hi anon! i..kinda forgot to save the brush setting for that one ahahah, but i think it should be something like this! and i used SAI :)
and aaah thank you so much!! ;___;
The second part of the cloud tutorial is done :D You can view the full version on my dA. And for the new followers, you can find part 1 here. I’ll be working on part three next :3
Here’s the free download to the Photoshop brushes.
I was very happy to see all the beautiful skies people have been painting while using my tutorial :) I hope to make lots of other tutorials on lighting and colors as well. Now I just need time between my busy schedule orz…
-Gif of the tutorial work progress (best viewd in Firefox)
- jfashion magazines!!! (helps out with outfits/people)
- paint online!!
- some paint tool sai brushes!!
- mypaint (its free and its kinda like photoshop its really cool!)
- firealpca (its free and its pretty much a mix between sai and photoshop!!)
- photoshop (for free! and other stuff)
- more textures
- more brushes
- bonus: a kawaii photo editor
Stretches that improve different aspects of your body.
Notice how Shan Yu doesn’t even question it or make a comment about “BUT YOU’RE A GIRL” he just instantly goes into a “I’LL TEACH YOU TO KILL MY MEN AND STEAL MY VICTORY” rage and I think about this a lot sometimes
((Well that might have to do with the fact that he’s a Hun. Women among the Huns had higher status than their Chinese counterparts and even some of their own men. Women were free to hunt and fight along side of the men, could choose their own husbands and divorce him if she choose to. There were even records of clans being led by women leaders. So for Shan Yu Mulan is just another soldier))
#i’m glad someone pointed that out#nomads have pretty much always been better about equality than their sedentary neighbors#also fun thing: one of the reasons the mongols never really assimilated was that mongol women flat out refused to participate in misogynistic chinese culture
LITERALLY CANNOT BELIEVE WHAT I JUST FUCKING READ
"misogynistic chinese culture" WHO THE FUCK DO YOU THINK YOU ARE. THIS MOVIE IS SO GROSSLY ORIENTALIST AND RACIST LIKE??? NEWSFLASH: MULAN WASN’T SHAMED OR SHIT ON WHEN SHE REVEALED HER TRUE GENDER. SHE WAS OFFERED A REWARD AND CHOSE TO GO HOME. THAT’S IT. IF YOU DON’T KNOW TWO SHITS ABOUT CHINESE CULTURE SHUT THE FUCK UP. SHUT THE FUCK UP AND STOP CALLING MY CULTURE "MISOGYNISTIC" WHEN IT OPERATES SO DIFFERENTLY FROM WESTERN CULTURE
HEY AND FUCK YOU—PEOPLE WERE FUCKING SLAUGHTERED UNDER MONGOL RULE. PARTS OF OUR CULTURE SHITTED ON. But nah, who fucking CARES. CUZ THE MONGOLS WERE SOOOOO GREAT AT GENDER EQUALITY LOLOLOL give me a FUCKING break
And stop stop what the FUCK this movie literally portrays Mongols as gray hulking evil beasts—look, their form is barely fucking human. Like how is this not racist?
You’re going to shit on MY culture and call it sexist for shit YOU KNOW 0 ABOUT, and then you’re gonna just ignore the blatant racism perpetrated against the people you’re praising.
Shut the fuck up. Mulan is not and never was about feminism. Stop trying to fucking warp my culture into your fucking little white feminism views. Sit the fuck down.
A+++++++ COMMENTARY ABOVE
i hv much to say but i am on my phone
i will deal w this hot steaming mess later
i’m on my computer and at home now so i can deal with this cesspit of misinformation and assholery
first, you are wrong. you are wrong, you are wrong, and i sincerely hope the lot of you and any other ignorant asshat sits on an anthill.
