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[–][S] 166 points167 points  (11 children)

This is the “modular surface” of z²+1. Modular surfaces are a great way to visualise complex functions. The base is the input, which is the complex plane. The height is the absolute value of the output. The colour represents the phase. The points where the function is equal to zero is at the bottom. z²+1 has two imaginary solutions at I and -i. This render was made for this little video: https://www.youtube.com/watch?v=nT3WYFxvPLk

[–] 24 points25 points  (8 children)

What does phase mean? Whats the definition?

[–][S] 51 points52 points  (0 children)

Sorry that wasn't clear, phase is more of a physics term I guess. The colour is arg(z) (the polar angle). If you need more info, I explain the graphic in the 2nd part of this video: https://www.youtube.com/watch?v=jU7QW6AjUf4

[–] 25 points26 points  (0 children)

Instead of complex plus imaginary coordinates which would be cardinal coordinates, you can represent a complex number through polar coordinates, using an angle and magnitude. The phase would refer to the angle of the polar coordinate, i.e. if you think about the angles as a cycle, the phase would be where you are along that cycle.

[–] 7 points8 points  (5 children)

Phase or phases may refer to:

== Science ==

=== Physics === State of matter, or phase, one of the distinct forms in which matter can exist Phase (matter), a region of space throughout which all physical properties are essentially uniform Phase space, a mathematical space in which each possible state of a physical system is represented by a point — this equilibrium point is also referred to as a "microscopic state" Phase space formulation, a formulation of quantum mechanics in phase space Phase (waves), the position of a point in time (an instant) on a waveform cycle Instantaneous phase, generalization for both cyclic and non-cyclic phenomena Polyphase system, a means of distributing alternating current electric power in multiple conducting wires with definite phase offsets Single-phase electric power, distribution of AC electric power in a system where the voltages of the supply vary in unison Three-phase electric power, a common method of AC electric power generation, transmission, and distribution Three-phase, the mathematics of three-phase electric power Phase problem, the loss of information (the phase) from a physical measurement Phase factor, a complex scalar used in quantum mechanics in Continuous Fourier transform, the angle of a complex coefficient representing the phase of one sinusoidal component

=== Other sciences === Archaeological phase, a discrete period of occupation at an archaeological site Color phase, in biology, a group of individuals within a species with a particular coloration Gametic phase, in genetics, the relationship between alleles at two chromosomal loci Lunar phase, the appearance of the Moon as viewed from the Earth Planetary phase, the appearance of the illuminated section of a planet Phase separation, in physical chemistry, the separation of a liquid mixture into two immiscible liquids above and below a meniscus Phase (syntax), in linguistics, a cyclic domain (proposed by Noam Chomsky) Development of the human body, in cognitive psychology, occurs in 9 phases by age In biology, a part of the cell cycle in which cells divide (mitosis) and/or reproduce (meiosis)

== Entertainment == Phase (band), a Greek alternative rock band Phases (band), an indie pop American band, formerly known as JJAMZ Phase (album), a debut studio album by an English singer Jack Garratt Phases (The Who album), a box set of albums by The Who Phases (I See Stars album), 2015 Phases (Angel Olsen album) "Phases" (Buffy the Vampire Slayer), a 1998 episode of the TV series Buffy the Vampire Slayer Phases (.hack), fictional monsters in the .hack franchise Phase IV, a 1974 science fiction movie directed by Saul Bass Phase, an incarnation of the DC Comics character usually known as Phantom Girl Phaze, an alternate world in Piers Anthony's Apprentice Adept series

== Other uses == Phase 10, a card game created by Fundex Games Phase (video game), a 2007 music game for the iPod developed by Harmonix Music Systems Phase (combat), usually a period of combat within a larger military operation A musical composition using Steve Reich's phasing technique

== See also == FASOR (disambiguation) Faze (disambiguation) Phase 1 (disambiguation) Phase 2 (disambiguation) Phase 3 (disambiguation) Phase 4 (disambiguation) Phase 5 (disambiguation) Phase space (disambiguation) Phaser (disambiguation) Phasing (disambiguation) Phasor (disambiguation)

More details here: https://en.wikipedia.org/wiki/Phase

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Good bot

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More than good, that bot was truely beautiful!

[–] -1 points0 points  (1 child)

will somebody pls explain this to a real analysis nerd who has never touched complex analysis bc i am very lost lmao

[–] 0 points1 point  (0 children)

What would you like explained? Fellow real analysis nerd

[–] 54 points55 points  (8 children)

Was this made with Blender? I wanted to start making some visualizations like this. What was you process?

Edit: grammar

[–][S] 55 points56 points  (7 children)

Yes Blender. I build it using python code within Blender.

[–]Commutative Algebra 36 points37 points  (2 children)

Do you mind sharing your code!

