The twistor flow on real 3-space, induced by unit complex scalar multiplication on the 3-sphere.

It's less ... focused ... than most of my mathematical portraits both visually and conceptually, but maybe more visceral and evocative?

HyperRogue 13.1: https://zenorogue.blogspot.com/2025/08/hyperrogue-131.html
#roguelike #mathart #NonEuclideanGeometry #NonEuclidean

Look at this brot!

start wiθ a cubic honeycomb
split each cube into 6 pyramids
merge each pair ðat shares a 4gon into irregular octahedrons
& u get ðe almost regular octahedral honeycomb

(yes, someone made ðis a technical term, rly)

Pavement tiling, Ouarzazate, Morocco
#TilingTuesday #geometry #tiling #MathArt #photography #design #TravelPhotography #octagon

And one more, this time three-quarters-cut.

Another visualization of Costa's original surface (torus with three punctures).

The parametrization is as attributed to Alfred Gray:

https://mathworld.wolfram.com/CostaMinimalSurface.html

The \(\wp\)-function is evaluated numerically using Wikipedia's expression in terms of \(\vartheta\)-functions, summing cosines over \(1 \leq n \leq 4\):
https://en.wikipedia.org/wiki/Theta_function#Jacobi_theta_function

https://en.wikipedia.org/wiki/Weierstrass_elliptic_function#Relation_to_Jacobi's_theta_functions

My code to evaluate complex line integrals is fairly simple-minded; line segments and parabolic sums ("Simpson's rule") with 16 pieces (subintervals) sufficed.

With these conventions, the unit square \([0, 1] + i[0, 1]\) is a fundamental domain. The points \(1/2\) and \(i/2\) are the catenoid ends; \((1+i)/2\) is the planar end. I split the domain into four squares of side length \(1/2\), three centered at the punctures, the fourth centered at \(0\). The colors correspond to these squares.

To parametrize these squares "nicely" I used the inverse elliptic function
\[
f(z) := \int_{0}^{z} \frac{dw}{\sqrt{1 + w^{4}}},
\]
which maps the unit disk conformally to a square with sides parallel to the coordinate axes. Scaling, translating, and precomposing with polar coordinates (for the center) or with the exponential map (to avoid the punctures), and a bit of ad hoc fiddling to shrink the disks (avoiding the singularities of the integrand/corners of the image of \(f\)), gives the image/animation shown.

I first saw plots of Costa's complete, embedded, minimal surface of finite total curvature around 1985, and was intrigued (wry understatement) by both the shape and topology.

In 2022-23, when I was starting to design classroom models, having the surface 3D printed came to mind, an idea independently suggested by a former colleague. For various reasons, I'm only now able to plot/model the surface in C++/LaTeX. The animation loop shows the surface appearing to blossom (and unblossom), with a pause at the equatorial plane of symmetry. The low frame rate is intended to clarify what happens as the topology changes.

I've understood since the mid-1980s how cross sections change with height, but it's nice to see the sublevel sets plotted accurately. A 3D-printed model should arrive in a few weeks.

Base of a Celtic cross (c. 1890), Glasnevin Cemetery (Irish: Reilig Ghlas Naíon), Dublin, Ireland

#FractalFriday (It's already Saturday)

Elephant valley of the Mobius Mandelbrot set.

Formula: \(z_{n+1}=f(z_n^2+c,1)\)

where \(f(z, a)\) is defined as (C-like pseudocode):

Complex f(Complex z, Real a) {
Real dist = abs(re(z)) - a;
if (re(z) > a) {
re(z) = -a + dist;
im(z) = -im(z);
} else if (re(z) < -a) {
re(z) = a - dist;
im(z) = -im(z);
}
return z;
}

Coloring is done by using three very similar colormaps, mapped to triangle inequality average values.
The final color is produced by interpolating between these three colors using the so called "atom domains".

Generated using my GLSL Shadertoy fractal generator: https://www.shadertoy.com/view/33sSRf

Look at my new sketch on #openprocessing, ColourCurtain.

https://openprocessing.org/sketch/2695752

The items' colours are altered resulting from the counter value mod 3; thus, only one of the three colour channels is changed, causing the effect of an irregular colour gradient.
Try out generating more pattern by reloading the page.

New free pattern on Ravelry:
https://www.ravelry.com/patterns/library/pixie-square

The Pixie Square is a new type of crochet granny square designed for detailed, high-complexity, pixellated motifs. (Great for mathematical patterns!)

4d shapes
9d colors

"The Great Splash". Based on the golden ratio.
#fractal #mathart #algorithmicArt #GoldenRatio

Ulugh Beg Observatory Museum, Samarkand, Uzbekistan

The Nilgeometric Geodesic Cone in horizontal view