Well that might have to do with the fact that he’s a Hun.
he is not a hun. disney might say a thing, but that does not mean a thing is correct. FIRST, the closest relationship the huns have to chinese history are the xiongnu, but the evidence for that is flimsy at best and is far from being widely accepted
Women were free to hunt and fight along side of the men, could choose their own husbands and divorce him if she choose to. There were even records of clans being led by women leaders.
yeah, and? you seem to be under the frankly birdbrained idea that chinese notions of gender roles are fixed, and based entirely around the strict neo-confucianist gender roles of the song dynasty, but that’s far from true and chinese society is far from static, even during the song.
tang and song dynasty women were allowed to own businesses, own capital, divorce their husbands, let’s not fucking pretend like china doesn’t have female rulers who, though only one of them was a legitimate sitting emperor, wielded significant amounts of power
that’s not to mention the fact that china has a loooooooongass motherfucking tradition of female warriors, most of whom, UNlike hua mulan, did NOT disguise their genders yet STILL rose to distinction, and most of whom, i might add, ARE VENERATED
ha. ha. funny thing, that. esp given that in most CHINESE depictions of hua mulan, she is the subject of the northern wei court. who ruled the norther wei? you might ask. well, friend, it was the tuoba! which is why the dynasty is also sometimes called the tuoba wei. fun fact, their capital was actually in my hometown of datong, which is where (and when) they carved the yungang grottoes. before they moved it to luoyang. fuck luoyang. now who were the tuoba, you might ask? well, friend, they were a NOMADIC MONGOLIC PEOPLE.
or or or! it could have something to do with the fact that north chinese (along with other vanquished people like the khitans of the liao dynasty the jurchens of the jin and the koreans of the goryeo) were treated as the second-lowest rung of the mongol social ladder! OR the fact that south chinese were treated like the LOWEST rung of that ladder! and god, why would the people at the top of that ladder want to have anything to do with the people at the bottom? but yo, what the hell do i know, i’m just a descendent of those second-lowest people and a willing female participant of this “misogynistic chinese culture” you clearly have so nuanced and wholistic an understanding of.
also, just as a by the way. the khans did, to some extent, assimilate.
but hey! thanks for weighing in on a history you clearly have no education in, a culture you know nothing about, and a folktale that has nothing to do with and which has been destroyed by people like you!!!!
The results were 10 days. Not sure what they did, I hear this clinic has a really good reputation when it comes to skin
Clinic name is 맑은얼굴 참진한의원/JinMedi(i think)
Website for the clinic (Beware of ads)
LET ME TELL YOU ABOUT THIS NEAT THING OKAY?
Just to preface this, I don’t know if anyone else has posted this, so if you have, I’m sorry! I’m posting for anybody with any anxiety, stress, want to cool down from a rough day, or just need background noise to function!
This is a really neat site because you have more than one noise to choose from to listen to. If rain isn’t really your thing, they have crackling fire and breaking waves, and it’s just really relaxing. And you know what’s the coolest part about each noise? You can change the levels. You can slide and switch levels around to have the perfect amount of thunder, or light rain, or crackles in your fire, or foamy sea goodness! Each noise (to my knowledge) has 10 sliders for different sounds within said noise, so you can mix and listen for as long as you need! And if you don’t feel like mixing it yourself, there’s a neat button called “Animate” which allows the noise to evolve and change itself, so it gives it a little flavor.
It even has some that are specifically catering to mental health and sound therapy.
But really, I encourage everyone to at least try it out, it’s just super neat and it calms me down and serves as a nice low noise in the background if I need it for sleep or working on homework.
More FREE online courses to take, & ways to earn your degree, without leaving your own home!!
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Fuck college. All education should be free no matter what.
Unless you wanna just pay dozens of thousands of dollars for a piece of paper you can hang in a fancy frame and bragging rights.
If knowledge is what you want, take it.
For anyone who feels lost about what to study, check, explore!
This post made me happy *nerd*
FOR ALL YOU STUDENTS THAT ARE BAD TRIPPING LIKE I AM
A much more accurate “now” image for direct comparison would be a photograph. Abstract art came about as a direct result of the invention of photography, as paintings no longer need to be representative, as photographs are inherently better at that task. Accurately representative paintings were just the photography of the pre-photography era. Portraits of nobility have been replaced by Sears Portrait Studio. Self portraits have been replaced by Instagram selfies. Paintings of historical events have been replaced by photojournalism. Photography is the democratization of art and abstract paintings and sculpture are the most evolved form of art possible. These are not things to complain about.
