8 ms·

The shadows lurking in the equations

308 points by calebm 11mo ago

IAmBroom 11mo ago
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.
d-us-vb 11mo ago
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.
aDyslecticCrow 11mo ago
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.
- willguest 11mo ago
  Taking it a step further, how would simple algorithms behave when viewed in this way? Rather that just the outcome, we could observe a possibility space... Michael Levin has talked about interesting dynamics with the bubble sort algorithm, which is only a few lines of code, that have parallels in biological processes, suggesting there is a more nuanced logic to nature that we are not seeing
  - bigmadshoe 11mo ago
    This sounds a lot like the programs encoded by neural networks.
  - moi2388 11mo ago
    Isn’t that just done in a higher level language, tweaking the algorithm to allow duplicates, and then being surprised there is clustering? I mean, I don’t see why that is special? Correct me if I’m wrong. I like his research and views on biological electric spaces, but this I did not understand.
    - willguest 11mo ago
      the clustering isn't surprising? are you saying that it is an artefact of the higher level representation? special - perhaps not by itself, but when the same strategy is also expressed by single cell organisms, at least intriguing
      - moi2388 11mo ago
        It randomly gives types to cells. Certain cells move left if the left value is bigger. Other cells move right if the right value is smaller. Others randomly move back and forth. I fail to see how it’s surprising you don’t end up with a complete sort, yet still with clusters. That’s exactly what I’d naively expect to happen.
jessep 11mo ago
Really beautiful. I bet Ramanujan just “saw” and felt these.
ethanlipson 11mo ago
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 https://gods.art/fuzzy_graphs.html
- almogaver07 11mo ago
  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 11mo ago
    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.
oulipo2 11mo ago
Isn't that what mathematicians have always done with "level lines"?
- calebm 11mo ago
  I did recently learn of the https://en.wikipedia.org/wiki/Level_set https://en.wikipedia.org/wiki/Level_set concept, and it is a very similar concept.
  - xico 11mo ago
    Those were popular in the 90s for image processing: e.g. https://shape.polymtl.ca/lombaert/levelset/ https://shape.polymtl.ca/lombaert/levelset/
  - andrewflnr 11mo ago
    Dude, it's fine to be learning stuff and even writing about it. But if you're still discovering basic stuff like level sets, then maybe hold off on declaring that you've discovered, after centuries of mathematical development, a completely new form of graphing?
    - ducttapecrown 11mo ago
      It reads more like mysticism than a serious claim of novelty, chill out.
      - andrewflnr 11mo ago
        > a new type of graphing called "fuzzy graphing" > For all the history of computational mathematical visualization, graphing equations has been done in binary mode These are very concrete, non-mystical claims. But do you really think "mysticism" is better here?
        calebm 11mo ago
        To be fair, yes, there are some places where non-binary graphing has been done (like error gradient graphs in AI), but as far as I know, this is the first app where you can type in a basic x/y equation and get a non-binary graph.
arturventura 11mo ago
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.
roywiggins 11mo ago
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/pubs/amm_complements/complex.html https://web.archive.org/web/20120208174423/https://maa.org/p... https://observablehq.com/@rreusser/complex-function-plotter https://observablehq.com/@rreusser/complex-function-plotter
fouronnes3 11mo ago
Very cool! This is also known as signed distance function in computer graphics, or implicit form equations in maths.
- refulgentis 11mo ago
