#mathart
#ArtAdventCalendar Day 7
Most of us learn the equation for a circle: (x-a)² + (y-b)² = r². But how many of you learned an equation for a square? Just need some absolute value trickery: |(x−a) + (y−b)| + |(x−a) − (y−b)| = r.
I like to include equations in the margins or on the back when I print my math-art, and admit I had to look this one up. This is another vector field image, very similar to the one I posted two days ago, but with the initial coordinates on a square.
Fractal Chicken 🖤 Wall Art Prints & more: https://matthias-hauser.pixels.com/featured/fractal-chicken-02-colorful-dream-matthias-hauser.html Cluck into the fractal world, where every feather tells an endless story! 🐔🌀✨ #chicken #bird #birds #animal #animals #fractal #fractalart #art #mastoart #fediart #mathart #wallart #ShopEarly #Fedigiftshop #AYearForArt #BuyIntoArt #ArtMatters
Without judgment:
A lot of math art involves writing, either algebraic expressions and equations (quadratic formula, Pythagorean theorem, ...), or an illustrated quote referring to doughnuts and coffee cups, or (misattributing Erdős instead of Alfréd Rényi) coffee and theorems.
To me, the power and beauty of mathematics is insight, and the older I get, the less insight feels naturally expressed in symbols (though symbols may be a particularly efficient tool to convey insight to others with suitable training).
With age as well, mathematics has come to seem a human invention and a thin veneer on reality. Mathematics may be by far the most precise and flexible description we have of physical phenomena, but fundamentally mathematics is not reality. I fear that mistaking mathematics for reality -- at levels from fundamental physics to the social sciences -- underwrites a certain disregard for Life and the Mystery of Existence.
"Imagination, Forge" is an attempt to capture this illusory reality of mathematics, with sympathy and as a cautionary reminder.
https://diffgeom.com/products/imagination-forge-wall-art-poster
Orange peel Nudibranch- I’m processing loss with slug doodles. I suppose we all cope differently.
It's a dark and dreary rainy day, here in San Francisco. I've been wanting to crawl back into bed since I got up.
Drawing a simple fractal with a little more chaos than I usually play with, in today's #inkDays.
Here's a work-in-progress photo.
As usual, the finished drawing is posted for my supporters on Patreon and Ko-fi.
#ink #drawing #art #Fractal #MathArt #GenerativeArt #flowers #wip #wipwednesday
Happy Color Wheel Wednesday!
This color wheel is made with three primary watercolor pigments, a bit of colored pencil, white gel pen, black ink, and six mica paints that shimmer in the light. Indanthrone blue is a beautiful denim that’s much more lightfast than natural indigo. Quinacridone magenta and nickel azo yellow are also wonderfully lightfast. I’m curious how lightfast the mica paints are as they get their color in a way that’s different from pigments.
Weierstrass's continuous, nowhere-differentiable functions are uniform limits of (smooth!) trig polynomials. In the image, virtual mountain ranges lie beneath the graphs of successive approximations to a Weierstrass function, translated horizontally (added: and vertically) for visual effect.
https://diffgeom.com/products/nowhere-differentiable-desert-mug
Day 5 of #ArtAdventCalendar
I made a bunch of these vector field images this past summer. Define a field x -> f1(x, y); y -> f2(x, y), and a set of initial values (in this case all the points on a circle). Trace the points along that circle through the field a few dozen times, while changing color at each step.
Early in 2024 at the Differential Geometry math art shop, we anticipate introducing 3D printed pendants, earrings, and pins. (To start these will only be offered in the US.)
The plated brass pieces shown are cyclides of Dupin, flat Clifford tori in the three-sphere mapped to Euclidean three-space by stereographic projection. Each cyclide is divided into diagonal bands bounded by two "parallel" Villarçeau circles. Unlike the meridians and parallels of a torus, Villarçeau circles are images of great circles (here, Hopf circles) in the three-sphere.