YES YES IT IS okay i have a whole lot of feelings about abstract and non-representational art
because yeah, once photography became more common and widely available, people who painted started to question what they painted for. They started to wonder what made something art, what distinguished a painting from a photograph - if photographs could depict “objective” reality (insofar as such a thing even exists), then maybe the strength of painting lay in what photography, in that day and age, couldn’t capture, like feelings or impressions or the tricks the eye plays when seeing an object.
and so they asked, well, why does this portrait feel so comfortable and warm and this one feel threatening and stiff? what elements of the picture suggest that? is it the lighting? the way people are posed? can you play up those elements, exaggerate them, make the figures express the feeling instead of the other way around? what would happen if you did?
and for that matter, people continued to ask, why do we find a certain change of lighting comforting? why do we respond to someone wearing a red shirt or a blue dress differently to someone wearing a white one? what is it about red? or black? or green? why does the shape that people are standing in, the way the figures are placed in a frame, change how we feel about it?
art in a way started to become about psychology - it became about thinking and about why we think and how. because photographs (again, at the time) weren’t engaging with that nearly as much, art started to move towards a “why” of photographs. why that pose? why that color?
that’s when you start to get art like the one under “art now”, right, because look at it. REALLY LOOK, okay, don’t just shrug and walk away because “anyone could do it”. look at that exact shade of orange. do you ever see just a big swath of one color in nature? no, you see hints of it. where have you seen it? what does it remind you of? for that matter, what about the little stripes on it? does that make you feel like there’s depth to the painting - something inside it? why? after all, the painting is a flat plane*, so if you do get a sense of depth from it that’s your brain interpreting signals its familiar with. isn’t that incredible, that all it takes is a few little lines on a single color? isn’t it strange how one person will see depth and another won’t?
*and for that matter it ISN’T a flat plane, there are variations in the height of the paint on the canvas and how much it’s built up, and it protrudes slightly from the wall instead of being recessed into it - does that do anything to the sensation of depth? while we’re on that note, do you ever look at a representational painting and think about how you, the viewer, are looking into it and see it as having space and depth when it really doesn’t - only it does, but not the same space and depth as is represented in the picture?
and that’s without even getting into larger cultural shifts like the World Wars - and it’s hard to overestimate the effect that WWI and WWII had on even the “mainstream” art world - and the greater voice of underrepresented and oppressed groups like women, POC, and LGBT artists and the increasing technological sophistication of photography and the advent of video and widely-available audio recording and the increasing use of galleries to display art rather than private residences and it is still art, okay, representative art is art too but that doesn’t mean this isn’t it’s just focusing on something different and if you dismiss non-representational art as lazy or a con i will sit your ass down in the nearest chair and yell at you about marcel duchamp for an hour
I have a lot of feelings about this, so I’m gonna just spew them everywhere.
Most critically! The red piece isn’t art now. It’s art 60 years ago — 1950, they great heyday of abstract expressionism in the USA! All that abstract shit you hate, all that stuff that’s just splatters and giant dots? 1950-1960. The United States. A small, elitist movement shaped by maybe a dozen artists and two or three very influential critics. In a decade abstract expressionism had pretty much said all there was to say about the action of painting and the canvas as an object rather than a representation, and it got stuck in the museum for people to be bewildered at.
The Rembrandt piece above it? Also a snapshot of a very particular time and place. Our view of art 400 years ago is blinkered by what we’ve bothered to preserve and focus on. When people think “old-timey art” they think of bright white marble statues with no limbs and Da Vinci and Dutch still life. Which is such a tiny fraction of things that have happened in art history, you know? That’s like, three things! Most of them done for rich dudes in Western Europe!
I call such bullshit on someone trying to knock down all of contemporary art by comparing something made for the cultural elite in 1650 to something made for the cultural elite in 1950.