  With a fuzzy graph, what the essay shows, we plug each point into a recipe, the equation, and see how badly it fails. Big failure → dark region (like a bitter taste). Small failure → light region (almost right). It’s just showing the raw mistake you get after plugging in x and y. With a signed distance function, instead of looking at the recipe’s mistake, we measure how far the point is from the perfect curve—like pulling out a ruler and measuring the nearest distance to the “correct” line. It always has units of length and behaves nicely (positive outside, negative inside, zero on the curve). So the fuzzy graph is about “how wrong is the equation here?” A signed distance function is “how far away from the exact solution am I?” They’re related ideas (both start from equations written as something = 0), but they’re not the same thing. Reasonably, I ask myself: "but isn't dark region far distance and light region close distance?" Sitting with that: In the fuzzy graph, we’re coloring by equation error—how badly the equation is satisfied at each point. In a signed distance field, we color by actual geometric distance to the curve. Those two numbers aren’t the same unless the equation is written in a very special way. If you multiply an equation by some huge factor (say, multiply everything by 1,000, or divide by something small), the shape of the solution curve doesn’t change—distance hasn’t changed—but the equation’s error suddenly becomes 1,000 times larger (or smaller). That would completely change the shading in the fuzzy graph while leaving the real distances untouched. Signed distance is measured with a ruler (pure geometry). The fuzzy graph is measuring algebraic error (how close the formula comes to zero). Both can get lighter near the curve and darker away from it, but they’re doing it for different reasons, so they’re not interchangeable.
- mkl 11mo ago
  This is unsigned distance; it's |f(x)-g(x)|, not f(x)-g(x). The traditional line graphs are the graph implicitly defined by the equation.
willguest 11mo ago
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.
- roywiggins 11mo ago
  It's just a matter of subtracting the two functions, taking the absolute value, and putting that number through a color ramp. If you want to see the result in 3D you can subtract the functions and throw that into a 3D graph plotter. Building a 3d surface plotter would be the hard part, but they already exist, eg plug "abs(y/(x^2+y^2) - (x+1)/(x^2+y^2))" in here: https://c3d.libretexts.org/CalcPlot3D/index.htmlT https://c3d.libretexts.org/CalcPlot3D/index.htmlT This viewer also has a "2d" mode that produces a colored 2D plot.
  - willguest 11mo ago
    trouble is, i'm more engineer than mathematician, so while i appreciate that this is an entirely solvable problem, assembling it from scratch would likely mean many errors, and less fun the 3d plot is nice but not what i would call "spatialised", since it's still a flat render, and I'm exactly thinking about the meshing of the thing. i am familiar with delaunay and marching cube strategies, at least enough to get a machine to hook them up to a spatial plotter
    - roywiggins 11mo ago
      Desmos has a nice renderer too: https://www.desmos.com/3d https://www.desmos.com/3d
WhyOhWhyQ 11mo ago
Does he say how the fuzzification is defined?
- calebm 11mo ago
  I need to add more details about that. But it's simply: abs(left-right)^fuzzyValue
CGMthrowaway 11mo ago
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.
- bigmadshoe 11mo ago
  Presumably they just render the absolute error between the lhs and rhs of the equation for every pixel in the plot.
  - calebm 11mo ago
    Yep - it's just |left-right|^fuzzyLevel
  - deleted 11mo ago
    [deleted]
- deleted 11mo ago
  [deleted]
clickety_clack 11mo ago
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.
- calebm 11mo ago
  I've been calling it "error" (the difference between the left and right side of the equation). But if there is a better term to use, I'd like to know it.
  - mkl 11mo ago
    I think "difference" is better, as "error" implies one side is correct and the other is an approximation. Specifically, you're plotting absolute difference z = |f(x, y)-g(x, y)| (or, worse, absolute error), not error.
Aloyjahat 11mo ago
[flagged]
baruchel 11mo ago
Shameless plug: eight years ago, I created the following website for posting plots of complex functions using similar gradients: https://kettenreihen.wordpress.com/ https://kettenreihen.wordpress.com/
- Syntonicles 11mo ago
  Those are really cool to look at. I kept trying to click them to learn more, I wish some of them were mini blog posts to give a little bit of grounding.
- oersted 11mo ago
  Aesthetically this color map is much nicer!
taeric 11mo ago
Isn't this essentially how many fractals are colored?
- calebm 11mo ago
  It is similar, except with a lot of fractals, the numbers being colored represent how many iterations are required to get outside of a set threshold (which indicates divergence).
  - taeric 11mo ago
