high school, not particularly, but I have a passion for both of those. I keep meaning to pursue those subjects further but life as always gets in the way. For example, I’d love to research high capacity super cap batteries, develop quiet VTOL craft (as quiet as possible!), or help research room temperature superconductivity or even fusion. One of the people I look up to is Elon Musk – someone who tried to think what technology could most help the world – and go full steam ahead in that direction.

MM: What is your impression of the fractals generated today?

Daniel White: So many amazing works being produced! I took the opportunity to browse through thousands on the DeviantArt site over the past few days, and you can see my favorites here: … I saved around 100 new fractals from artists such as Batjorge, Les-Monts and plenty more. If I had one criticism or something I would love to see more of, it would be fractals with proper global illumination for better lighting. Programs such as Mandelbulb 3D produce many incredible pieces, but the lighting only offers a couple of bounces from what I can tell, which can sometimes make scenes look a little flat or more mis-colored than they otherwise might. The problem is pictures rendered with GI could take days or weeks instead of minutes or hours because the complexity to render fractals is much trickier than most other types of scenes! Anyway, something to look forward to in the future.

What I haven’t explored fully yet are the incredible videos out there. Well I mean, over the years I have occasionally looked on Youtube and found some really interesting ones, but I just bet there are countless new gems waiting to be discovered.

MM: You had the opportunity to engage with many of the pioneers during the early development of 3D fractals. Do you still communicate with any of them?

Daniel White: Rarely, but occasionally! It would be nice to return to the scene in a greater capacity at some point, whether it’s creating more pictures, teaming up with others, programming a Mandelbulb 3D type program, or trying to find the ‘Grail’. Who knows what the future holds.

MM: How did you feel when you first saw the higher power versions of the Mandelbulb? (e.g. the z^8 done by Paul Nylander)?

Daniel White: Very excited, and desperate to see more of what we just stumbled across. It inspired me to create a 3D renderer, something which I would have never dreamt of before. It was so intriguing, I found myself staring at the screen, watching the renders build up slowly, pixel by pixel, line by line, in anticipation of seeing how it would appear in full.

MM: What would you do differently today if you could go back and start fresh?

Daniel White: No regrets really. I was thinking about giving talks to universities at one point, something I might’ve followed up if I had more time. But yeah, I can’t really think of anything I’d do differently.

MM: Do you have a copy of Mandelbulb 3D on your computer and if so do you create any art with it, even if it is for your own pleasure?

Daniel White: I hadn’t tried recent versions of Mandelbulb 3D, but gave it a quick look just a couple of days ago. Like 3D ray-tracing programs tend to be, it’s quite daunting at first glance! I could see myself becoming addicted quite easily 🙂

MM: Do you have any fractal art hanging in your home and if so who created it?

Daniel White: I have a few of my own pieces hanging around, some of which I originally intended to sell!

MM: Have you seen Guardians of the Galaxy 2? Did you know that Hal Tenny was commissioned to provide fractal images later used to conceptualize some of the fractal work that was produced for the film?

Daniel White: Yeah I love GOTG 2! (and its prequel). And yes I heard someone was picked to help Marvel with the development of the 3D fractals that were featured. Actually, some of Hal Tenny’s work on Deviantart is quite exquisite, especially how he combines fractals with nature. I’d love to have the opportunity to work in the same capacity for something like that. Good for him! I wonder if his rendered images or objects were directly used in the film, or if they rather used his images as a ‘blueprint’ or foundation for the actual images.

MM: It has to be said that Hal Tenny’s work was used as a foundation for the future artworks produced for the Guardians of the Galaxy 2 movie. I’m fairly confident we can all agree that Hal’s work dramatically affected the overall look and feel of the movie!

MM: How do you feel about the progression that 3D Fractals have taken since your involvement?

Daniel White: Let’s see. The Mandelbox discovery is obviously a tremendous development and most imagery today is based off that. We have galleries of fractal works (like Jérémie Brunet), and people are even creating jewelry by printing such shapes in 3D (e.g: Jérémie and Johan Andersson). In the future, I expect we’ll be exploring the Mandelbulb/box and other inspirational fractal worlds in virtual reality with full global illumination at 200 frames per second in 8k video. How magical would that be?

Thanks for the opportunity for this interview and hope it’s of interest to your readers!


MM: What a joy this interview has been. I want to give a Huge Thanks to Daniel for trusting me to do this and for his total cooperation! Stay tuned as more interviews are in the works.



For deeper reading on the topic check out these articles:

From Wikipedia, the free encyclopedia

The Mandelbulb: first ‘true’ 3D image of famous fractal –

Geeky Math Equation Creates Beautiful 3-D World –

The Mystery of the REAL, 3D Mandelbrot Fractal Daniel White’s site

Hal Tenny’s Deviant Art Page


This interview was conducted by Ricky Jarnagin, creator of the Facebook group Mandelbulb Maniacs   March 28, 2018