For this particular design, I played around with a tangent plot. It’s so easy to plot polar graphs with it, and they usually have a lot of symmetry. I’ve been using Desmos to brainstorm window designs. This technique is good for smaller pieces, and designs that have a lot of detail. I’ll show you how to make a window using the copper foil technique. By iterating this process, we end up with shapes that are, in my opinion, quite interesting.In a earlier post, I explained the steps involved in making an artglass window using lead came. This isn’t necessarily a proper stellation, but it’s a similar notion.

This is equivalent to adding a new vertex orthogonally a certain distance from the center of each face and connected it to that face’s vertices. By triangulating the set of original vertices together with the vertices of the dual graph, we can get a mesh representing a new shape. Long story short, there’s no nice answer to this problem (which explains why it isn’t written or documented anywhere!) but at least this graph gives a way of approximating it.Ĭatenary Rope graph: Stellated IcosahedronĪ polyhedron can be represented as a planar graph (fittingly, a polyhedral graph). The integral would not be computable in terms of standard functions, and Desmos evaluates integral expressions numerically anyway. The solution to this can be written explicitly with an integral in terms of the Heaviside step function, but there’s no point in that. There’s no clean way to solve this, so the graph uses a regression to approximate a solution. They are dependent on the solution to the transcedental equation As it turns out, the curve is a scaled and shifted hyperbolic cosine, and the scaling and shifting factors are highly nontrivial to compute. However, nobody ever says WHICH catenary, or how to find a formula for the curve describing a rope of a given length hanging between two points. It’s relatively well-known that the shape of a rope hanging between two points, for instance attached to two fenceposts, is not a parabola, even though it kinda looks like that. This is analogous to the construction of a tesseract, but with Bernard as the base shape instead of a square. I formed it by extruding the traditional Bernard prism along each of its faces. This is the only 4D manifestation of Bernard to my knowledge. I have made countless Bernard-related graphs, one of which is on YouTube, and I even have a solid aluminum laser cutout of Bernard, and a crocheted/stuffed version made by a friend.

He’s the unofficial mascot of Desmos at this point. Many people in the discord, on the Desmos subreddit, and certainly elsewhere have tried their hand at an artistic rendition of Bernard. If you want to know more details about Bernard and why he exists, join the discord server that’s where all the experts on this subject are. If you don’t know about Bernard (I pity you), he is the artifact caused by Desmos’s quadtree descent algorithm for drawing implicit equations. Mandelbrot set domain coloring on the Riemann sphere: Fractal Grassįor those active in the Desmos community online, Bernard is a familiar name. Mobius Transformation on the Riemann sphere, with occlusion: This graph was inspired by the (extensive) work on domain coloring by a member of the Desmos discord server (who I’ll name here, pending permission). The resulting diagram, when projected to the sphere, allows us to see the effect of the function over the entire input space! These colors are determined by the modulus (magnitude) and argument (angle) of the number when in polar form. Domain coloring works by assigning a color to each number in the complex plane, and moving each input to its output. Complex functions with one input and one output are 4-dimensional, so they’re non-trivial to visualize. Domain coloring is a method for visualizing complex functions. The Riemann Sphere is a model of the complex plane that uses inverse stereographic projection to fit the entire infinite plane onto a surface with finite surface area. Domain Coloring on the Riemann Sphere domain coloring on riemann sphere I’ll try to avoid graphs that I posted on YouTube. Nevertheless, here are eight of my favorites, in chronological order from oldest to newest. As such, attempting to comb through and select only a few graphs that represent the year is an exercise in futility. From harmonographs and obscure coordinate projections to artistic animations, this has probably been my most active year with Desmos, which is really saying something, since it has been my primary hobby for quite some time now. But it’s not 2023 yet (hours away!) so nobody can say I didn’t do it. In a Desmos Global Math Art Contest-induced fit of madness, I almost forgot to do the annual Desmos graph showcase.

Desmos desmos polar graphing art