    Right, I just meant more that you plot more than just the equality to get some visualizations. Is also a common way to visualize game theory stuff, I thought. You want to know where the expected equilibrium is, but you also want to see, essentially, what the strength of getting there is.
danbruc 11mo ago
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).
- abtinf 11mo ago
  > 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) If f(x, y) = 0, wouldn’t using f(x, y) for the height just result in a flat graph?
  - roywiggins 11mo ago
    They're really two different types of equal signs. f(x,y) = x+y might be better written as f(x,y) := x+y where := means "is defined as". Then f(x,y) = 0 is an equation that expands to x+y = 0, or in familiar intro algebra form, y=-x. g(x,y) := 0 really is a flat plane.
    - wholinator2 11mo ago
      I'd seen the := in programming for years but always thought it was basically just =. Thank you for your explanation!
      - roywiggins 11mo ago
        I will say that in programming it's commonly used as assignment, which isn't quite the same thing as definition. golang uses it to declare variables so that's pretty close
  - mitthrowaway2 11mo ago
refulgentis 11mo ago
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".
141205 11mo ago
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.
mmaunder 11mo ago
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.
virtualbluesky 11mo ago
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
keyliejener 11mo ago
[dead]
cognisent 11mo ago
> 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.
rustybolt 11mo ago
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)".
- calebm 11mo ago
  Why would you say it's not a graph of y/(x^2+y^2)=(x+1)/(x^2+y^2)? I would argue that a conventional/binary graph is also not a "pure" representation of the equation, but rather one possible representation - one that runs it through a "left_side == right_side?" boolean filter. In fact, there is no way to visualize an equation with doing something to it.
  - wholinator2 11mo ago
    There's an equal sign in the equation. That means it is true when y = x + 1. There's no filter we're applying, that's literally what the equation says. What you plot is f(x,y) = (y-x-1)/(x^2+y^2). The line plot is when that equals zero, the fuzzinss of it is when it doesnt. But notice that f(x,y)=0 is exactly equivalent to y=x+1. They're exactly the same. Thus, when you're plotting the fuzzy graph it is definitively _not_ a plot of y=x+1, it's a plot of z=(y-x-1)/(x^2+y^2) and those are not the same thing. We'd only need to "apply a filter" to get the line graph if we started with z(x,y), but that's not what you wrote
  - chemotaxis 11mo ago
    I think the parent basically sensed that you're not a trained mathematician and is trying to throw their middle-school math textbook at you. The simplest definition of a "graph of a function" is that it's a representation of the points satisfying some underlying equality. Your plot isn't that. A more conventional name would be a heatmap: a plot of a function that takes two parameters - x and y coordinates - and then assigns a third value (color) to each. I don't think the distinction is all that interesting. They're both function plots.
BriggyDwiggs42 11mo ago
I wish the grapher had a radial mode. Would probably produce really cool symmetries.
- calebm 11mo ago
  It does - it's a hidden feature: https://fuzzygraph.com/?equation=r%3Dcos%282t%29&fuzzyLevel=80&colorMap=plasma&invertColor=true&xCenter=0&yCenter=0&yHeight=6&showAxes=true&minOverride=4.440892098500626e-16&maxOverride=5.1479962891692965 https://fuzzygraph.com/?equation=r%3Dcos%282t%29&fuzzyLevel=...
  - BriggyDwiggs42 11mo ago
    Ooooh thank you!
glitchc 11mo ago
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.
meindnoch 11mo ago
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.
layer8 11mo ago
I was surprised to learn there is a Slashdot equation. :)
anematode 11mo ago
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_coloring/ https://anematode.github.io/grapheme-math/demo/domain_colori...
zkmon 11mo ago
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.
peter_d_sherman 11mo ago
I nominate this work for a Fields Medal! (https://en.wikipedia.org/wiki/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 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-at-infinity-i-projective-geometry/ https://pointatinfinityblog.wordpress.com/2016/04/11/points-... https://pointatinfinityblog.wordpress.com/ https://pointatinfinityblog.wordpress.com/

