[HN Gopher] The shadows lurking in the equations
___________________________________________________________________
The shadows lurking in the equations
Author : calebm
Score : 239 points
Date : 2025-11-05 14:21 UTC (8 hours ago)
(HTM) web link (gods.art)
(TXT) w3m dump (gods.art)
| IAmBroom wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| This sounds a lot like the programs encoded by neural
| networks.
| moi2388 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| Really beautiful. I bet Ramanujan just "saw" and felt these.
| ethanlipson wrote:
| 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
| almogaver07 wrote:
| 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 wrote:
| 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.
| roywiggins wrote:
| People can play with graphing these in 3D and 2D here:
|
| https://c3d.libretexts.org/CalcPlot3D/index.html
|
| https://www.desmos.com/3d
| amavect wrote:
| 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 wrote:
| Interesting. I didn't know you could plot like this in
| Desmos. Thanks for doing the work to plug them in.
| amavect wrote:
| 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 wrote:
| 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 :)
| chemotaxis wrote:
| > 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.
| chaboud wrote:
| "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.
| pfortuny wrote:
| 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 wrote:
| Noted.
| jvanderbot wrote:
| 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.
| calebm wrote:
| 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 wrote:
| 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.
| oulipo2 wrote:
| Isn't that what mathematicians have always done with "level
| lines"?
| calebm wrote:
| I did recently learn of the
| https://en.wikipedia.org/wiki/Level_set concept, and it is a
| very similar concept.
| xico wrote:
| Those were popular in the 90s for image processing: e.g.
| https://shape.polymtl.ca/lombaert/levelset/
| andrewflnr wrote:
| 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 wrote:
| It reads more like mysticism than a serious claim of
| novelty, chill out.
| andrewflnr wrote:
| > 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 wrote:
| 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.
| munificent wrote:
| The first app that lets you type in an implicit curve and
| get a graph of its level set is a very different claim
| from "For all the history of computational mathematical
| visualization, graphing equations has been done in binary
| mode".
|
| The millions of brightly-colored fractal posters adorning
| walls in the 80s are a very clear counter-example to your
| claim.
|
| Your app is cool and the visualization is neat. The
| hyperbolic claims of originality really detract from
| that.
| spongebobstoes wrote:
| the author uses hyperbole, yeah. they're an artist, I think
| it's expected. I don't find anything personally offensive
| in the exaggerated framing of this art
|
| I think your critique would be more effective if you left
| out the part where you shame the author's lack of knowledge
| about level sets
|
| I think it hides your valid (if overstated) criticism of
| the author's exaggeration behind a non-constructive insult,
| and your comment would be better without that tone
| andrewflnr wrote:
| > Dude, it's fine to be learning stuff and even writing
| about it.
|
| The point about level sets is entirely that the author
| not only does not have the background to make claims
| about novelty, _they have all the clues they need_ to
| figure out that they don 't have the background.
| semi-extrinsic wrote:
| It's also what scientists have done to visualize solutions of
| PDEs since the 1960s. Author should download Paraview and give
| it a twirl, to get this perspective.
|
| First create a mesh (Sources -> Plane for 2D, or Sources -> Box
| if you want to do it in 3D). Set reasonably high values for
| Resolution on this source. Then use a filter to apply your
| function, either Filters -> Alphabetical -> Calculator for easy
| stuff, or Filters -> Alphabetical -> Python Calculator if you
| want complicated stuff. The "coordsX" etc. are your spatial
| coordinates on the mesh. Pick whatever color map you want
| (diverging types are good for this), change the limits on
| coloring, use a log scale, whatever.
|
| If you do this in 3D on a box, you can then use a slice to
| scrub through the result on an arbitrarily oriented plane. You
| could visualize translucent isosurfaces of constant "error" and
| raytrace them. Or you could take the gradient of your "error"
| and plot as a vector field. With a bit of leg work you can add
| a fourth coordinate (time) and make animations. And you can
| combine all of these. Sky is the limit.