Art is huge, poorly defined, and it has always been that way, has always had elements that are democratic and has always had a thick vein of nasty elitism. The carvings on the doors into Notre Dame tell the stories of the saints so that everyone could understand them, whether they had access to books or not. Comic books and photorealism and murals in urban areas and fashion spreads — all this stuff is made to wow everyone, independent of how much time they’ve spent studying the deep philosophical circle-jerk of art criticism.
I love art criticism, I love Frank Stella and Ad Reinhardt and Eva Hesse, and I am still incandescently furious when people try to reduce the evolution of art to simply justifying or condemning their work. Because that means we’ve fallen head-first into the trap of omission and framing that keeps art defined as only for the museum-attending. There’s museum art — cerebral and obtuse and annoying and demanding of effort and education and money to appreciate — and then there’s literally a whole world of more art. It is an appalling disservice to all the other artists making it out there (corporate designers and media hubs and scrappy little collectives and crafters and professional illustrators) to sweep them under the rug in favor of arguing about museum art as if it is the most important art, or, worse, the only art.
Don’t like Barnett Newman? Fuck Barnett Newman. Fuck his arrogance and his inaccessibility and his ego and his concept of the primitive.
But fuck you if you call him “Art now” while you do it. Don’t make one man the measuring stick for a century of modern creative works. That’s a bullshit premise and you know it.
isn’t captain hook and his crew suppose to be a lost boys who escaped and that’s why he’s trying to kill peter pan
…what the actual fuck
I NEVER TRUSTED PETER PAN
nah everything in Peter Pan was fucked up.
Tinkerbell and her fairy buddies were having an orgy when they found baby Peter. Tinks also extremely jealous, tricking one of the Lost Boys into shooting Wendy in the fucking chest.
Peter’s also crazy omnipotent. Like, he “make believes” he’s a doctor, and heals Wendy. When he’s hungry, he pretends to eat imaginary food and his stomach actually gets fuller.
He’s also a dick. He would teach children how to fly but never how to stop, so they’d fly for months on straight without rest or break, and they couldn’t sleep either or they’d stop flying. And when one of Wendy’s brothers actually fell asleep and plummeted into the ocean, Peter laughed his ass off. He only saved him when Wendy begged him too.
okay but that’s the point of Peter Pan. It’s not supposed to glorify never growing up, it’s supposed to show kids why growing up is not only good, but necessary otherwise they’d end up as fucked up as Peter. He never matured, never learned right from wrong, he never listened to his parents because - according to Peter - he ran away as an infant.It’s a tale to teach children that listening to their parents and growing up is good. As far as Tinker Bell goes, if you actually read Peter Pan you would know that fairies only feel one emotion at a time and they feel that emotion very strongly so the orgy? lust. Trying to kill Wendy? Jealousy. She embodies the seven deadly sins and what happens if you let your emotions get the best of you. (And as far as the new fairies series of films making her nicer it’s because you only see the jealous side of her in Peter Pan and you see other sides of her in the series because those movies are about her).
Rant over, you can go back to your regularly scheduled blogging now.
So if Peter Pan shows up in your window. Stab him in the fucking chest kids. You have school tomorrow
Reblogging because I believe this will be important to the Once Upon a Time fandom tomorrow.
It’s more complicated than that. Peter is kind of a tragic hero. He chooses not to grow up, he knows he is incomplete.
I mean, he cut off Hook’s hand because he thought it was a game. He clearly doesn’t know right from wrong. He also only knows the unconditional love of a mother to a child, which is why he thinks everyone wants to be his mother. He also switches sides in a fight just for fun, kill pirates for fun, and “thins” out the Lost Boys when they can’t fit in the tree anymore.