308 points by calebm 11mo ago

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.

d-us-vb 11mo ago

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.

aDyslecticCrow 11mo ago

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.

willguest 11mo ago
Taking it a step further, how would simple algorithms behave when viewed in this way? Rather that just the outcome, we could observe a possibility space... Michael Levin has talked about interesting dynamics with the bubble sort algorithm, which is only a few lines of code, that have parallels in biological processes, suggesting there is a more nuanced logic to nature that we are not seeing
- bigmadshoe 11mo ago
  This sounds a lot like the programs encoded by neural networks.
- moi2388 11mo ago
  Isn’t that just done in a higher level language, tweaking the algorithm to allow duplicates, and then being surprised there is clustering? I mean, I don’t see why that is special? Correct me if I’m wrong. I like his research and views on biological electric spaces, but this I did not understand.
  - willguest 11mo ago
    the clustering isn't surprising? are you saying that it is an artefact of the higher level representation? special - perhaps not by itself, but when the same strategy is also expressed by single cell organisms, at least intriguing
    - moi2388 11mo ago
      It randomly gives types to cells. Certain cells move left if the left value is bigger. Other cells move right if the right value is smaller. Others randomly move back and forth. I fail to see how it’s surprising you don’t end up with a complete sort, yet still with clusters. That’s exactly what I’d naively expect to happen.

jessep 11mo ago

Really beautiful. I bet Ramanujan just “saw” and felt these.

ethanlipson 11mo ago

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 https://gods.art/fuzzy_graphs.html

almogaver07 11mo ago
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 11mo ago
  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.

oulipo2 11mo ago

Isn't that what mathematicians have always done with "level lines"?

calebm 11mo ago
I did recently learn of the https://en.wikipedia.org/wiki/Level_set https://en.wikipedia.org/wiki/Level_set concept, and it is a very similar concept.
- xico 11mo ago
  Those were popular in the 90s for image processing: e.g. https://shape.polymtl.ca/lombaert/levelset/ https://shape.polymtl.ca/lombaert/levelset/
- andrewflnr 11mo ago
  Dude, it's fine to be learning stuff and even writing about it. But if you're still discovering basic stuff like level sets, then maybe hold off on declaring that you've discovered, after centuries of mathematical development, a completely new form of graphing?
  - ducttapecrown 11mo ago
    It reads more like mysticism than a serious claim of novelty, chill out.
    - andrewflnr 11mo ago
      > a new type of graphing called "fuzzy graphing" > For all the history of computational mathematical visualization, graphing equations has been done in binary mode These are very concrete, non-mystical claims. But do you really think "mysticism" is better here?
      - calebm 11mo ago
        To be fair, yes, there are some places where non-binary graphing has been done (like error gradient graphs in AI), but as far as I know, this is the first app where you can type in a basic x/y equation and get a non-binary graph.

arturventura 11mo ago

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.

roywiggins 11mo ago

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/pubs/amm_complements/complex.html https://web.archive.org/web/20120208174423/https://maa.org/p... https://observablehq.com/@rreusser/complex-function-plotter https://observablehq.com/@rreusser/complex-function-plotter

fouronnes3 11mo ago

Very cool! This is also known as signed distance function in computer graphics, or implicit form equations in maths.

