# #MathArt

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 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.

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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.

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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

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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

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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 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.

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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.

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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

A conformally-parametrized cyclide of Dupin.

Our holiday sale at the Differential Geometry math art shop continues one more week! Use the code maths23 at checkout to get 20% off all garments and wall art through December 3, 2023.

This curve, incidentally, is a projection to real three-space of the image of the unit circle under the twisted cubic mapping into complex three-space. If \(t\) is real and \(z = e^{it}\), then

\[

(z, z^{2}, z^{3})

= (e^{it}, e^{2it}, e^{3it})

\simeq (\cos t, \sin t, \cos 2t, \sin 2t, \cos 3t, \sin 3t).

\]

This particular curve is the projection to the second, third, and sixth coordinates:

\[

\gamma(t) = (\sin t, \cos 2t, \sin 3t).

\]

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)

The Riemann surface of \(\log\) may be viewed as the graph of \(\exp\) together with projection to the image plane, i.e., as the set of points \((z, w) = (z, \exp z)\) over the \(w\)-plane. An oblique projection, qualitatively taking the third coordinate to be the sum of the real and imaginary parts of \(z\), captures both logarithmic behavior as \(|w| \to 0\) and the infinite-sheeting covering as \(w\) traverses a circle about the origin.

Use the code maths23 at checkout to get 20% off all garments and wall art through December 3, 2023.

https://diffgeom.com/products/riemann-surface-of-log-wall-art-poster

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

Each element of the Cantor set may be viewed uniquely as a ternary number none of whose digits are \(1\), i.e., as a path through an infinite dyadic tree whose alternatives are \(0\) and \(2\) (base \(3\)), or algebraically as an expression

\[

\sum_{k=1}^{\infty} \frac{x_{k}}{3^{k}},\qquad

\text{\(x_{k} = 0\) or \(2\).}

\]

Each element of the unit interval may similarly (but perhaps non-uniquely) be viewed as a binary number, i.e., as a path through an infinite dyadic tree whose alternatives are \(0\) and \(1\) (base \(2\)), or algebraically as an expression

\[

\sum_{k=1}^{\infty} \frac{y_{k}}{2^{k}},\qquad

\text{\(y_{k} = 0\) or \(1\).}

\]

The map sending "Cantor set paths" to "unit interval paths" (or, "halve each ternary digit and interpret as binary") is onto, and two points of the Cantor set are sent to the same point if and only if they are "adjacent in the Cantor set," i.e., endpoints of a "removed middle third." This mapping consequently has a continuous, non-decreasing extension to the interval \([0, 1]\) that is constant on each interval complementary to the Cantor set.

Use the code maths23 at checkout to get 20% off all garments and wall art through December 3, 2023.

https://diffgeom.com/products/cantors-staircase-wall-art-poster

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

first breakday #mathart riffing on Vasarely in geogebra https://www.tumblr.com/mathhombre/734714158706278400/vasarely-hexes?source=share

I've always been curious about the growth model @nervous_jesse developed so I thought I'd explore it a bit. Here is the model but with expansion governed by grey-scott reaction diffusion.

#mathart #GenerativeArt #GenArt #algoart #algorithmicart #sculpture #algorithmicsculpture #SciArt

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

Cosmic Eye 01, Colorful Universe. Wall Art Prints & more: https://matthias-hauser.pixels.com/featured/cosmic-eye-01-colorful-universe-matthias-hauser.html 🧡 💙

This surreal and vibrant artwork is a cosmic voyage through the eye, the window to the soul, set against a backdrop of swirling galaxies and nebulous star fields. #eye #universe #galaxy #space #psychedelic #surreal #colorful #art #mastoart #fediart #mathart #wallart #ShopEarly #Fedigiftshop #AYearForArt #BuyIntoArt #ArtMatters

Doodling with Jessica is a lot of fun. She wanted to make it monochromatic. So we compromised and left out red and purple.

This illustrates a way to make a #Voronoi diagram. Start with nodes (points). Connect them to their neighbors with edges (line segments). Find the midpoint of each edge, and paint a blob of color around each node up to the midpoints on the edges that touch that node. Very meditative.

#Watercolor, black ink, cotton paper

#mathart #network #doodle #algorithmicart #algorithm

My porthales project is in the news today: https://www.salemreporter.com/2023/11/14/salem-artist-reimagines-the-citys-manholes-as-portals-to-another-world/

#porthales #naNoWriMo2023 #haiku #python #data #rpg #mathArt

I was unsatisfied with how handles were resolving, and I found a solution using quadrics... but then I happened upon another implicit form which is even better,The Darboux Cyclide. The Darboux Cyclide is like a super torus...

I have quite a bit more to learn, but it turns out least squared fitting of the Cyclide worked very similarly to fitting a quadric!

#mathart #GenerativeArt #GenArt #algoart #algorithmicart #sculpture #algorithmicsculpture #SciArt

Simulated in time steps of 4h, I only used used 1 frame every 1325 steps, at 10fps that is about 6 earth years passing per second. The planet's orbit seems rather stable, but it changes parameters at each breath in of the twin suns.

Great #mathart opportunity from Paula Beardell Krieg. Learn about Seminole patterns. https://bookzoompa.wordpress.com/2023/11/04/seminole-patchwork-with-paper/ and what you need https://bookzoompa.wordpress.com/2023/11/12/what-youll-need-to-join-in/

A wild new shape appeared, I call it a sqrt(2) ball, maybe it has a better name. It has square, triangular and rhombic faces. I built it with prisms and rhombohedra with sqrt(2) rhombuses, as well as square pyramids, tetrahedra, and regular triangular prisms. All with the recently updated @hedron app #Polyhedron #MathArt #Geometry #Hedron