Some hilarious 3D functions

Hello, intenter wanderer, it's me, Lepersonaf again. It's 22:05 here, August 27 of 2025.
As you can guess from the title, i've got something to share with you. On this site i will showcase my new discovery and tell you a small story about how i found it.
Anyway, let's start!

Today i decided to continue messing around with 3D graphics to see what i can find today. And oh boy, i've got some wild results. So, let's begin.

Hyperbolic Sine with a Trick

Even thought i didn't study this function yet, i decided to try to use it. And i've got this funny graph with help of following formule:

$${\displaystyle \begin{array}{c}z=\frac{\sinh \left(x\right)}{y}\end{array}}$$

This was the first graph I found, so you won't see anything special.

I've thought it was interesting, so i kept it. Nothing else to say here. Also, i will use Desmos this time to give you proper view of this graphs, since Copilot can't handle it, and I'm not smart enough to write code on Javascript for this, so yeah. Moving onto next function.

Unusual Sine

As you can guess, most of this functions are going to be trigonometric because they are the ones that give the most unusual results,
since just using x and y with some constants give you obvious results (not always, but usually yes).
Right after hyperbolic sine I've got this graph with following formule,

$${\displaystyle \begin{array}{c}\sin x=yz\end{array}}$$

which is exactly the same as previous, but sine is not hyperbolic anymore. Funny thing is that if you will rearrange it to make it look like previous formule, it will give you a bit different result for some reason.

I have no idea how to name it, so it will be on you. Anyway, let's continue.

First Wave, Sine Wave

Here you can see my first wave function I found (there will be more of them in the future, because I find them very attractive). Here is the formula that defines this pattern:

$${\displaystyle \begin{array}{c}\sin xy=z-1\end{array}}$$

Once again it looks like the previous formule, but it's sill different. Can't say anything else here.

Looks yummy :p

Sine-Cosine Hills

It looks exactly as it sounds: uniting sine and cosine gives you checkerboard-like pattern, but in 3D. Formule is fairly simple:

$${\displaystyle \begin{array}{c}z=\cos x+\sin y\end{array}}$$

Seems like this graph can be modified a lot with parameters, for example adding n before x, y and z will change how many "hills" are drawn. Can't really add anything here.

Just a nice graph.

The First Step into Madness: The Tangent of Three Variables

At one moment i decided to try out tangent to see how it will go, and i got this:

$${\displaystyle \begin{array}{c}\tan \left(xyz\right)=0\end{array}}$$

Looks fine, right? But it's graph.. This is just... I don't know how to describe this.

This function is interesting because the smaller the value of the tangent, the more detailed the graph becomes, while at a value of 25 the graph almost completely disappears.

$${\displaystyle \begin{array}{c}\tan \left(xyz\right)=3\end{array}}$$

$${\displaystyle \begin{array}{c}\tan \left(xyz\right)=6\end{array}}$$

$${\displaystyle \begin{array}{c}\tan \left(xyz\right)=25\end{array}}$$

Now we enter into more absurd functions.

Let's talk about factorials

Yeah, this one mathematical operation which creates insane graphs in 2D plane. But what if we extend this in 3D plot? Then we'll get following equation:

$${\displaystyle \begin{array}{c}x!+y!=z!\end{array}}$$

Since usual factorial is boring, we will use Gamma function instead, so we'll get following graph:

Yeah, looks scary yet fascinating. Moving on.

Wavy Pillars

This one is weird. Well, what else I expected from this type of equation?

$${\displaystyle \begin{array}{c}{x}^{\cos y}=z^{\sin x}\end{array}}$$

Yeah, sometimes i ran out of good ideas... Whatever

Actually this one might have some potential, if you will think properly. But i can't imgaine it, so movin' on.

Trigonometric Hell

"What do you usually do when something works fine? Correct, copy half of it and try to make even better thing by spamming it!"
This is how I was thinking while writing down this equation:

$${\displaystyle \begin{array}{c}{x}^{\sin x}+y^{\cos y}=z^{\tan z}+n^{\cot n}\end{array}}$$

It's parametric in it's basis, so let's assume that n = 5. Them we get this:

Looks funny anyway.

Tangent wtih a Trick

Yeah, i love tangent function a lot, how you guessed? Well, maybe because i tried a lot of different forms of this function to get a proper result. But it didn't really work. But i should admire that it's always forms same pattern of tangentoid over and over.

$${\displaystyle \begin{array}{c}\tan xzn=y-1\end{array}}$$

This function "shoots" very quickly because of the parameter, so we will assume that n = 1, then we will see the following:

If you want to take a closer look at these charts, here is a link to Desmos with all the functions from that site.

Cosine in Tangent

As it turns out, when you subtract tangent from a cosine of two variables, tangent overwrites cosine and creates a tangentoid with surface of cosine.
Too complicated? Yes. Nonsense? Also yes.

$${\displaystyle \begin{array}{c}z=\cos xy-\tan x\end{array}}$$

From this we can get more funny graphs like this in future.

Interesting interaction.

Caramel Falls

Finally, a worthy graph! This one was a lucky pick, not gonna lie.
When i typed it for the first time i got this yummy green color, which just fit the graph, that's why i called it like that.

$${\displaystyle \begin{array}{c}\sin \left(xyz\right)=\pi +x+y+z\end{array}}$$

You can modify it by changing sine to cosine or any other trigonometric function, it will give it's own unique result.

One of my favorites for sure. This pattern just looks amazing.

The Wall

This one is another lucky pick, which gave me a worthy graph. It makes two same patterns, one parallel to z axis and one patallel to x/y axis. And of course, it uses cosine.

It also can be modified by changing cosine to sine, and it will still give you a beautiful graph.

$${\displaystyle \begin{array}{c}z=\frac{\sin \left(xy\right)y}{x}\end{array}}$$

$${\displaystyle \begin{array}{c}z=\frac{\cos \left(xy\right)y}{x}\end{array}}$$

It's so fascinating how a simple trigonometric function can give you so much research potential in functional analysis. At least i think so.

Twisted Cosine

Here i tried to mess around with argument of cosine, and occasionally got this:

$${\displaystyle \begin{array}{c}z=\cos \left(\frac{xy}{\pi }\right)+\frac{\pi x}{y}\end{array}}$$

Overcomplicated as usual, I know. It forms this graph, which reminds me of my very first function with hyperbolic sine:

For me it's really cool when a regular plane gets this wave effect from trigonometric functions,
the graph seems to come to life and becomes more cheerful or something. It's not just me, right?..

Anyway, that's it for now. I hope you liked this functions as much as i did. I'm also thinking about making a homepage, but it's for future. Also, i will surely make a website dedicated to a very special function, which has charmed me to the depths of my heart. So yeah, expect more!
Peace.