refulgentis 11mo ago
With a fuzzy graph, what the essay shows, we plug each point into a recipe, the equation, and see how badly it fails. Big failure → dark region (like a bitter taste). Small failure → light region (almost right). It’s just showing the raw mistake you get after plugging in x and y. With a signed distance function, instead of looking at the recipe’s mistake, we measure how far the point is from the perfect curve—like pulling out a ruler and measuring the nearest distance to the “correct” line. It always has units of length and behaves nicely (positive outside, negative inside, zero on the curve). So the fuzzy graph is about “how wrong is the equation here?” A signed distance function is “how far away from the exact solution am I?” They’re related ideas (both start from equations written as something = 0), but they’re not the same thing. Reasonably, I ask myself: "but isn't dark region far distance and light region close distance?" Sitting with that: In the fuzzy graph, we’re coloring by equation error—how badly the equation is satisfied at each point. In a signed distance field, we color by actual geometric distance to the curve. Those two numbers aren’t the same unless the equation is written in a very special way. If you multiply an equation by some huge factor (say, multiply everything by 1,000, or divide by something small), the shape of the solution curve doesn’t change—distance hasn’t changed—but the equation’s error suddenly becomes 1,000 times larger (or smaller). That would completely change the shading in the fuzzy graph while leaving the real distances untouched. Signed distance is measured with a ruler (pure geometry). The fuzzy graph is measuring algebraic error (how close the formula comes to zero). Both can get lighter near the curve and darker away from it, but they’re doing it for different reasons, so they’re not interchangeable.
mkl 11mo ago
This is unsigned distance; it's |f(x)-g(x)|, not f(x)-g(x). The traditional line graphs are the graph implicitly defined by the equation.

willguest 11mo ago

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.

roywiggins 11mo ago
It's just a matter of subtracting the two functions, taking the absolute value, and putting that number through a color ramp. If you want to see the result in 3D you can subtract the functions and throw that into a 3D graph plotter. Building a 3d surface plotter would be the hard part, but they already exist, eg plug "abs(y/(x^2+y^2) - (x+1)/(x^2+y^2))" in here: https://c3d.libretexts.org/CalcPlot3D/index.htmlT https://c3d.libretexts.org/CalcPlot3D/index.htmlT This viewer also has a "2d" mode that produces a colored 2D plot.
- willguest 11mo ago
  trouble is, i'm more engineer than mathematician, so while i appreciate that this is an entirely solvable problem, assembling it from scratch would likely mean many errors, and less fun the 3d plot is nice but not what i would call "spatialised", since it's still a flat render, and I'm exactly thinking about the meshing of the thing. i am familiar with delaunay and marching cube strategies, at least enough to get a machine to hook them up to a spatial plotter
  - roywiggins 11mo ago
    Desmos has a nice renderer too: https://www.desmos.com/3d https://www.desmos.com/3d

WhyOhWhyQ 11mo ago

Does he say how the fuzzification is defined?

calebm 11mo ago
I need to add more details about that. But it's simply: abs(left-right)^fuzzyValue

CGMthrowaway 11mo ago

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.

bigmadshoe 11mo ago
Presumably they just render the absolute error between the lhs and rhs of the equation for every pixel in the plot.
- calebm 11mo ago
  Yep - it's just |left-right|^fuzzyLevel
- deleted 11mo ago
  [deleted]
deleted 11mo ago
[deleted]

clickety_clack 11mo ago

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.

calebm 11mo ago
I've been calling it "error" (the difference between the left and right side of the equation). But if there is a better term to use, I'd like to know it.
- mkl 11mo ago
  I think "difference" is better, as "error" implies one side is correct and the other is an approximation. Specifically, you're plotting absolute difference z = |f(x, y)-g(x, y)| (or, worse, absolute error), not error.

Aloyjahat 11mo ago

[flagged]

baruchel 11mo ago

Shameless plug: eight years ago, I created the following website for posting plots of complex functions using similar gradients: https://kettenreihen.wordpress.com/ https://kettenreihen.wordpress.com/

Syntonicles 11mo ago
Those are really cool to look at. I kept trying to click them to learn more, I wish some of them were mini blog posts to give a little bit of grounding.
oersted 11mo ago
Aesthetically this color map is much nicer!

taeric 11mo ago

Isn't this essentially how many fractals are colored?

calebm 11mo ago
It is similar, except with a lot of fractals, the numbers being colored represent how many iterations are required to get outside of a set threshold (which indicates divergence).
- taeric 11mo ago
  Right, I just meant more that you plot more than just the equality to get some visualizations. Is also a common way to visualize game theory stuff, I thought. You want to know where the expected equilibrium is, but you also want to see, essentially, what the strength of getting there is.

