# sine function

The sea.

Each color is a sine function. The height is constant because sin(t+0*2*pi/5 )+ sin(t+1*2*pi/5)+sin(t+2*2*pi/5)+sin(t+3*2*pi/5)+sin(t+4*2*pi/5)=0 for all t value.

In a column the i-th segment have length sin(t+i*2*pi/5)+1, so the height is always 5.

I write sines not tragedies

the outline of the wave starts and ends in the middle meaning its a sin wave, and when I checked the pixels I found that roughly the amplitude of the wave is 132 pixels while the period is 2162 pixels. you can do the math and check the coding in the game if you really want to find the formula with the speed and number of oscillations. Of course me being the massive nerd I am, completely lost it when I saw a sin wave as an attack. You’re gonna have a bad amplitude vs time graph. If anyone is a math wiz and wants to figure out the frequency, distance, etcetera and graph it or convert it to sound be my guest.
credit goes to NoHitRuns on Youtube for the video capture of the attack.

Does Johnson the Metaphysical Goalie have any words of advice for young tadpole Chowder?

heeeeey, Johnson here hahaha

tho like i should totally say that i haven’t graduated yet lol. it’s like still March or somethin in my completely fictional reality. (well, not like your reality’s ne realer than mine tbh hahaha—when u think abt it. woooaaaaah!!!)

anyway, chow-chow. chowds.

bro u gotta remember that ur gonna have good days in the net and bad days. the good days all the boys are gonna love u, and on the bad days…well, theyll still love u—(#1 rule of SamwellHockey: u always got ur bros back), but like, YOU’RE the one who s gonna feel frustrated & it totally sucks.

but bro lifes a fucking sine function describing the oscillations of our existence. pucks are gonna go in. just forget them. think of the puck like its a grain of sand. let it disappear into the waves of ur ocean if that makes sense.

the forwards have their line, the dmen have their partners, but you—-u bro, u got the poles and ur resilience.

being a goalie’s hard. so go with the flow. :-)

so there you go New England Clams. just stuff to think about in the crease.

-J

Hi! Is there anywhere where you explain the math behind your work? I love it and it's beautiful and I'm a math major in college right now. It would be very interesting to me.

I haven’t explained much so far but I can mention some interesting math concepts I use a lot, (with a quick explanation for non math major)

- sine function
(describes the motion of a simple smooth oscillation, which can be seen as “the more I move away from the center, the more it attracts me, the more I go close, the less it attracts me” https://betterexplained.com/articles/intuitive-understanding-of-sine-waves/) and its parameters : amplitude (the farthest it moves before going back), period (the duration of one oscillation), phase (delay/timeshift)

- linear interpolation
(allows to regularly distribute n numbers between a max and a min value, for example : 5 numbers between 3 and 6 gives 3; 3.75; 4.5; 5.25; 6)

- harmonic oscillations
(describes the way a guitar string vibrates : see the video “Aesthetics of the vibrating string” made a few years ago : https://vimeo.com/87357998 )

- perlin noise
(allows dots to move randomly while keeping close to the previous position and to the neighborhood)

- fractal
(auto-similar structure that is made of parts looking like itself : trees, snowflake, lightening, mountain… https://www.google.com/search?q=fractal+nature&tbm=isch)

- above all, there is this fact that in mathematics, all the fields you can explore are never compartmentalized, that everything’s linked somehow, so there’s always a bridge between two different points, which is something I find beautiful and try to illustrate in my work too

3

## Alternating wave pattern on Ropes

I saw someone working out with Battle Ropes the other day and this wonderful pattern emerged was absolutely fascinating. Of course, the waveforms are not purely sinusoidal but it helps us to understand why you see such patterns

Sine wave.

2017.04.25 // hidden school life hack [ april study challenge, 25/30 ]

i’m like the last person u should ask about life hacks but anyway what i’ve found that works really well for organization is to pick something that you like/you can stick to, whether it be four-inch binders idk how ppl carry those around, or accordion folders or digitalizing practically everything but then u just have to stay w/ it and this got longer than i expected i will stop now.

2017.04.26 // how do you go to school/uni [ april study challenge, 26/30 ]

the parental units, bc i’m barely old enough to get my permit and so i do not have a permit, nor do i have a car.

pictured: math notes on sine functions, aka me trying not to fail

The sine function starts life in a triangle (year 9?) then becomes intimate with a circle at A level, as y-coord of a point rotating on a unit circle.  Later it is about infinite series or, even, differential equations.

Nico’s suffering (1/?)

NSFW, Nozonicoeli
Nico with a D
Words: 1k

for @noelclover89 and @nozoroomie lmao thanks for listening to my screaming :,))

“Nicocchi!”

Nico gulped, tightening her grip on the rail. Summer could only mean two things, the summer homework and the sweltering heat. Which certainly didn’t help in matters when they still had physical ed lessons at this time of the year.

More Than You Ever Wanted to Know About Math: Euler’s Formula

We’ve looked at how you can translate a sinusoidal voltage or current between trigonometric, polar, and rectangular coordinate forms.

Being able to do this makes it way easier to deal with these mathematically: rather than remembering a whole bunch of trig formulas and identities, we can just stick with regular addition, subtraction, multiplication, and division.

Euler’s formula is the reason we can do this. Euler’s formula is both simple and powerful and it comes up over and over again in electrical engineering. It relates the sine and cosine functions to both the imaginary number j (or i, if you’re not dealing with electrical engineering) and the constant e. It looks like this:

Looks like the way we did rectangular coordinates, right? That’s exactly what it is.