<>
Algebraically, view the real three-sphere as the set of complex pairs \((z, w)\) satisfying \(|z|^{2} + |w|^{2} = 1\). A Clifford torus may be viewed as the set of pairs where \(|z|^{2} = |w|^{2} = \frac{1}{2}\), or may be rotated in either complex axis to arrange that the three-space image is not a circular torus. The multiplicative unit circle acts by scalar multiplication: \(\chi_{u}(z, w) = (uz, uw)\). This action preserves Clifford tori, which are unions of great circle orbits. The geometric effect, if realized, would "slide the diagonal bands along themselves," a cross between a barber pole and a smoke ring.
Contemplated life’s beauty in its details tonight. #nudibranch #symmetry #mathart
Weird flow field! The code to make this is even more of a shambles than the output.
Kind of makes me want to invest in a plotter though :-)
The image below is an experimental crossed-eyes stereogram for a poster of the chaotic attractor for the ODE system
\begin{align*}
\frac{dx}{dt} &= 10(y - x), \\
\frac{dy}{dt} &= x(28 - z) - y, \\
\frac{dz}{dt} &= xy - \tfrac{8}{3}z.
\end{align*}
If you're willing to comment, it would be helpful to know whether the stereo effect is difficult to achieve and/or maintain.
An open disk and a radially-slit open disk are conformally equivalent: There is an invertible holomorphic mapping from one to the other. The grid lines are the images of concentric circles and rays from the origin.
Day 3 of #ArtAdventCalendar
Here's a fun piece I created for the last Genuary, an annual creative coding challenge.
In this one, I'm playing with sine curves and evolving colors.
I made a short post about it: https://www.patreon.com/posts/genuary-2023-day-75979244
The conformal parametrizations
\[
\operatorname{cat}(u, v) = (\cosh u \cos v, \cosh u \sin v, u)
\]
of a catenoid and
\[
\operatorname{hel}(u, v) = (\sinh u \cos v, -\sinh u \sin v, v)
\]
of a helicoid are well-known to be not just locally isometric, but deformable through a one-parameter family of minimal surfaces
\[
\mathbf{x}_{\theta}(u, v) = \cos\theta \operatorname{cat}(u, v) + \sin\theta \operatorname{hel}(u, v).
\]
The animation loop is a stereogram following one turn (\(-\pi \leq v \leq \pi\)) of the helicoid. It may be easiest to cross your eyes to fuse the initial stereogram of two catenoids, then hover the mouse to start the animation loop.
<>
The 2023 holiday sale at the Differential Geometry math art shop runs through Sunday, December 3, at 11:59 PM eastern time. Visit through the link https://diffgeom.com/discount/maths23 or use the code maths23 at checkout to get 20% off garments and wall art.
This is doing something.
Trying separate the growth axis. The reaction diffusion pushes in the normal direction and I increase edge length in the transverse direction.
I'm weighting the growth using the principle directions of curvature that I get from local quadric fitting. It works OK.
#mathart #GenerativeArt #GenArt #algoart #algorithmicart #sculpture #algorithmicsculpture #SciArt
Fractal Women, Yin and Yang 💙 🧡 Wall Art Prints & more: https://matthias-hauser.pixels.com/featured/fractal-woman-01-yin-and-yang-matthias-hauser.html 🖤 #woman #women #female #fractal #fractalart #art #mastoart #fediart #mathart #wallart #ShopEarly #Fedigiftshop #AYearForArt #BuyIntoArt #ArtMatters
There's a new blog post about compound circular motion at the Differential Geometry math art shop. The intended readership includes adults who have not seen trigonometry in a long time, students of calculus, and anyone who enjoys rotational symmetry.
https://diffgeom.com/blogs/about-math/epicycles-compound-circular-motion
"Nebulabrot" of the mandelbrot set + burning ship + IFS hybrid
#fractal #mastoart #mathart #creativecoding #rainbow #generativeart #noise #ghost #space
The image below depicts the Riemann surface of arcsine, a.k.a., the complex graph \(w = \sin z\) viewed "as a function of \(w\)." Accordingly, \(w\) is the horizontal plane; the real axis of \(z\) is vertical and the imaginary axis of \(z\) is projected away.
