Every time I try to get some deeper insight about Newton’s gravitational law I stumble upon the geometrical properties of the ellipse.

After many years of these *strange and challenging encounters* I really think that the ellipse is a rich and wonderful trove of geometrical nuances and subtleties…

# Category Archives: Beautiful math

# Mathematical flowers

# Apollonian gaskets: beautiful math can be simple

I’ve recently discovered the beauty, symmetry and mathematical richness of Apollonian gaskets.

Here’s a very short code (under 128 character’s length) that I’ve made with *Wolfram Mathematica* guided by the saying “*Beautiful math can often also be very simple*“.

`Graphics[{Purple,Circle[],Disk@@@Flatten[Table[1/(k^2+2) {{(-1)^r (-k^2+1), -2 (-1)^j k},1}, {k,0,9}, {j,0,1}, {r,0,1}],2]}]`

And here’s the twit published by @wolframtap (*Wolfram Tweet-a-Program).*

```
```@sfera314 (More info: http://t.co/HBUuaZOB3w) #wolframlang pic.twitter.com/1aP7Wceob4

— Tweet-a-Program (@wolframtap) April 6, 2015

Well, actually that is not a complete Apollonian gasket, but it can give the idea.

To produce a full gasket the code should be longer than that allowed by a twitter length, but I think that a basic one could be done in about 500 characters or less.

## Some links on Apollonian gaskets:

- https://en.wikipedia.org/wiki/Apollonian_gasket
- Apollonian_gasket A Tisket, a Tasket, an Apollonian Gasket (pdf), Dana Mackenzie, American Scientist, 2010
- The Circles of Descartes (Wolfram Demonstrations Project) – Contributed by:
**Ed Pegg Jr**(you’ll need to have the Wolfram CDF Player installed to play with this full simulation)

# Raindrops

Sometimes math can be completely useless, but amazingly simple and beautiful…

Another possible example of this fact is the following animation , that could be created with a very short code in *Wolfram Mathematica* (just 221 characters in total):

Animate[With[{r := RandomReal[]}, Graphics[BlockRandom[ Table[With[{z = r}, {, GrayLevel[2 (t - z)], Thickness[0.03 (0.20 - t + z)], Circle[{1.7 r, 0.82 r}, Max[0, t - z]]}], {k, 1, 45}]], PlotRange -> {{0, 1.7}, {0, 0.82}}]], {t, 0, 1}, DefaultDuration -> 20]

Too much long to be posted in in the twitter @wolframtap (*Wolfram Tweet-a-Program)*. But short enough to show how some basic mathematical ideas can be very simple and yet beautiful (even if, maybe, useless). Here’s the video posted on youtube:

# A twisted Eiffel tower (useless but beautiful math)

Sometimes math can be completely **useless**, but amazingly **simple** and **beautiful**.

A possible example of this is the following image, that could be created with a *twitter*-sized code in *Wolfram Mathematica* (123 characters in the present case):

Graphics3D[Table[Rotate[Cuboid[{-0.9^k, -0.9^k, (1/20)*k}, {0.9^k, 0.9^k, (1/20)*(k + 1)}], k*0.1, {0, 0, 1}], {k, 0, 60}]]

This *mini*-program was published (and favorited) in the twitter @wolframtap (*Wolfram Tweet-a-Program).*

Here‘s the twit.

Another interesting thing about the fancy building depicted in the image is that, although it might have infinite height, it’ll still have a finite volume.

There’s also a small extension in this interactive demonstration (in which it’s possible to change the angle between consecutive parallelepipeds.

(Thanks to BV for suggesting me this beautiful idea)