[–] 1 point2 points  (1 child)

RemindMe! 1 Week "sweet, sweet blender graphs"

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I will be messaging you in 7 days on 2020-12-09 21:08:12 UTC to remind you of this link

1 OTHERS CLICKED THIS LINK to send a PM to also be reminded and to reduce spam.

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[–]Number Theory 5 points6 points  (0 children)

That's awesome! You are making me want to implement this on shadertoy!

[–] 8 points9 points  (0 children)

Sweet! I think graphs will be my next project in blender. Thanks for the inspiration!

[–]Differential Geometry 0 points1 point  (1 child)

What are your computer specs like? This looks like a fun thing to play around with but I worry it'd take hours on my laptop

[–][S] 0 points1 point  (0 children)

Umm, I actually have one of the fastest AMD CPUs on the market. It takes around 2min to render a 4k frame. That said, I do play around on my basic Surface Pro 4 laptop, if you are happy to make only still images it would be fine. A surface that is opaque, not glass, is also less computationally expensive.

(A graphics card in a modest desktop would also give you reasonable render times)

[–] 35 points36 points  (0 children)

Mathematica RTX On

[–] 38 points39 points  (1 child)

I think there should really be more visuals on this subreddit

[–] 12 points13 points  (0 children)

If I had tools like this when I was first learning math I like to think I would be a lot better at it.

[–] 12 points13 points  (3 children)

It looks great. It's very attractive, and makes a clear easy to read shape.

The phase is tricky though... I mean, even if there was a key to which colours meant what, it's not always easy to tell what colour we're looking at - because it is so shiny and reflective. The colours seem to shift a bit as it rotates. I guess if we were trying to seriously read the values from this, it might be better if it was less reflective... but then it might not look as cool. :p

[–] 7 points8 points  (1 child)

I agree, it's great and the phase coloring could be improved. Two ideas:

1. Phase-color the underlying plane too? That could serve as a kind of color key.

2. Use a perceptually uniform cyclic color map.

[–] 1 point2 points  (0 children)

That colour map information was very interesting. Thanks for that.

[–][S] 4 points5 points  (0 children)

Yes, I used a glass shader because it looks cool. To visualise values, I use a non-transparent shader with labelled contour lines overlayed as a texture. In Blender the glass is ok though, because the image is clear enough top-down when you are not looking through more than piece of coloured glass.

[–] 10 points11 points  (0 children)

This is beautiful

[–] 18 points19 points  (4 children)

3b1b did a video about this but he used a 2d graph with brightness indicating the absolute value, so the zero of a function would be a black spot.

[–] 7 points8 points  (3 children)

Welch labs did similar video series about imaginary numbers. Climax is with this kind of diagram.

[–] 7 points8 points  (2 children)

That series of videos made me climax, myself.

[–] 2 points3 points  (1 child)

Completely understandable. Imaginary Numbers are known for the beauty.

[–] 1 point2 points  (0 children)

As an electrical engineer I get to work with imaginary numbers all day!

[–] 8 points9 points  (1 child)

Since complex functions are transformations of the plane, my favorite way to visualize them is a warped 2D grid.

[–] 0 points1 point  (0 children)

Hm. I wonder if this has something to do with why the gridlines on a nichols plot are all warpy.

[–] 5 points6 points  (1 child)

Of course visualizations like this look nice, but never did I have the feeling of really getting more insight on the behavior of (holomorphic) complex functions by inspecting them.

[–] 2 points3 points  (0 children)

Right, I'm looking at this and admiring the aesthetics, but... now what? What am I supposed to have learned? What mathematics can be done because of this visualization? What are other people getting from this, that I'm missing?

[–] 4 points5 points  (0 children)

wow this is impressive, i know a little math a little python and a little blender too, lets see what we can do

[–] 2 points3 points  (1 child)

I always found these colour plots to be confusing. What angle is red for example? Or how what is the angle between red and green? It doesn't seem intuitive.

[–][S] 0 points1 point  (0 children)

You're right, you need a legend to read the colour. This is explained in the linked video series. Red it the positive real axis. Cyan is the negative real axis.

[–] 2 points3 points  (0 children)

I love this! As an electrical engineer it annoys me when people try to display real and imaginary parts separately. It has its uses, but mag and phase are so much more intuitive for telling me generally where it goes in a particular direction!

[–]Undergraduate 2 points3 points  (0 children)

I absolutely love posts that beautify math like this

[–] 1 point2 points  (0 children)

Damn that glassy, see-through look. Looks pretty nice.

[–] 1 point2 points  (0 children)

Seems related to the approach to complex functions in Wegert's book.

[–] 1 point2 points  (0 children)

This deserves an award but i have none

[–]Math Education 1 point2 points  (0 children)

For those that want some more visual representations of complex functions, here's a plotter (albeit a 2D one) that lets you graph an arbitrary complex function:

http://davidbau.com/conformal/#z

[–] 1 point2 points  (1 child)

And what does that even mean?

