[ethanlipson]

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

[amavect]

I thought the same, so I programmed the examples into Desmos 3D (click the show/hide buttons on the left).

https://www.desmos.com/3d/3divdux6jh

Dropping the absolute value makes a better visualization. The 3D graph for Example 4 Shadow Line has an established name, a hyperbolic paraboloid. The color graph for Example 5 Phi Equation doesn't capture the odd symmetry F(x,y)=-F(-x,y). The color graph for Example 6 Underwater Islands looks far inferior to the 3D surface.

[calebm]

Interesting. I didn't know you could plot like this in Desmos. Thanks for doing the work to plug them in.

[amavect]

No problem. While fuzzy graphing has existed for a long time as contour plots, I'd still like to encourage you to experiment more. The colors look pretty. :)

I have a suggestion. Calculate z=left-right without absolute value, and try mapping [negative limit, 0, positive limit] to [black, red, white], or possibly [negative limit, 0, positive limit, outside of limits] to [red, white, blue, black]. And of course, do variations and have fun!

[calebm]

I'm not sure I've tried that - might good idea. I've actually been playing with all kind so of visualizations - I've done most of my art as stand-alone Python scripts. Here's an open-source starting point for doing this kind of math art in Python: https://github.com/calebmadrigal/truthygraph.py. You might want to try your idea yourself :)

[calebm]

Thanks for the feedback (you and all the others - both positive and negative). HN is such a great way to present ideas to smart people.

I may have been a little hyperbolic in the opening sentence, and may try to tighten up that language. It's true that I've encountered things like visualizing error gradients (in the ML space) in a non-binary way. And yes, mapping 3D to color (and 3D graphing) is nothing novel.

But, I think that the idea of visualizing equations using the question "How different are both sides?" instead of "Are both sides EXACTLY equal - Yes or No?" is a new way to think about it - I don't know of any other graphing app/calculator (besides https://fuzzygraph.com) that does it.

I think some would say that the "Fuzzy graph" of an equation is not a true graph of that equation because a "visualization transform" is being applied, and that only the conventional/binary graph is a valid graph of the equation. But even conventional graphing apps must apply a "visualization transformation" to the equation - something like: 'black' if Boolean(left(x, y)==right(x, y)) else 'white.

[nyeah]

Yeah, the opening sentence is very easy to disprove. That's going to spark distrust among people who are "more mathy."

Otherwise ... the plot is obviously engaging, based on the discussion here. For people who haven't seen it before, it's a new way to think about it. It may well be a new "plot type" to most people relying on standard plotting software to visualize equations.

Maybe consider saying it's grounded in "more advanced" math than what others have brought to plotting software. That's still impressive, but it can be made pretty much true. You can cite cool math references about "implicit function theorem" or whatever. Stuff that your target audience really weren't going to just find on their own.

[chemotaxis]

> Neat, but I think it's deceptive

I think your comment is insightful, but it's also a terrible choice of words (and something we probably do too often here). I very much doubt that deception was the intent.

Sometimes, someone just reinvents the wheel (or improves on it). And if it serves to teach several thousand people about a new visualization technique, I think that's a net positive.

[pfortuny]

You are right but the author's first claim jumped to my eyes (I practice geometry) and burned them... The author seems to be just plotting the values of the implicit function y-f(x,y)...

So, good for him but some historical perspective is needed when making such sweeping claims.

[calebm]

Noted.

[chaboud]

“For all the history of computational mathematical visualization, graphing equations has been done in binary mode...”

Intentional or not, the linked article opens with a comically untrue statement, that, because it is verifiably false, doesn't even escape as puffery. When I encounter this sort of grandstanding in framing (generally from junior engineers or fresh-from-school product managers), I spell out just how negatively such misstatements harm the point being made.

It's a turn-off for readers, and it's unnecessary.

"You may be used to seeing graphs like..." or "In grade school, we learned to graph like..."

would probably be more useful than dismissal of the history of visualization of implicit functions. Hopefully, next time, the author will be a bit less grandiose.

[ziofill]

Something can be deceptive without malice.

[roywiggins]

People can play with graphing these in 3D and 2D here:

https://c3d.libretexts.org/CalcPlot3D/index.html

https://www.desmos.com/3d

[jvanderbot]

There are some truths though that are worth showing.

1. plotting in this way shows areas that are nearly solutions which is really cool.

2. Not mentioned, but this shows gradient around the solution as well, which helps understand attractor/repulsor a bit intuitively

I mean I have generated these plots before to visualize things like error or sensitivity, but this is clean and very cool. So, credit where credit is due for spreading the idea.

[almogaver07]

I'm wondering if there are topological tools to find the hyperplane of self-intersection from that surface, which is actually the solution of the equation? Or if given a fuzzy graph z=|F(x,y)| we can use differential geometry to find 0=F(x,y)? Does any of the these questions make sense?

[ethanlipson]

For a general function F, finding the points (x, y) with F(x, y) = 0 has no closed-form solution. The entire field of mathematical optimization is largely dedicated to finding solutions to F(x, y) = 0, in one form or another.

When F has a special structure (say, low-order polynomial), we can actually find the exact solutions. More general structure (e.g. convexity, differentiability) doesn't give us the exact solution, but it lets use use clever numerical algorithms to find them. There are techniques we can use when F has little to no structure, known as "black box" methods, and they work particularly well when we have few variables. In the case of "fuzzy graphs", there are only two variables, so this software takes the approach of computing F(x, y) for every pixel on the screen. In general this doesn't work due to the curse of dimensionality, but it creates good visualizations in low dimensions :)

To answer your question directly, yes we can use differential geometry to speed up optimization. As an example, you've probably heard of gradient descent. Preconditioned gradient descent leverages the geometry of the surface to speed up convergence. In the language of differential geometry, if we're optimizing f(x), then x is "contravariant" but grad(f) is "covariant", so technically we can't just add grad(f) to x since they have different types. We first have to multiply grad(f) by a rank-2 tensor (the "preconditioner") that encodes the local curvature of f around x. This technique is used by the Adam optimizer, with the assumption that the preconditioner is diagonal.