"With this simple system, the Sierpinski fractal triangle emerges. Isn't that amazing?"
"The Mandelbrot set: an infinite amount of complexity."
"Fractals: Impossible mathematics that give rise to stunning shapes."
"He didn't invent fractals but he took a very crude fractal science and advanced it radically."
"It is, in fact, a fractal—arguably the most famous fractal is the Mandelbrot set."
"A structure that repeats itself on smaller and smaller scales is called a fractal."
"Finally, a rainbow Julia set: adding vibrancy to the fractal world."
"We live in a fractal universe, and it's fractal not only in terms of space but in time as well."
"By far the most popularized example of a rational function that you might study like this and the fractals that can ensue is one of the simplest functions, z^2 + c."
"Fractals inspired by the patterns in the natural world."
"Fractals are images of dynamic systems, the pictures of chaos."
"Mandelbrot's lifelong Quest was to find a simple mathematical basis for the rough and irregular shapes of the real world."
"As above, so below is the fractal universe, the holographic universe."
"Markets are fractal, meaning the same patterns can be seen on different time frames."
"Nature is fractal, what's true at the micro level is true at the macro level."
"I'm watching the whole world in terms of like the blueprint of what is in our physical reality, which looks a little bit like fractals."
"Mountaintops the way they split is fractal the way they branch and branch into branches and so on."
"Clouds have a fractal nature to them."
"The physical world contains things that are proximate fractals."
"Lovely and ferns are fractal because you have a stem and it has and it has growing off the side stems which have growing off the sides of those stems which have growing off the side and so on and it repeats on smaller and smaller scales."
"Nature often has patterns that are fractal in nature, similar at all different scales."
"Fractals are found quite frequently in nature... the overall shape is repeated in smaller and smaller versions."
"The only thing that exists is that dot, that the dot can be subdivided and subdivided, and there's like some fractal nature to the particle."
"We are a fractal of God, and last I checked, God doesn't have any limitations neither do we."
"The universe is structured like a neural network, raising the possibility that this fractal universe is indeed alive."
"Not only can anyone participate in this network, but it makes it easy for anyone worldwide to create new fractal sub-networks."
"What's the Mandelbrot set? Well, it looks like a set of points, but actually, it's a set of numbers."
"In the limit, if you don't pick your weights carefully, you get something that isn't smooth anywhere. In fact, it's a fractal."
"To teach scale you have to teach fractals, but to teach fractals you have to go all the way to the first principle."
"Now, whether you compress or expand an image depends on the same fractal principles, and Barnes's theories have now become a commercial reality."
"What you see are fractal textures, fractal creations that mimic the missing data... they're done using fractal geometry."
"...the most important fractal of all in the human body: a small portion of the incredibly complex wiring circuit of the brain."
"Fractal geometry has enormous commercial value."
"...will there be new devices, new extraordinary devices based on the principles of fractal geometry that will emerge over the next centuries?"
"The world truly is fractal in the most profound sense."
"Identifying fractal patterns and being able to take advantage of them."
"Remember, Elliott Wave has a concept of fractal; it relates to this idea that different degrees always consist of smaller waves."
"The fractal nature of wave means that smaller patterns can be identified within bigger patterns."
"An infinitely small ant walking around the perimeter of the Mandelbrot set would never complete the journey."
"Fractals often exhibit self-similarity, which means they have patterns that repeat themselves at different scales."
"Thinking in fractals is going to be the most abstract lesson so far."
"By looking for patterns at different scales... you learn to think in terms of fractals."
"Fractals appear everywhere in nature."
"Your heartbeat follows a fractal pattern."
"The Chinese discovered the fractal nature of time that the rules of expression of temporal elements which govern the rise and fall of dynasties also govern the rise and fall of love affairs and moods."
"Financial markets data consists of trends and follows fractal patterns."
"The shape that emerges is this: a map of points that belong to the Mandelbrot set."
"It's a very interesting shape, and it's made up of smaller shapes."
"Fractals, those mesmerizing patterns that repeat infinitely, offer us a glimpse into the underlying order and beauty that pervade creation."
"The intricate fractals serve as a cosmic mirror, reflecting the profound unity that binds the microcosm and the macrocosm."
