# Ellipse: geometric proof of the equivalence of two definitions

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…

# Apollonian gaskets: beautiful math can be simple

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

An Apollonian Gasket of type -1_2_2_3

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]}]`

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.

# 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)