https://diffgeom.com/products/riemann-surface-of-arcsine-wall-art-poster
<>
The blog at the Differential Geometry math art shop aims to explain images and objects we sell, for readers of all ages who have perhaps seen Cartesian plane coordinates and functions, but whose comfort zone may not include these topics. Alec Wilkinson's "A Divine Language" provides continual guidance and inspiration.
https://us.macmillan.com/books/9781250168580/adivinelanguage
<>
There is a more advanced blog post about Riemann surfaces, treating the square root in detail, aimed at undergraduate students who have seen vectors and parametric surfaces:
https://diffgeom.com/blogs/about-math/square-roots-with-a-complex-twist
![A blue-and-gold periodic surface against a dark purple background, made of two intertwining helicoid-like halves that meet in the middle along the real sine graph as a bright green curve. Mathematically, the surface comprises an infinite sequence of sheets, each a branch of arcsine on the complex plane with two slits. The interval [-1, 1] traces a half-arch of the real cosine running vertically. Each sheet is cross-attached to its downward neighbor along the one slit and cross-attached to its upward neighbor along the other slit.](https://files.mastodon.social/cache/media_attachments/files/111/499/566/527/214/109/small/34bd654d254261cd.png)
A little #mathart in geogebra https://www.tumblr.com/mathhombre/735363718809092096/schleifer-turning?source=share after a new-to-me artist, Fritz Schleifer.
my mathober piece - exchange - is featured in this weeks codepen spark!
Happy Color Wheel Wednesday!
Here’s what you get when you use two similar magentas, two similar blues, and two similar yellows in every m-b-y combination. You get 8 similar color wheels. It surprised me when I painted them, but now it seems pretty obvious. It just goes to show that if you don’t have exactly the right color to match what you’re doing, close might be close enough.
For mathematicians: Put three equally-spaced loxodrome bands on a globe, and rotate the globe about its axis. Now conjugate this loop of isometries by a Möbius transformation that moves the poles to a pair of non-antipodal points.
For non-mathematicians: Think of a sphere dancing with boas.
<>
One challenge of communicating mathematics to non-technical readers is the precise, recursive compression inherent in, for example, the first paragraph; the webs of concepts, connections, and idioms that must be carefully unpacked to make contact with non-experts' language and experience.
A complementary vexation is the superficial imprecision of even a vivid qualitative description.
<>
An image at the Differential Geometry math art shop starts with a mathematical idea, usually one that people struggle to learn in technical detail, such as uniform convergence, Möbius transformations, group actions, quotient topology, or Riemann surfaces. One aim is to capture a mathematical portrait that is immediate and visually-appealing, inviting the interested viewer to learn more, yet honest (as far as it goes) so that further study only reinforces the piece.
https://diffgeom.com/products/mobius-flow-pond-wall-art-poster
Another coded poem:
//just enough for beauty
https://sophiawood.medium.com/just-enough-for-beauty-fa7eeda0d91d
#wccchallenge #obscure #mathart #p5js #codedpoem #generativeArt #creativeCoding
... and an animation loop where the values of the fields rotate counterclockwise through a full turn.
medium post (no paywall) with poem for dichotomies painting: https://medium.com/p/5c403a18bf31 #haiku #mathart
A plane vector field assigns a vector to each point of a plane region. Near an isolated zero, a vector field is characterized qualitatively by an integer, the index, which counts how many times the value rotates upon tracing a small circle about the zero.
The formula \(e^{i\theta} = \cos\theta + i\sin\theta\) gives a pleasant explanation of the shapes in the poster. If \(k\) is an integer and \(r\) a positive real number, the field whose value is \(re^{ikt}\) at the unit complex number \(e^{it}\) models a field of index \(k\) along the unit circle. The tips of the arrows are the vector sum,
\[
e^{it} + re^{ikt} = e^{it}(1 + re^{i(k-1)t}),
\]
a path having \(|k - 1|\)-fold rotational symmetry about the origin.
https://diffgeom.com/products/topological-index-wall-art-poster
⚠️ It's #TilingTuesday everyone(yes that's a thing now)
🌐 Here's the Globe™️
#tiling #mastoart #mathart #blackandwhite #stars #owo #mandala #限界曼荼羅
Done! A painting without orientation (hang it as you wish).
