The Mandelbrot Set - Numberphile

Famously beautiful, the Mandelbrot Set is all about complex numbers. Featuring Dr Holly Krieger from MIT.
More links & stuff in full description below ↓↓↓

Since this was filmed, Holly has become a mathematics Lecturer at the University of Cambridge and the Corfield Fellow at Murray Edwards College.

NUMBERPHILE

Videos by Brady Haran

[michaelbauers]

Those super deep zooms of the set never get old. I especially love the zooms that don't move...they just descend. And I think, it's pretty amazing how much complexity you get in such a small number space

[AlanKey]

Allegedly, when Benoit B. Mandelbrot used to be asked what the "B" in his name stood for, he would reply:

_The B? It stands for Benoit B. Mandelbrot!_

Legend.

[TacomaPaul]

At about 2:55 she says, "1+1=2".  I got that !
The rest... ?   Yikes.
Fascinating stuff.

[lollertoaster]

That presenter is very good at explaining. I love how she reiterate on the things that can be more difficult for some people.

[tomolonotron]

I hear words, but I'm not understanding them

[Infinite_Omniverse]

Math can be very beautiful... The Mandelbrot set proves this.

[gwenmcardle]

i did a presentation on fractals last year and the mandelbrot set was my big finale. this video helped me a ton! i actually kind of understand it now, but my classmates didn't. im not the best teacher.

[SlyMaelstrom]

I'm too much of an iterate to understand this...

[billlson]

I think we need more Holly on Numberphile

[noradosmith]

I feel stupid for only just understanding this. And I feel doubly stupid for knowing they're trying to dumb it down for people lik me to understand. I like the idea though, of being on the cusp between 'blowing up' and not 'blowing up'. Pretty much summarised my brain watching this.

[michaelk]

I love when Dr. Holly does anything with Numberphile.  She doesn't sound like she's droneing, but rather is excited to teach and loves that which she is teaching.  She's the type of teacher I bet more people wished they had had growing up when teaching Math or other subjects.  I'll never understand the teachers that don't have passion for what they are teaching.

[rkerner]

Probably my favorite Numberphile ever. It's certainly amazing how such a simple function can lead to the most wonderful art... I've never been a fan of science fiction nor art for only just the sake. Rule fact here is so much more beautiful and amazing because it is absolutely so very honest to the core. Dr Krieger, I very much appreciate your patient explanation. Thanks!

[hweigel]

Someone asked, "why is two the bound after which everything blows up?", which is a very good question.  The reason becomes more intuitive if you know a few important properties of complex numbers, namely that |u*v| = |u|*|v| for all complex numbers u and v, and that |u + v| >= |u| - |v| for all complex numbers u and v.  

Using these two properties, consider the magnitude of a given number going through this procedure.  Given that z has magnitude |z|, f(z) = z^2 + c has magnitude |f(x)| = |z^2 + c| >= |z^2| - |c| = |z|^2 - |c|.  

Now we can consider a function based on some |c| >2.  Clearly f(0) = 0^2 + c = c, and so |f(0)| = |c| > 2.  Next, f(c) = c^2 + c = c*(c+1), and so |f(c)| = |c|*|c+1|, and since |c|>2, |c+1| >=|c|-|1|>1.  Therefore |f(c)|=|c|*|c+1|>|c|.  Now, assume that we have done this procedure enough times to reach some arbitrary number z, such that |z| > |c| > 2. (We already know that we reach a number with this property after two steps).  |f(z)| = |z^2 + c| >= |z|^2 - |c| > |z|^2 - |z| = |z|*(|z|- 1). Since |z| > 2, |z| - 1 > 1, and therefore |f(z)| > |z|*(|z| - 1) > |z|.  Since this is true FOR ALL |z| > |c|, we know that |z| < |f(z)| < |f(f(z))| < |f(f(f(z)))| <... Since the magnitude always increases as we perform more iterations (and does so in a way that does not converge, since |z|-1 increases in size as |z| gets larger), the number "blows up", as Brady would say. QED.

[tacchinotacchi]

Isn't there any video from this girl outside of Mandelbrot, julia set, and -7/4?
Her voice is so relaxing

[ralfoide]

Keeping watching this particular video over the years and it's still the best Mandelbrot set explanation I've seen to date. Dr. Krieger is remarkable, and the series of Numberphile videos on Mandelbrot with Dr. Krieger are all extremely clear and interesting. Would be nice to see Dr. Krieger return to lecture us on whether the Mandelbrot set is local connected and what it means if it is.

[aidabit]

If you turn on subtitles @5:05 to 5:06 you see "[evil giggle]" lol

[henrikwannheden]

This is perhaps the most enlightened description of what the Mandelbrot set is that I've ever heard, and I've been listening to explanations for at least 25 years. Very good!

[axianerve]

You know I suck ass at any kind of math, but for some reason I love watching these vids.

[TheJuan]

why I didn't have a teacher like her ?

[KarlFFF]

for once a club that accepts zeros (7:40)