We don’t normally write polar notation with the exponential, but that’s just what it is. When we write the polar expression A /_ φ, it’s just a shorthand for Ae^(jφ). Knowing this, you can see why the translation between polar and rectangular coordinates makes sense:

Euler’s formula will come up again for us numerous times. There’s other stuff you can do with it, but for now just keep in the back of your mind that sines, cosines, and exponentials are related and that you can switch back and forth between them at will.

Do you know anything about the irrational number e (2.71828...)? Like its meaning in the nature? BTW I really enjoy your blog!

Hi nerdyyfun,

Since your question is quite general, I’ll give you some remarkable properties of Euler’s magical constant. First of all, its definition. In calculus, e is the unique real number so that the exponential function with base e equals its own derivative.

The exponential function is the only function with this property, except for the trivial constant zero function. You can thus figure out why it plays a central role in calculus.

Just like π, this number e pops up in various mathematical formulas.

Rather surprisingly, there is a strong connection between the exponential function and the sine and cosine. Their relation doesn’t show up immediately between the real numbers: you should look in the complex plane. The complex exponential function, complex sine and complex cosine (the unique extensions of the real-valued ones) are related as follows.

This intriguing identity is known as Euler’s identity. When you plug in the constant π, you get a formula which is commonly known as the most beautiful identity in whole mathematics:

This results make the exponential function (and the number e in particular) a nifty way for dealing with complex numbers, sines and cosines, and harmonic oscillations. The latter are very common in physics and nature. Indeed, differential equations governing harmonic oscillators are frequently solved by exponentials.

Another interpretation of the exponential function says that the rate of growth is proportional to the current size, which makes it a natural model for population growth (Malthus’ model).

However, there are also manifestations of e in nature which are more “visual”. In particular, the inverse function of the exponential, called the logarithmic function, can be transformed into a spiral using polar coordinates. This logarithmic spiral is very common in for instance seashells, because again, it models the idea that expansion is proportional to the current size, and thus expresses the (linear) growth between the animal and the proportional growth of its shell.

I hope you find this useful and interesting.
Have a nice day, and thanks for the kind words!

Mathematics

## Fourier series

The Fourier series is used to represent a periodic function in terms of an infinite sum of sines and cosines.

In its most general form the Fourier series of a function f(x) with period P is represented by

f(x) = ½ a0 + Σ∞
n=1 [an · cos (2nπx/P) + bn · sin (2nπx/P)]

where an and bn are the Fourier coefficients defined such that

an = 2/P · ∫P/2
P/2
f(x) · cos (2nπx/P) dx

bn = 2/P · ∫P/2
P/2
f(x) · sin (2nπx/P) dx

and a0 is the average of the function, calculated by

a0= 1/P · ∫P/2
P/2
f(x) dx

There is a wonderful representation of this I found on Wikipedia that perfectly visualises the effect of using the Fourier series for increasing values of n to approximate a square wave.

(Source: Wikimedia, Fourier Series (via Wikipedia) http://en.wikipedia.org/wiki/Fourier_series)

We will go through this is more detail by looking at examples of approximating different waveforms using the Fourier series.

Prompt #14: Kiss on the Neck
Summary: Fluff. And homework.
Word Count: 404
Notes: Sorry it’s not Ladynoir – I’m working my way to number 8 though, and that’ll be Ladynoir. (Working my way up to it, more like *shudders* The big, scary one…) ;) Also, I hope I wrote the right Math curriculum for France. :3 Actually, I hope Algebra 2 is high enough for their age?

Marionette sighs, stretching to get the crinks out of her neck after bending over the design of the shirt she’s been working on for hours. She still can’t get it right, no matter how many times she’s redesigned it.

That annoys her.

Arms wrap around her, but she doesn’t jump because she recognizes the black leather on them. She smiles as his lips press against the side of her neck, and then her lips. Her heart still does that staccato thing, in the way it did the first time he’d kissed her (as Chat Noir, not Adrien), and her cheeks feel warm.

“Hi.” Adrien says, still not letting go of her. She tips her head back to look at his masked face and reaches up to run her fingers through his hair. His eyes close, and his lips curve up.

“Don’t you know it’s easier to knock, Kitty Cat? My dad loves you, but I’m not sure that I’d be able to explain how you got into my room without going through the bakery.”

His smile turns impish, and he opens his eyes. “That takes all the fun out of it though.”

She giggles and pushes him away from her, before bringing him back. The transformation wears off on him, and Plagg floats above her computer with Tikki.

“Where’s my cheese?”

Adrien sends him an exasperated glance, but digs through his pocket and holds it out. Marionette leans away from him, plugging her nose up because that is really smelly cheese.

“Right.” He says airily, sitting down on the other spiny-chair, crossing his legs somehow. “Mari, can I look at your notes for today’s Math lesson? Pretty please? With cherries on top?” He gives her puppy – well, kitten – eyes, complete with the pouting lip. “Mrs. Gunner was talking too fast and I only managed to write the part about angles greater than three-hundred and sixty degrees, not the trigonometric functions.”

“Sine, cosine and tangent?”

“Yeah, that.”

She kicked her chair back and glided effortlessly toward her bed, where she had thrown her book bag. When she pulled her notebook out, she kicked off her bed and sailed back to her desk, oomphing softly when she collided with it harder than she expected.

“It should be in here,” she said, handing it over to him. He grinned widely and flung his arms around her, squeezing her too tight.