by ethanlipson
6 subcomments
- 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.
[1] https://gods.art/fuzzy_graphs.html
- 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).
- Shameless plug: eight years ago, I created the following website for posting plots of complex functions using similar gradients: https://kettenreihen.wordpress.com/
by rustybolt
2 subcomments
- 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)".
by virtualbluesky
0 subcomment
- 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
by roywiggins
0 subcomment
- People who like these types of charts will probably also like domain coloring plots of complex functions:
https://web.archive.org/web/20120208174423/https://maa.org/p...
https://observablehq.com/@rreusser/complex-function-plotter
by aDyslecticCrow
1 subcomments
- 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.
- 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.
by fouronnes3
2 subcomments
- Very cool! This is also known as signed distance function in computer graphics, or implicit form equations in maths.
by clickety_clack
1 subcomments
- 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.
by willguest
1 subcomments
- 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.
by arturventura
0 subcomment
- 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.
- Isn't that what mathematicians have always done with "level lines"?
- Love this!
Some years ago I made an online demo for complex domain coloring, which is related to this idea:
https://anematode.github.io/grapheme-math/demo/domain_colori...
- 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.
by CGMthrowaway
2 subcomments
- I don't pretend to understand the method by which the "error == 0 surface" is calculated (do they explain it?).
But I am curious if these plots can/have been empirically validated with real world data.
by refulgentis
0 subcomment
- 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".
- While this perspective has merit, it is hampered by the fact that all of the examples used are polar equations, and the illustrations are therefore unnecessarily dramatic. Given that a Cartesian representation of a polar relationship is always a planar projection of the underlying conic, extremums near valid points are to expected.
It would be more useful to visually demonstrate linear relationships but of course the errors there would not make for such a punchy blog post.
- I was surprised to learn there is a Slashdot equation. :)
by WhyOhWhyQ
1 subcomments
- Does he say how the fuzzification is defined?
by peter_d_sherman
0 subcomment
- I nominate this work for a Fields Medal! (https://en.wikipedia.org/wiki/Fields_Medal)
It's seriously that good!
Also, related to the idea of visualizing old equations in new ways:
YT Video: "Putting Algebraic Curves in Perspective":
https://www.youtube.com/watch?v=XXzhqStLG-4
>"Ever wonder what happens when you combine graphing algebraic curves with drawing in perspective? The result uncovers some beautiful relationships between seemingly different shapes, and all because of what happens when you
include infinity through projective geometry ."
...and the following might be of interest as well:
https://pointatinfinityblog.wordpress.com/2016/04/11/points-...
https://pointatinfinityblog.wordpress.com/
by BriggyDwiggs42
1 subcomments
- I wish the grapher had a radial mode. Would probably produce really cool symmetries.
- > In this case, there is absolutely nothing to show on a conventional graph, as there are actual solutions to this equations.
I feel like this must be missing a "no", but also I'm bad at math, so maybe not.
- Similar to the mapping of complex number plane with z = f(z'), that is, for a point in complex plane, new z is some function of current z for that point.
- Isn't this essentially how many fractals are colored?
- Really beautiful. I bet Ramanujan just “saw” and felt these.
- Ummmm... You're just plotting a function of 2 variables (R^2 → R) as a heat map.
"Note that the Shadow Circle is invisible in the conventional graph. In fact, the conventional graph looks identical to a conventional graph of the x=0 equation (as if the denominator was not there)."
Ummm... Yeah, because the equation x / (x^2 + y^2 - 1) = 0 simplifies to x = 0. Your "fuzzy graph" is actually just a plot of the function z(x, y) = |x / (x^2 + y^2 - 1)|, where z is encoded as a color.
- 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.
I'm impressed.
by keyliejener
0 subcomment
- [dead]
- [flagged]
- 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.
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