| arturventura wrote:
| 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 wrote:
| 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
| fouronnes3 wrote:
| Very cool! This is also known as signed distance function in
| computer graphics, or implicit form equations in maths.
| refulgentis wrote:
| 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.
| willguest wrote:
| 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 wrote:
| 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
|
| This viewer also has a "2d" mode that produces a colored 2D
| plot.
| willguest wrote:
| 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 wrote:
| Desmos has a nice renderer too:
|
| https://www.desmos.com/3d
| WhyOhWhyQ wrote:
| Does he say how the fuzzification is defined?
| calebm wrote:
| I need to add more details about that. But it's simply:
| abs(left-right)^fuzzyValue
| CGMthrowaway wrote:
| 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 wrote:
| Presumably they just render the absolute error between the lhs
| and rhs of the equation for every pixel in the plot.
| calebm wrote:
| Yep - it's just |left-right|^fuzzyLevel
| clickety_clack wrote:
| 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 wrote:
| 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.
| baruchel wrote:
| Shameless plug: eight years ago, I created the following website
| for posting plots of complex functions using similar gradients:
| https://kettenreihen.wordpress.com/
| Syntonicles wrote:
| 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.
| taeric wrote:
| Isn't this essentially how many fractals are colored?
| calebm wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| > 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 wrote:
| 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 wrote:
| I'd seen the := in programming for years but always thought
| it was basically just =. Thank you for your explanation!
| roywiggins wrote:
| 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 wrote:
| When we say "f(x, y) = 0" in this context, we also usually
| have a separate definition for f(x, y) provided, where that
| f(x, y) is not necessarily 0 at for all x,y. And so this
| constraint "f(x, y) = 0" means "find pairs of x and y such
| that it makes f(x, y) become 0".
|
| If "f(x, y) = 0" is actually the definition of f(x, y), then
| yes, it would be a pretty boring graph.
| layer8 wrote:
| _f_ ( _x_ , _y_ ) = 0 is true only for _some_ combinations of
| _x_ and _y_. It's an equation to be solved, not a universal
| statement like [?] _x_ , _y_ : _f_ ( _x_ , _y_ ) = 0, nor a
| definition like _f_ ( _x_ , _y_ ) [?] 0 (or "[?]"). The
| solutions to the equation are the points ( _x_ , _y_ ) where
| the graph has height 0. Which points these are depends on how
| _f_ is defined.
|
| For example, _f_ might be defined as _f_ ( _x_ , _y_ ) [?]
| _x_ 2 + _y_ 2 - 1. Then the points ( _x_ , _y_ ) for which
| _f_ ( _x_ , _y_ ) = 0 are those on the unit circle (those for
| which _x_ 2 + _y_ 2 = 1). The graph will have height 0 only
| for those points.
| refulgentis wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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
| cognisent wrote:
| > 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| andrewflnr wrote:
| The "graph" of a function is formally defined as exactly
| those points that make the equation true.
| https://en.wikipedia.org/wiki/Graph_of_a_function Granted,
| the graph is only one _visualization_ of a function, and not
| the only valuable one.
|
| Of course we also have to remember that functions are not the
| same as equations, and a given function, or more generally
| relation, can be represented by multiple different equations.
| For a trivial example, multiply both sides of your slashdot
| equation by a constant, or add x*y.
| soVeryTired wrote:
| It confused me a lot - I cancelled the denominators in my head
| too.
|
| But then I realised they're just plotting
|
| y/(x^2+y^2) - (x+1)/(x^2+y^2) = c
|
| and colouring by c (i.e. a heatmap, as others have mentioned in
| the thread).
|
| That's why you get a more interesting image than you'd get with
| y - (x + 1) = c
| BriggyDwiggs42 wrote:
| I wish the grapher had a radial mode. Would probably produce
| really cool symmetries.
| glitchc wrote:
| 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 wrote:
| 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 wrote:
| I was surprised to learn there is a Slashdot equation. :)
| anematode wrote:
| 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...
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