Zoom in. Rotate. Repeat. Forever.
The Vicsek fractal keeps the same pattern at every scale, this is what a fractal dimension of 1.465 looks like in motion.
Coded with Python & OpenCV.
YouTube version (4K 60fps) : Vicsek Fractal
Most people expect chaos from a double pendulum. But if both arms rotate at constant angular velocity, the result is pure deterministic symmetry. Change the ratio, change the number of loops : it's that simple. Coded with Manim & Python.
More on the channel : Visualizing Mathematics
Thanks !
A recursive algorithm, iterated as the curve winds through the entire volume of a cube.
Each step replicates the previous shape 8 times : order 5 already generates 32,767 connected segments, all from a single continuous line.
Full video : Generative 3D space-filling curve
I’ve been trying to understand the exact geometry of the panels of the Adidas “Trionda” football.
From what I can tell, the design is based on a tetrahedral structure mapped onto a sphere (or at least strongly tetrahedral symmetry). This seems fairly consistent across visual evidence.
However, I’m struggling with determining the exact shape of the panel seams.
I don’t understand the geometry behind the images circulating online, such as this one : picture
In particular, I cannot derive:
- the exact mapping used from the polyhedral structure to the sphere,
- the analytical form (if any) of the seam curves on the spherical surface,
- nor the planar development (2D pattern of a single panel).
My suspicion is that many of the SVG / vector reconstructions online are approximations rather than the true underlying construction.
Does anyone know if there is a known mathematical model for these panels) ?
Thanks ! ⚽
Remove the center. Repeat forever.
The Sierpiński carpet starts with a single square and, with one recursive rule, punches an infinite number of holes into it.
The Mandelbrot set, zoomed out thirty-six octillion times.
At this depth, the boundary keeps revealing new structure, the same chaos echoing at every scale.
I animated an epicycloid in Manim with a fixed circle of radius R=3.5 and a rolling circle of radius r=0.7 (R/r=5), and counted 7 cusps instead of the 5 I expected.
Working through the parametrization, the relative angular speed between the traced point and the rolling circle's center comes out to (R+2r)/r = 7, not R / r = 5 or even (R+r)/r = 6. The extra +r seems to come from the direction convention for the point's rotation on the rolling circle (I parametrized it rotating in the same direction as the orbit rather than the more standard opposite direction).
Is there a clean geometric way to see why this specific convention shifts the cusp count from (R+r)/r to (R+2r)/r? Or is this a known distinction between different epicycloid parametrizations that I'm just not aware of?
x(t) = (R+r)cos(t) − r*cos((R+2r)/r*t)
y(t) = (R+r)sin(t) − r*sin((R+2r)/r*t)
This is the "Butterfly Curve", discovered by T. Fay. I used Python and Manim.
This curve is defined by a polar equation involving exponential and trigonometric functions.
The animation plots the curve as a function of t between 0 and 2π
By rotating a square around one of its vertices, we create a perfect circular shape. Each image adds a new layer and a shifting hue, forming a color gradient that fills the void. This is where simple geometry meets the beauty of a portion of the color spectrum.
If you're interested in more math-based animations, I post them here 📺 Visualizing Mathematics
Built using Robinson triangle decomposition in Python/Manim.
The two rhombus types inflate recursively at each step, producing the characteristic non-periodic 5-fold structure.
More visual math : Visualizing Mathematics
A Penrose tiling built using Robinson triangle decomposition.
Two rhombus types (thick and thin) are substituted recursively at each iteration, producing a non-periodic structure with 5-fold symmetry.
The animation reveals the tiling growing radially from the center outward.
More visual math experiments on my channel : Visualizing Mathematics
The Seed of Life is not just a symbol : it is a geometric algorithm.
Starting with a simple circle, each new centre is placed at the intersection of the previous ones, following a simple hexagonal pattern that repeats outwards.
A square is rotated step by step around a fixed vertex.
Each frame is recorded and accumulated, creating a continuous geometric trace.
A simple geometric rule unfolding over time.
Here’s a simple geometric process :
A square is rotated incrementally around one fixed vertex.
Each frame is recorded and accumulated, creating a visible trace of the transformation.
I used Python (Manim) to generate the animation.
Full animation and other mathematical visualizations : Visualizing Mathematics
360 frames generated in Python/PIL, one per degree of rotation of c = 0.7885e^(iα). 300 iterations, float64, smooth color banding.
If you're interested in more math-based animations, I post them here 📺 Visualizing_mathematics
If you're interested in more math-based animations, I post them here 📺 Visualizing_mathematics
An object with a constant width is called a circular object. A circle is obviously an object with a constant width ; this width corresponds to the length of its diameter. However, there are other objects that possess this property.
An interesting example is the Reuleaux triangle (named after a 19th-century German engineer).
This object, which is not strictly speaking a triangle since it is curved, is constructed from an equilateral triangle on which arcs of a circle have been drawn, centered at one vertex and with their endpoints at the other two vertices of the triangle.
This object has a constant width, meaning that the distance between two parallel tangents (as measured with a caliper) is always the same.
In San Francisco, manhole covers are shaped like this. The advantage is that, while preventing them from rotating, this shape also keeps them from falling into the drain no matter which way they are inserted.
British 20- and 50-pence coins aren't round either ; they, too, are Reuleaux polygons (heptagons).
These "triangles" are covered in my book (written in French) Géométrie dans l'espace et impression 3D.
You can find an excerpt from the book here : Read an excerpt.
A link to a complete GeoGebra model (It's easy to extend this to pentagons, heptagons, and so on !) : Simple Reuleaux Triangle
