New fractal.. needs a name

[Tglad]

Thanks Syntopia, useful info.
Here you can see the surface with height of -0.25 up to 1:


It really explodes out like a volcano when height increases past 1, so I think I'll call it a volcanic surface fractal.
Around 1.75 the fractal occupies all of space and has a Hausdorff dimension of infinity!

[cKleinhuis]

infinite hausdorff dimension ? please elaborate!

[DarkBeam]

Works - managed to disable that ANGLE!

[kram1032]

There's mathematically no way it could have a Hausdorff dimension above 3. - unless you embed the entire thing in higher-dimensional space. - particularly in infinite-dimensional one.
Besides that, it's really nice stuff.

If you somehow calculate the Hausdorff Dimension with some sort of algorithm and you get infinity, that's either because of numerical instabilities or your algorithm is wrong.

[Tglad]

It is because there is overlap (for height > 0.55), so some points in space have double cover, or triple cover etc. You can do the same thing with the Levy C curve, for 90 degree bend angle it has Hausdorff dimension 2 (log(2) / log(sqrt(2))), for bend angle > 90 degree it is log(2)/log( < sqrt(2) ) so the dimension is > 2.
If you don't allow multi-cover of the space then the max Hausdorff dimension is the dimension of the space.

Here's a higher res of above image:

[kram1032]

ah, so multiple-covering counts for adding dimensions. That sort of makes sense. It's like you need more space than you actually have, so you multi-cover to get there.

[cKleinhuis]

i get this, but in my eyes it is not really special, because such curves exists ... basically the trivial curve with no movement ... the dot at the center of the c oordinate system would have as well unlimited hausdorff dimension

this brings us to the space filling curves and why it is such a good name but as far as i know the definition of space filling curves is NOT limited to the topografical dimension !?

[Tglad]

I agree, higher dimension from overlap is a bit meaningless. If we don't allow multi-cover of the space then this fractal goes from 2d and approaches 3d as height goes from 0 to about 1.75. We'd also have to accept that Levy C curve has Hausdorff dimension a little less than 2.

Here are the first 3 iterations if you hadn't guessed:

[kram1032]

that iteration looks pretty wild. nice.

[Tglad]

Here's a variation, you reverse the face normal of the 6 sub-triangles, it gives a shape that looks the same from both sides of the surface:



So is a bit like how the Koch curve differs from the Levy C curve.
The pics are the largest height without self intersection, about 0.35.

Here's a challenge that is open to everyone... 
ready?...

is there a fractal like this that doesn't self-intersect but folds up into a space-filling surface like the Von Koch curve does?

[Tglad]

There's an interesting critical point, when the height is 0.55209 then all the triangles are isosceles. This is also about the point where the shape is at its most rough without self-intersecting. I'm not sure if those two are exactly the same point but they could well be.

[kram1032]

it's pretty likely that they are.
Would be a strange rare math stunt if they weren't.