My process:
-feel the feelings
-start painting
-watch the feelings unfold
-contemplate when to be done
-sign it ( this time in 2 places)
Will post poem on medium later. (Acrylic 15x30x1in on canvas)
Drying and in poor lighting, but obviously requires two signatures because it has no orientation. Will seal and post better pics tomorrow.
Anyone else have tree and circle obsessions?
Acrylic, 15x30in
Fantasy Alien Landscape. Wall Art Prints: https://www.pictorem.com/841976/Fantasy%20Alien%20Landscape%2002%20Romantic%20Night.html?refer=BM8DNL7TQN Free Shipping (US and Canada)! #fantasy #planet #landscape #psychedelic #art #mastoart #fediart #mathart #wallart #ShopEarly #Fedigiftshop #AYearForArt #BuyIntoArt #ArtMatters
Mushroom Fantasy Landscape 🧡 Wall Art Prints and other Home Decor products: https://matthias-hauser.pixels.com/featured/mushroom-fantasy-landscape-01-matthias-hauser.html 💙 #mushroom #mushrooms #psychedelic #fantasy #landscape #art #mastoart #fediart #mathart #wallart #ShopEarly #Fedigiftshop #AYearForArt #BuyIntoArt #ArtMatters
My nautilus with golden rectangle print for Fibonacci Day. November 23 if written in MM/DD format recalls the #mathematician Leonardo Bonaccio of Pisa (c. 1170 - c. 1240 or 50) aka Fibonacci’s sequence (1,1,2,3…) where each number is the sum of the previous two. He used it to describe rabbit populations, but the sequence is commonly observed in nature, 🧵1/2
#linocut #printmaking #mathematics #mathart #sciart #nautilus #GoldenRatio #Fibonacci #fibonacciday2023 #FibonacciDay #MastoArt
made a generative knitting pattern toy to contemplate some crafting - please share if you find some cool ones or knit it up!
https://codepen.io/fractalkitty/full/MWLVqqM
#mathart #generativeArt #creativeCoding #knitting #patterns #p5js
made a generative knitting pattern toy to contemplate some crafting - please share if you find some cool ones or knit it up!
https://codepen.io/fractalkitty/full/MWLVqqM
#mathart #generativeArt #creativeCoding #knitting #patterns #p5js
Here's another output from the summer's abandoned project. I didn't have any real direction with these except I wanted to do something with colour and texture. Now I have a big set of images like this that I don't know what to do with!
Would they work as prints? They look maybe more like postcards to me.
Some people like the Riemann sphere, stereographic projection, Möbius transformations, and holomorphic vector fields. Go figure....
Hello #mathematics #mathart #introductions
Labels over-simplify and distort. That said:
By current profession I'm a freelance mathematician who creates (non-generatively) and sells (without surveillance capitalism to the extent possible) mathematical images and objects. Other labels include author and blogger, mathematical illustrator, digitizer for Project Gutenberg, hobbyist coder, former math professor, and deeply concerned human citizen.
Regarding mathematical art, my primary inspiration is form (shape), accessible without technical training but an open invitation to deeper encounter with mathematics.
My aspirations include sharing and promoting the beauty of shape and structure for pure contemplative enjoyment, and helping to cultivate public understanding of math as a participatory, inclusive, wonder-filled activity.
Psychedelic Space Cat. Wall Art Prints & more: https://matthias-hauser.pixels.com/featured/colorful-psychedelic-cosmos-cat-01-matthias-hauser.html 💜 🧡
This whimsical artwork is an enchanting depiction of a cat, whose vibrant, multicolored fur is a psychedelic tapestry that merges with the surreal cosmic environment it inhabits. #cat #cats #catsofmastodon #animal #animals #colorful #surreal #psychedelic #art #mastoart #fediart #mathart #wallart #ShopEarly #Fedigiftshop #AYearForArt #BuyIntoArt #ArtMatters