OK, I was expecting some sort of marketing BS at the start, but ... it's geniunely providing a lot more information than the "binary", black-and-whte conventional chart does.
Seems like this is one way of visualizing the solutions to many closely related equations simultaneously. I wonder what the graph looks like if instead coloring based on error, one composited all the solutions within a range of values of of the coefficients.
This is brilliant and oddly obvious in hindsight. Measured valuable almost always have noise, and equations rarely solve to true zero. Setting a small delta is common practice, but these graphs show that some equations may have odd behaviour when you do that.
Neat, but I think it's deceptive for the website to claim this is a "new type of graphing" [1]. The fuzzy graph of F(x, y) = 0 is simply a 3D plot of z = |F(x, y)|, where z is displayed using color. In other words, F(x, y) is a constraint and z shows us how strongly the constraint is violated. Then the graph given by F(x, y) = 0 is a slice of the 3D graph. If you're claiming that you've discovered visualizing 3D graphs using color, you're about 50 years too late.
Is it possible to run this in a chaotic function? I would be interested to see what patterns emerge. I haven't found any code or model to generate this.
My first thought was "how can i do this in 3d and walk around it in VR?"
I can do the VR part - any chance you can share the algo, so I can get the machine to lift it? I can imagine a 3d graphing tool would need spatialisation in order to be properly appreciated.
It took me a second to figure out what these are showing because I usually fit plots to data and the “low error” areas are the areas where, if there was a datapoint, it would be in an area where there would be a wide confidence interval, ie low confidence and more likely to be high error in the model.
The dark areas in the plot seem to be the features driving the shape of the plots. That means that these would be the areas the plotter should be most sure about, otherwise the plot would have a different shape. The bright “low error” areas are the areas where the model seems least likely to be correct.
I might be missing an interpretation that makes much more sense, but I think “error” might be the wrong terminology to use here. It doesn’t just mean “difference between A and B”, it includes some idea of being a measure of wrongness.
Shameless plug: eight years ago, I created the following website for posting plots of complex functions using similar gradients: https://kettenreihen.wordpress.com/
From school you are used to think of function in their explicit form y = f(x) but you can easily turn that into the implicit form f(x) - y = 0 or more generally f(x, y) = 0. With that you can plot the graph of f(x, y) either as a 3D surface with f(x, y) being the height at point (x, y) or encode the function value at (x, y) into some color at (x, y). Where that surface is equal to zero, i.e. where it intersects the z = 0 plane, that are the points of y = f(x). Points (x, y) at which the value of f(x, y) has small non-zero magnitude are what the article calls low error points or regions, points or regions that almost satisfy y = f(x).
Reading it twice and sitting on it, I have an uneasy feeling.
It feels like it distracts more than it illuminates. ex. Quasar Equation. I don't know what it capital-M Means that at (X, Y) = 0, there's a region where there's higher differences between y and x/x^2+y^2.
But counterpoint to myself:
I'm looking at a toy example.
I'm sure there's been plenty of times I was genuinely comparing two equations and needed to understand where there'd differ.
Its just harder for me to grok when one of the equations is "y".
This is cool to look at, but isn't this just obtained by taking the absolute value of the first equation minus the second? These are very pretty visualizations—but trying to present them as some kind of "sea change" in perspective feels unhelpful.
Computers waste a ton of time being perfect when good enough would work just as well. If we get better at mapping what mostly right means, we can make more software faster by trading exactness for speed. You see this kind of thing in quantized LLMs and jpeg compression.
It's the heat map of the error surface of the equation... Fairly well understood as a concept in the land of optimization and gradient descent.
Interesting, what's being visualized there is actually a failure mode for an unidentifiable equation - the valley where the error is zero and therefore all solutions are acceptable. Introduce noise into the measurements of error and that valley being too flat causes odd behaviour
Ouch, this hurts to read. It's not novel and lacks a very basic understanding of math.
The graph of y/(x^2+y^2)=(x+1)/(x^2+y^2) by definition contains the points that satisfy this equation. This is exactly the set of points for which y = x + 1.
The "fuzzy" graph is just coloring the difference between the left hand side and right hand side. This is very basic, not new, and it's definitely not "the graph of y/(x^2+y^2)=(x+1)/(x^2+y^2)".
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[ 3.9 ms ] story [ 52.1 ms ] threadI'm impressed.
[1] https://gods.art/fuzzy_graphs.html
https://web.archive.org/web/20120208174423/https://maa.org/p...
https://observablehq.com/@rreusser/complex-function-plotter
I can do the VR part - any chance you can share the algo, so I can get the machine to lift it? I can imagine a 3d graphing tool would need spatialisation in order to be properly appreciated.
But I am curious if these plots can/have been empirically validated with real world data.
The dark areas in the plot seem to be the features driving the shape of the plots. That means that these would be the areas the plotter should be most sure about, otherwise the plot would have a different shape. The bright “low error” areas are the areas where the model seems least likely to be correct.
I might be missing an interpretation that makes much more sense, but I think “error” might be the wrong terminology to use here. It doesn’t just mean “difference between A and B”, it includes some idea of being a measure of wrongness.
It feels like it distracts more than it illuminates. ex. Quasar Equation. I don't know what it capital-M Means that at (X, Y) = 0, there's a region where there's higher differences between y and x/x^2+y^2.
But counterpoint to myself:
I'm looking at a toy example.
I'm sure there's been plenty of times I was genuinely comparing two equations and needed to understand where there'd differ.
Its just harder for me to grok when one of the equations is "y".
Interesting, what's being visualized there is actually a failure mode for an unidentifiable equation - the valley where the error is zero and therefore all solutions are acceptable. Introduce noise into the measurements of error and that valley being too flat causes odd behaviour
I feel like this must be missing a "no", but also I'm bad at math, so maybe not.
The graph of y/(x^2+y^2)=(x+1)/(x^2+y^2) by definition contains the points that satisfy this equation. This is exactly the set of points for which y = x + 1.
The "fuzzy" graph is just coloring the difference between the left hand side and right hand side. This is very basic, not new, and it's definitely not "the graph of y/(x^2+y^2)=(x+1)/(x^2+y^2)".