danbruc 11mo ago

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).

abtinf 11mo ago
> 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) If f(x, y) = 0, wouldn’t using f(x, y) for the height just result in a flat graph?
- roywiggins 11mo ago
  They're really two different types of equal signs. f(x,y) = x+y might be better written as f(x,y) := x+y where := means "is defined as". Then f(x,y) = 0 is an equation that expands to x+y = 0, or in familiar intro algebra form, y=-x. g(x,y) := 0 really is a flat plane.
  - wholinator2 11mo ago
    I'd seen the := in programming for years but always thought it was basically just =. Thank you for your explanation!
    - roywiggins 11mo ago
      I will say that in programming it's commonly used as assignment, which isn't quite the same thing as definition. golang uses it to declare variables so that's pretty close
- mitthrowaway2 11mo ago

refulgentis 11mo ago

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".

141205 11mo ago

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.

mmaunder 11mo ago

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.

virtualbluesky 11mo ago

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

keyliejener 11mo ago

[dead]

cognisent 11mo ago

> 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.

rustybolt 11mo ago

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)".

calebm 11mo ago
Why would you say it's not a graph of y/(x^2+y^2)=(x+1)/(x^2+y^2)? I would argue that a conventional/binary graph is also not a "pure" representation of the equation, but rather one possible representation - one that runs it through a "left_side == right_side?" boolean filter. In fact, there is no way to visualize an equation with doing something to it.
- wholinator2 11mo ago
  There's an equal sign in the equation. That means it is true when y = x + 1. There's no filter we're applying, that's literally what the equation says. What you plot is f(x,y) = (y-x-1)/(x^2+y^2). The line plot is when that equals zero, the fuzzinss of it is when it doesnt. But notice that f(x,y)=0 is exactly equivalent to y=x+1. They're exactly the same. Thus, when you're plotting the fuzzy graph it is definitively _not_ a plot of y=x+1, it's a plot of z=(y-x-1)/(x^2+y^2) and those are not the same thing. We'd only need to "apply a filter" to get the line graph if we started with z(x,y), but that's not what you wrote
- chemotaxis 11mo ago
  I think the parent basically sensed that you're not a trained mathematician and is trying to throw their middle-school math textbook at you. The simplest definition of a "graph of a function" is that it's a representation of the points satisfying some underlying equality. Your plot isn't that. A more conventional name would be a heatmap: a plot of a function that takes two parameters - x and y coordinates - and then assigns a third value (color) to each. I don't think the distinction is all that interesting. They're both function plots.

BriggyDwiggs42 11mo ago

I wish the grapher had a radial mode. Would probably produce really cool symmetries.

calebm 11mo ago
It does - it's a hidden feature: https://fuzzygraph.com/?equation=r%3Dcos%282t%29&fuzzyLevel=80&colorMap=plasma&invertColor=true&xCenter=0&yCenter=0&yHeight=6&showAxes=true&minOverride=4.440892098500626e-16&maxOverride=5.1479962891692965 https://fuzzygraph.com/?equation=r%3Dcos%282t%29&fuzzyLevel=...
- BriggyDwiggs42 11mo ago
  Ooooh thank you!

glitchc 11mo ago

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.

meindnoch 11mo ago

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.

layer8 11mo ago

I was surprised to learn there is a Slashdot equation. :)

anematode 11mo ago

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_coloring/ https://anematode.github.io/grapheme-math/demo/domain_colori...

zkmon 11mo ago

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.

peter_d_sherman 11mo ago

I nominate this work for a Fields Medal! (https://en.wikipedia.org/wiki/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 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-at-infinity-i-projective-geometry/ https://pointatinfinityblog.wordpress.com/2016/04/11/points-... https://pointatinfinityblog.wordpress.com/ https://pointatinfinityblog.wordpress.com/