Toortle of toorteellography -- steell only a toortle

What's cooking in the oven?

@user430580 I did the original in POV-Ray. I have several other ray-traced versions, but not on my phone. Here's a Python / Sage script that draws a rhombic dodecahedron in 3D, and shows how it's related to the cube & octahedron.

sagecell.sagemath.org/…

Another option is to draw it as a parametric dodecahedron. Set the parameter h to 1.

sagecell.sagemath.org/…

Set h > 1 or h < 0 to get strange non-convex shapes.

Nice links, thanks!!! I like this Sage language, because it looks similar to Python

Yes, Sage is built on Python. But it lets you do a whole lot more. It's not quite as powerful as Mathematica, but it's free. And I find it easier to read.

Sage sounds like an interesting tool, especially if it's built on Python. It's always nice when things are free and accessible. I'm curious about how it compares to something like Mathematica in terms of ease of use. Do you have any favorite shapes or oddities you've created with Sage?

If you can already read & write standard Python, it only takes a few hours to learn the important differences between Python & Sage, and you can start doing all sorts of cool things with it, like algebra & calculus, almost straight away. But it's huge, so it takes a while just to learn what it's capable of.

There's an intriguing triply-connected minimal surface called the gyroid, which was discovered by a NASA scientist in the 1970s. The true gyroid requires elliptic integrals, but you can approximate its shape using simple trig functions. Here's a quick demo of a pseudo gyroid.

sagecell.sagemath.org/…

The gyroid surface partitions space into a pair of congruent labyrinths. It has become popular in recent years as a way of filling space in 3D printing, although they mostly use the pseudo gyroid because it's easier to calculate. But the real gyroid has better properties, because it's a minimal surface, and IMHO it looks nicer. I'll look through my collection for a nice example...

Here's a unit cell of the true gyroid, coloured to show how it's derived from the tessellation of 6 hexagons meeting at a point in the hyperbolic plane. Click on the "gyroid.html" link to open it in its own tab / window.

sagecell.sagemath.org/…

@Amit rofl

Here's one with some translucency

sagecell.sagemath.org/…

Here's a much simpler parametric surface: a Klein bottle. Unfortunately, it's a bit squished at the bottom. It uses a custom colour palette.

sagecell.sagemath.org/…

@user430580 Topological pancakes

@PM2Ring Is it comparable though symbolically with Mathematica? Can it do integrals etc?

Regardless, you share some really cool stuff... I'm a junkie of this kind of stuff

a bit of fun experiments with Rindler coordinates in Geogebra... I just recently started using it. Surely not as flexible as SageCell , but cool

As someone who wrote his fair share of code, I like to think myself now as a lazy retired old guy who won't write a code for something that a visual tool can do :P Is that bad? Yea, I think it's bad. Don't feel bad for me. I'll do it for my own self

lol wut

@Amit Um, I just said it can do calculus. But yes, it can do symbolic integration (& differentiation) of lots of functions. And of course it has a couple of numerical integrators as well.

Those gyroid programs create vector function objects from symbolic elliptic integral expressions. Then those functions are fed complex number parameters to generate the fundamental parametric surface element (a curved triangle). Matrix algebra is used to transform the fundamental element in 12×8 ways to create the unit cell.

@PM2Ring cool, I was just curious, then I have extra motivation to learn how to use it :) I sometimes struggle when I have to do symbolic stuff

Take a look at the tour. sagemath.org/tour.html

Sage does symbolic stuff by default. You have to explicitly tell it not to do that (in various ways) if you just want it to work numerically.

That can be a bit annoying when you forget. Especially if you're doing stuff in a loop. You wonder why it's taking ages to do some relatively simple calculation. Then you print out some term and see you've created a symbolic expression with hundreds of trig functions or nested square roots. :)

/convert 1 Amit to Celsius

@PetəíŕdtheWizard 1 Amit is approximately -17.222 Celsius. (source)

rofl

Here's a very simple example of symbolic integration:

sagecell.sagemath.org/…

@Amit Geogebra can do interactive stuff that's impossible in SageMathCell. The GUI capabilities of SageMathCell are limited to providing inputs to your Sage code. It does that via JavaScript running in your browser. The 3D plotting (which is done using three.js) has some interactivity. You can also create 3D animations.

@PM2Ring Thanks! I'll give it a look for sure

@PetəíŕdtheWizard =O How did it know I like the cold

@Amit Petəíŕd the Wizard is away: again

There's a 3D anim example at the bottom of this page: doc.sagemath.org/html/en/reference/plot3d/…

Here's an "interactive" polar function plot. Sadly, that slider doesn't work well on mobile.

sagecell.sagemath.org/…

Here's a little demo of symbolic algebra. It creates a Padé approximant of arctan(x) for x near zero. Then it uses that Padé to approximate pi, using tan(pi/8) = sqrt(2) - 1. Then it makes a simple plot to show the error in the Padé near zero.

sagecell.sagemath.org/…

The Padé has a much larger region of convergence than the truncated Taylor series that it's built from.

If you set num to 11 (or greater) the graph goes a bit wonky because its only using standard double-precision floats for the numeric evaluation, which have 53 bits of precision. But we can easily use much higher precision, if we want.

One of my favourite things to do with SageMathCell is using it to access the JPL ephemeris via their Horizons system. Eg, from astronomy.stackexchange.com/a/44903/16685

From ssd.jpl.nasa.gov/horizons

The JPL Horizons on-line solar system data and ephemeris computation service provides access to key solar system data and flexible production of highly accurate ephemerides for solar system objects (1,417,978 asteroids, 3,974 comets, 293 planetary satellites {includes satellites of Earth and dwarf planet Pluto}, 8 planets, the Sun, L1, L2, select spacecraft, and system barycenters). Horizons is provided by the Solar System Dynamics Group of the Jet Propulsion Laboratory

Horizons data for planetary barycentres spans 20,000 years. Detailed motion data for the planets & their moons have a smaller timespan due to chaos. ssd.jpl.nasa.gov/horizons/time_spans.html

I mostly just use plain Python for Horizons stuff, but Sage is occasionally handy, eg when I need to do vector arithmetic, eg

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@PM2Ring do you have some interesting simulations relating to hyperbolic motion / acceleration in SR? :P

Just 'cause it's something I'm studying at the moment

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🎵🎶 I'm sending you a letter, because I don't think there's much time 🎵🎶

@Amit Hi sending you a letter, I'm Oak!

@Amit Hey, sorry for slow response, but I haven't tried it until right now! I tend to avoid trying out promising suggestions if I happen to be in a bad mood, because I want to give the music a chance as fair as possible to make a good first impression. I like this one better than the previous one, but still La Foi En L’eau wins! =P

@user430580 =] it really is quite good

it's fine, i hope your mood is good

thank you, I hope yours is good too

@Amit this song is also cool!

I had a hunch you'll like it

@user430580 First time you've heard this band?

@user430580 They have very interesting stuff

This for example is quite a cheery tune, considering it's about the Apocalypse lol

also the tape recording is of that crazy cult guy, a story no less grim

@user430580 Thanks!! I missed that message earlier 🐢😍