In my AP Art class, there was a girl who was just starting to experiment with mixed media. At this point she was still playing around, trying to decide what direction she wanted to go with her portfolio. So one critique day, she brought in an abstract canvas with some rhinestone highlights and painted and real peacock feathers. She loved sparkles and peacock feathers so she thought she’d try introducing them a *little*. And after everyone had given some input, the teacher gave her his advice, VERY roughly paraphrased here:

“So here’s the thing… I do not like this style. These are just elements that do not speak to me personally, but I see that you like them, and you’re doing interesting things with them.

“My biggest critique is, I only merely *dislike* this piece. I want you to make me HATE it. Go crazy with the things that you like. Don’t hold back trying to make it palatable to people like me. Because I am NEVER going to like it. And if the audience does not like it, it should drive them crazy seeing how much YOU love it.”

Her portfolio was chock full of neon colors and glitter and rhinestones and splashes of peacock feathers and it was a delight. Our teacher despised every piece lol, but she got great marks and I think even won some awards. And more importantly, she was happy and proud of the results. Because she didn’t limit herself by trying to appeal to people who were never going to enjoy what she enjoyed.

From the second equation we get that cheeseburger plus fries-squared is five. Subtracting cheeseburger, which is six, from both sides, we get that fries-squared is negative-one. Math fans will know that there are two solutions to this; either fries are the "imaginary unit" 𝒾 or they are its negative, -𝒾. We'll do the rest of the problem with 𝒾, keeping in mind that at the end we should also take the complex conjugates as solutions.

Finally, we have that cup to the power of fries, minus cup, equals three. Replacing fries with 𝒾, and moving a cup to the other side, we get that cup-to-the-𝒾 is equal to cup-plus-three.

Now, the weird part about this is the cup-to-the-i. The problem with this is that complex exponentiation is technically not a thing. That is to say, there is no one function which is mathematically equal to "input-to-the-power-of-𝒾". In fact, there are infinitely many such functions.

Fortunately, due to reasons that take about six pages to explain (trust me I've done it), there is one particular function that many people have agreed is "the most reasonable one". This is not a mathematical notion, but a human preference. Seeing as this question was presumably written by a human, I am comfortable with using this function.

So, what function is this? Well, given a complex number r∠θ written in polar form (if you don't know what that means don't worry), where -π < θ ≤ π, then (r∠θ)^𝒾 = e^(-θ)∠ln(r).

Applying this to our problem a value r∠θ will be a possible solution for cup if e^(-θ)∠ln(r) = r∠θ + 3. Splitting this into real and imaginary parts, we get two equations: e^(-θ) cos(ln(r)) = r cos(θ) + 3 and e^(-θ) sin(ln(r)) = r sin(θ). We can graph these equations on Desmos:

let the value of the glass = z, for z ∈ ℂ:

z^(±i) = 3+z

let z = r·e^(iθ) for r,θ ∈ ℝ, -π < θ ≤ π

∴ z = r·e^i(θ+2πn) for all n ∈ ℤ

∴ (r·e^i(θ+2πn))^(±i) = 3+r·e^i(θ+2πn)

distribute powers (apologies for the use of ∓):

r^(±i)·e^(∓(θ+2πn)) = 3+r·e^i(θ+2πn)

convert to the same base:

e^i(±ln(r))·e^(∓(θ+2πn)) = 3+r·e^i(θ+2πn)

split into real and imaginary components:

re: cos(±ln(r))·e^(∓(θ+2πn)) = 3+r·cos(θ)

im: sin(±ln(r))·e^(∓(θ+2πn)) = r·sin(θ)

What awaits me?

And people like "wtf why" looking out their hotel room window and there's this. This dragon just curled up in the pool chilling, literally, cooling itself down

materialist crystal shop: these crystals will make you feel better because they're a cool part of nature and pretty to look at :)