[–] 0 points1 point  (0 children)

It's just a graph of a function.

The same as y=mx+b.

Except linear functions like the one above have a single input and a single output. So they're really easy to graph in 2D space. Input gets a dimension and the output gets a dimension.

Complex valued functions have two inputs and two outputs. That's because complex numbers are two dimensional. They have a real part and an imaginary part, typically represented by the x and y axes, respectively. Alternately you can think of them as having a magnitude and phase.

[–] 0 points1 point  (1 child)

[–] 0 points1 point  (0 children)

[–] 0 points1 point  (0 children)

Damn, thats sick

[–] 0 points1 point  (0 children)

That's a shiny function

[–][🍰] 0 points1 point  (3 children)

Are you a math major/professor smth? Like how did you even visualise that. I am asking because these 3d graphs arent readily available on the internet.

[–] 0 points1 point  (2 children)

Computers can visualize it for you, very easily.

I could make this exact graph in maybe 10 or 15 minutes. But I wouldn't have the beautiful movie to visualize it from different angles.

[–][🍰] 0 points1 point  (1 child)

Wow, can't wait to be this good.

[–] 1 point2 points  (0 children)

Aw. I'm nothing special. The computer is.

There are a couple really cool computing programs. The one I use the most these days is matlab. It has a ton of built in functions that make stuff like this super easy.

But ya, I think looking at this made me realize how far I've come in math. It's kind of cool! Holy cow I remember struggling with logarithms... So cool.

[–] 0 points1 point  (0 children)

Impressive, very nice.

[–] 0 points1 point  (4 children)

Asking as some who doesnt know math. Can you make any 3d shaoe using a function? Or to what degree of advanced can it go?

[–] 1 point2 points  (3 children)

Yep. The only rule is the vertical line rule. So if you can imagine a surface of some sort, as long as it passes the vertical line test, there is some function that describes it.

If it doesn't, then you have to start breaking it up into sections and trying to come up with functions for each one.

[–] 0 points1 point  (2 children)

Så technically, you could write a function that would look like a bike?

[–] 1 point2 points  (1 child)

Nope. That wouldn't pass the vertical line test.

The vertical line test means that if you ha a vertical line going through the shape, it would only pass through once.

There are lots of places where a line could be drawn that would pass through the bike in more than one place.

[–] 0 points1 point  (0 children)

Oh okay! thanks for clarifying. Very helpfull.

[–] 0 points1 point  (0 children)

that is patrick

[–] 0 points1 point  (0 children)

I ask my digital signal processing students to construct surfaces in the complex frequency plane. Just polynomial addition over a complex plane usually centered about the origin. It provides significant insights regarding for instance the behaviors of complex poles and zeros.

[–] 0 points1 point  (0 children)

Could you by any chance use this software to visualize the propagation of an electromagnetic wave using Maxwell's wave equation?

[–] 0 points1 point  (0 children)

squints at my calc books on the shelf

Welp.... time for a refresher...

[–] 0 points1 point  (0 children)

Pants!

[–] 0 points1 point  (0 children)

looks like the lower half of patrick lol

[–] 0 points1 point  (0 children)

Math is beautiful.

[–] 0 points1 point  (0 children)

I'm looking at this and admiring the aesthetics, but... now what? What am I supposed to have learned? What mathematics can be done because of this visualization? What are other people getting from this, that I'm missing?

[–] 0 points1 point  (1 child)

Can you make s domain graph of a laplace transform? That will be cool

[–][S] 0 points1 point  (0 children)

One day yes. I plan to cover the Laplace transforms at some stage on the channel.

[–]Number Theory 0 points1 point  (1 child)

This is very appealing.

I'd like to talk to you about producing a similar visualization for modular forms. These are inherently "just complex functions", but they're nontrivial to compute. But in two recent papers, I study how to compute modular forms and various visualizations of modular forms.

I'm knowledgeable about various 2d plotting, but I don't actually know anything about 3d plotting. I'm aware that blender exists and that shaders exist, but that's the extent of my knowledge. This is a major aspect of complex function visualization that I'm missing.

But I suspect the process is first to compute the intended complex function on some grid/mesh, to somehow insert this computation/mesh into blender (the step which I know the least about), and then to color the surface (maybe with a shader). Individually, these points don't seem too bad... but of course I don't actually know what I'm talking about.

Is this something that you would be interested in talking about more? If so either PM me or let me know and I'll PM you.

Cheers!

[–][S] 0 points1 point  (0 children)

Hi there...

Yes, I'm interested to discuss this. Please reach out via PM or email. (email is better) mathstownchannel (at) gmail (dot) com.

Simply put, if you can give me a Python function that returns a complex value, I can build it. eg...

def calc(z):
z =  cmath.cos(z)
return z