Fantastic Fractals lets you dive into the mesmerizing world of complex dynamics and mathematical art right in your browser. Switch instantly between classic and modern fractals—like Mandelbrot, Julia, and Newton—and explore them in real time with fluid pan and zoom. As you navigate, the app overlays precise complex coordinates and live view ranges, helping you connect visual patterns to their underlying math.
Whether you’re zooming into a Mandelbrot filament, reshaping a Julia set via a complex control point, or studying Newton basins with precise parameter control, Fantastic Fractals makes the deep structure of mathematics both approachable and beautiful.
Have you ever looked at a snowflake up close, or noticed how a fern’s leaves look like tiny versions of the whole plant? That repeating, “pattern inside a pattern” idea is what fractals are all about.
Long before anyone had the word fractal, mathematicians were stumbling onto strange shapes that didn’t fit the neat rules of geometry. In the 1800s, people like Georg Cantor, Helge von Koch, and Giuseppe Peano started drawing odd curves and sets. These shapes were puzzling because they were jagged, infinitely detailed, or didn’t seem to follow the normal rules of lines, squares, and circles. At the time, most mathematicians thought of them as “monsters” or curiosities rather than something useful.
It wasn’t until the 1970s that these wild shapes got their modern name. A mathematician named Benoît Mandelbrot studied these “monsters” and realized they were more than curiosities. He called them fractals, from the Latin word fractus, meaning “broken” or “irregular.”
Mandelbrot showed that fractals appear everywhere in nature: the branching of trees, the outlines of mountains, the swirls of clouds, even in your own blood vessels. Instead of being useless, fractals turned out to be one of the best ways to describe the messy, rough patterns of the real world.
Before computers, drawing fractals by hand was almost impossible, since they repeat patterns over and over. But with computer graphics, Mandelbrot and others could “zoom in” endlessly and see the shapes blossom into stunning, colorful designs. The most famous of these is the Mandelbrot Set—a shape so rich that even today, people are still discovering new details when they zoom in.
Fractals aren’t just pretty pictures. Scientists use them to understand things like coastlines, lightning, galaxy clusters, and even the stock market. Engineers use fractal ideas in antennas and image compression. Artists use them to make mesmerizing digital art.
Fractals show us that the world isn’t always smooth and simple. Nature likes to repeat itself in surprising, beautiful ways, and fractals help us see that order inside the chaos.
This application allows you to explore the intricate beauty of various fractals. You can interact with the fractals in two main ways: by manipulating the view on the main fractal view and by adjusting parameters in the control panel.
The fractal type and color scheme can be changed using the dropdown menus in the header navigation bar.
The control panel, located on the left (or top on phones in portrait mode), provides the following options:
You can use mouse or touch screen to pan, zoom, and reset the fractal view. Some fractals have complex parameters that are shown as control points in the fractal view. You can change those parameters by moving them around. A few fractals even let you add or remove complex paraemters.
Welcome to Triangular Geometry! This app is a fun way to learn about the special points, lines, and circles associated with triangles. It's like a virtual geometry playground.
With this tool, you can see how different "centers" of a triangle move as you change the triangle's shape.
This app makes learning geometry a hands-on experience. Enjoy exploring the beautiful world of triangles!
A long time ago, people noticed that triangles seemed almost magical. They are the simplest shapes made of straight lines, yet they keep showing up everywhere: in the beams of bridges, the roofs of houses, the sails of ships, even in nature’s crystals and mountain peaks. But what makes triangles even more fascinating is what happens inside them.
Over two thousand years ago, in ancient Greece, mathematicians like Euclid and Pythagoras studied triangles carefully. They measured sides, compared angles, and proved amazing theorems. But some wondered: if you draw special lines inside a triangle, where do they meet? Do they meet at all? And if they do, is that point special?
Over the centuries, mathematicians discovered that triangles don’t just have one “center”—they have several, each with its own meaning.
The Centroid (The Balancing Point) Imagine you cut out a triangle from cardboard. If you tried to balance it on your finger, the exact spot where it balances is called the centroid. It is found where the three medians (lines from a corner to the middle of the opposite side) meet. Fun fact: The Greeks already knew about this, but it was carefully studied in later centuries when scientists were exploring balance, levers, and physics.
The Circumcenter (The Circle-Maker) Long ago, people loved circles. They noticed that every triangle can fit perfectly inside a circle, with all three corners touching the circle. The center of that circle is called the circumcenter. You find it where the perpendicular bisectors of the sides meet. Ancient astronomers and sailors cared a lot about this, because circles and triangles helped them map the stars and navigate the seas.
The Incenter (The Secret Circle Inside) Later, mathematicians asked: can we also fit a circle inside a triangle, just touching all three sides? Yes! The center of this circle is called the incenter, where the angle bisectors meet. Builders and artists in the Middle Ages found this fascinating, because it gave perfect symmetry for designs.
The Orthocenter (The Perpendicular Meeting Place) Another curious point is where the three altitudes (the “height” lines dropped from each corner) meet. This is the orthocenter. It doesn’t always sit inside the triangle—it can wander outside! This strange behavior puzzled mathematicians like Euler, an 18th-century genius who studied it deeply.
By the 1700s and 1800s, mathematicians like Euler and Gergonne discovered that these centers are not random—they are connected in surprising ways. For example, Euler found that three of them (the centroid, circumcenter, and orthocenter) always lie on the same straight line, now called the Euler line.
Later, people realized there are not just four, but hundreds of possible triangle centers, each with its own story. Today, mathematicians have named and cataloged thousands of them!
By the 18th century, mathematicians like Leonhard Euler began connecting these centers in surprising ways. One of the most astonishing discoveries is the nine-point circle.
What is it? For any triangle, if you mark:
you get nine points. Amazingly, all nine lie perfectly on one circle!
Even more magical: the center of this nine-point circle lies on the same line (the Euler line) as the centroid, circumcenter, and orthocenter. It’s as if the triangle is whispering, “All my secrets are connected.”
Studying triangle centers wasn’t just a game. These ideas helped with architecture, navigation, astronomy, physics, and even computer graphics today. Triangles are the building blocks of 3D models, and their centers help with calculations of balance, rotation, and symmetry.
This application lets you explore the fascinating geometry of triangles. You can interact with the triangle by moving its vertices and adjusting the view.
The type of geometric property to display can be changed using the dropdown menu in the header navigation bar.
You can use mouse or touch screen to pan, zoom, and reset the triangle view. You can also change the shape of the triangle by moving its three corners (vertices). Just tap or click a vertex and drag it to a new position.