These visualizations are cool, but I feel only a little misleading. Any 2D relation can be plotted in 3D, and setting z=0 shows the original line in 2D. The whole notion of fuzzy is really just saying "let's make a heat map of how far z is from 0". That said, I can see how for some people, thinking about the roots of a function in 3D this way _could_ help unlock deeper understanding of where your roots are at than just staring at 3D graph.
Also, the visualisations are to some extent an artifact of how you arrange the equation: rearranging terms between the left and right side won't change the line but it will change these visualisations. The first example illustrates it nicely: the 'hole' only exists because of a redundant division by (x^2 + y^2) on either side of the equation.
My gut tells me that exploring mathematics in this way has very high potential for bringing on future advancements. (1) Simple elements that (2) reveal coherent structures, and were (3) hidden to historic geniuses due to requiring exhaustive computation.
They’re basically just taking equations of the form f(x,y)=k where k=0, and asking “ah but what about when k!=0”?
Indeed, looking at the constant as a variable by looking at the equation f(x,y)=z IS quite useful. (I’m reminded of Feynman favorite trick for solving integrals). That generalization trick is actually useful in MANY contexts. But there’s nothing novel here.
These visualizations are cool, but I feel only a little misleading. Any 2D relation can be plotted in 3D, and setting z=0 shows the original line in 2D. The whole notion of fuzzy is really just saying "let's make a heat map of how far z is from 0". That said, I can see how for some people, thinking about the roots of a function in 3D this way _could_ help unlock deeper understanding of where your roots are at than just staring at 3D graph.
Also, the visualisations are to some extent an artifact of how you arrange the equation: rearranging terms between the left and right side won't change the line but it will change these visualisations. The first example illustrates it nicely: the 'hole' only exists because of a redundant division by (x^2 + y^2) on either side of the equation.
I agree. But it’s cool to explore these things and if it’s the first time someone sees it, that’s nice!
My gut tells me that exploring mathematics in this way has very high potential for bringing on future advancements. (1) Simple elements that (2) reveal coherent structures, and were (3) hidden to historic geniuses due to requiring exhaustive computation.
It’s not so interesting I think.
They’re basically just taking equations of the form f(x,y)=k where k=0, and asking “ah but what about when k!=0”?
Indeed, looking at the constant as a variable by looking at the equation f(x,y)=z IS quite useful. (I’m reminded of Feynman favorite trick for solving integrals). That generalization trick is actually useful in MANY contexts. But there’s nothing novel here.
The same trick is used in f(R) theories, where the Ricci scalar in General Relativity is replaced with a linear function. https://en.wikipedia.org/wiki/F(R)_gravity
Reminds me of TRIZ methods too, trying to find assumptions or constraints and playing with variations of them.
So yeah, nothing novel here.
earlier discussion here: 89 comments, 11 months ago. https://news.ycombinator.com/item?id=45823141
Isn’t the “error in the equation x = 0” just…x?
Yes.
Seems similar to fractals.
That has another name but I forget. Essentially contour plotting in high res.
Do you mean Marching Squares? Or Signed Distance Fields? Both are related, but not exactly the same.
Level sets?