"The Mandelbrot set is a special fractal invented by Benoit Mandelbrot."
"Fractals: The infinite intelligence of God's design."
"Endless fractals spun slowly in different directions on the ceiling."
"The Mandelbrot set... it's created by mathematical equations which produce these spectacularly beautiful fractal structures."
"Fractals are simple, complex, and open-ended."
"Fractals, which is the word that I coined for a certain kind of orderly roughness, occur in mathematics, sciences, engineering, and finance."
"If you come back to the same theme and you make friends with fractals, you will be able to keep people's attention for a long time."
"We're fractals of the same source."
"Romanesco... prized for its mind-blowing fractal appearance."
"An immersive audio-visual journey through fractal dimensions."
"Scale invariance means that if you zoom out little pieces of the image sufficiently, you get the original picture."
"Drawing fractals is fun and everything, but it's not really a game you can win; there's no goal other than to make a cool picture."
"One of the most beautiful examples of a fractal is the fern."
"No system will emerge into life or gravity or self-awareness that is not fractal in both space and time."
"This type of structure that repeats infinitely on smaller and smaller scales is called a fractal."
"The Mandelbrot set is the set of all numbers for which this other number Z remains small according to this formula."
"Anywhere you see intersecting structures, you'll get a baby Mandelbrot right in the middle of them, incredibly beautiful."
"Fractals are very useful because they remove some of the noise shown by these swings and allow us to identify key levels with more accuracy."
"We found a mono fractal scaling relationship in historical volatility."
"We're missing tons of history, but the pieces we have are not two-dimensional puzzle pieces; they're fractals."
"Markets are fractal, so whatever is happening on a higher time frame is also occurring on a lower time frame."
"The unifying characteristic that all these fractals have is that their dimension is a bit funky."
"The formal mathematical definition of a fractal is a space where the Hausdorff dimension is not the same as the topological dimension."
"...the universe is a fractal is one of the first things that you encounter because it's a natural thing. I mean, symbols are usually fractals."
"Fractals are the pictures of chaos and complexity theory."
"Nature is full of fractals, these never-ending patterns that copy themselves at different scales."
"Fractals are the math that describes nature; it's math made fun and relevant."
"Fractals represent the creative force of nature."
"A strange attractor is an attractor whose local structure is fractal as opposed to smooth."
"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
"Fractals are a big topic that impact so many different fields."
"What happens at the critical point is a kind of self-similar fractal structure."
"The entire formula to create a Mandelbrot is Z equals Z squared plus C."
"To fully capture the detail of these kinds of fractals you pretty much need a computer."
"I'm showing the connection between fractals and coastlines and how you can think of a coastline in terms of a fractal."
"I'm taking two methods to find the fractal dimension and I'm testing both of these methods on Great Britain."
"Everything in the markets is fractal."
"The market moves in waves, and it doesn't matter what time frame we're talking about; the market is fractal by nature."
"The Julia set is named after the French mathematician Gaston Julia."
"The micro reflects the macro and the macro reflects the micro; we're all fractals of a general human construction."
"The Mandelbrot set is a 2D representation of these points which do not go to infinity using this method."
"The more space filled by your fractal, the closer its value of D will be to the number two."
"The universe is eternally existing self-reproducing fractal."
"The Cantor set is the limit as n goes to infinity of C sub n."
"This fractal structure never stops; it's just going to keep containing copies of itself all the way down to arbitrarily small scales."
"All these fractal shapes that we've created, if we make those objects in an array list, then we can start to apply things to them."
"This is a fractal; you could do that over and over again."
"Everyone's fond of saying that coastlines and forest distributions and all the stuff are fractal."
"This entire realm is a fractal of a whole; as above, so below."
"Elliot discovered and identified fractal structures in the financial markets in the 1930s."
"Elliot waves provide greater market detail because they recognize the fractal nature of the markets."
"The Julia set is often a fractal."
"The deeper you delve into a Julia set, the more you see the same structure over and over again."
"The fractal nature of a market means these patterns repeat on top of each other to create price movements and markets."
"Nature is self-similar across scales of distance or energy."
"Nature at any scale, it repeats itself, which is the essence of fractals."
"We have fractals in Python via graphics protocol."