But, like, it wasn’t a cautionary tale to tell you to listen to your parents, it’s a story about death and youth. Why can’t Peter grow up? One of the popular theories is that it’s because he’s dead. J.M. Barrie’s older brother died when Barrie was little and he dressed up in his brother’s clothes to please his mom. His mom - who was always distant, whose love Barrie craved like Peter craves a mom - started crying and said something like “At least my baby will never grow up” and that idea stuck with Barrie forever. Then, as an adult, it’s believed he never slept with his wife because Barrie was just a kid. He was Peter Pan. He was too innocent for that. He befriended the Llewelyn-Davies boys and based Peter Pan off of them and their games. (Fun fact: The boy Peter Pan was named after, Peter Llewelyn-Davies, threw himself under a train). There was also a bunch of stuff about Barrie being in love with The Llewlyn-Davies boys’ mother, but that’s not important here.
People think Peter’s dead because he literally cannot return home. He tried and the window was barred and his parents had replaced him with another baby. Why? Probably because they had lost Peter to the flu. Why does Peter come in through the window? Because of the joke “I once had a bird names Enza. I opened up the window and ‘influenza’.” Because lots of babies died back then form the flu. The Lost Boys are children who fell out of their prams. Odds are babies could not survive falling out of their prams. Peter is liked the pied piper ferrying the souls of young children to the neverland/afterlife. Barrie believed that all children were “gay and heartless” but he didn’t think that was a bad thing.
Also, Hook and his crew are not old lost boys trying to kill Peter. Hook was once a British gentlemen (hinted at to be associated with Charles II and attended Elton) and he is afraid of growing old. His biggest fear is growing old and dying - that is why his nemesis is the embodiment of eternal youth. That is why the crocodile that chases him swallowed a clock and ticks. That is why when Peter finally decided “It’s Hook of me this time” the crocodile has stopped ticking and Peter started (he’s trying to trick them into thinking he’s the croc). At that moment - Peter is time and time has ran out for Hook.
Also, it’s not so much that Peter is omnipotent. All kids basically are in the Neverland. Like, it states that the island looks different to every kid because it’s the land of their dreams and stuff. Also, the island legit freezes when Peter leaves and thaws when he comes back. He’s been there so long he’s not human anymore - but fey. (keep in mind being fey isn’t good, just chaotic neutral). Peter even secretes pixie dust now. The island is so fine tuned with him because he’s one of the only people that stay, that it caters to him. Most likely any child that stayed as long as he did would become omnipotent to an extent.
As for Tinker Bell, the above stated is true. Fairies are so tiny they can only have one emotion at a time - “Tink wasn’t all bad” - and they also have really short lifespans so, like, Tinker Bell isn’t even that important to Peter Pan. He forgets all about her and Hook by the time Wendy is grown up.And the orgies thing is because in the legends fey are known for their revelries.
And it wasn’t so much that Peter was a dick, he just doesn’t know when to stop. He’s a child. He doesn’t know right from wrong. He doesn’t know when to stop playing -cutting Hooks hand off was a game to him. He also has the memory of a child, so odds are he just forgot to teach kids how to stop flying or how to imagine food, etc. He is just carefree, like all children. Everything is a game to him, because he never learned anything else.
But like, no, Peter Pan is not a cautionary tale. Barrie loved his character and the story and brought up a lot of good things in it. He wrote Peter as an exaggeration of a cocky overconfident boy, but, like, Peter wasn’t afraid of death. It says “he felt scared, yet he felt only one shudder run through him when any other person would have felt scared up until death. With his blithe attitude towards death, he says, “To die will be an awfully big adventure”.” and with that Barrie is showing us both a naivety and bravery we possess as children but lose as adults and is basically telling us that we shouldn’t let that go. Like, the point is growing up is inevitable but you don’t have to lose everything.
And so yeah….I’m really passionate about Peter Pan.
The Golden Ratio
Also known by the Greek letter φ (phi), this curious irrational number has a closed-form given by:
φ = (√5 + 1)/2 = 1.61803398875…
From nautilus shells, the human body to spiral galaxies, the Golden Ratio seems to be everywhere in Nature, right?
Well, not really.
A very large portion of what you have probably heard about this number is just hype, widely propagated myths, extremely far-fetched analysis of data or, putting it mildly, just made-up nonsense.
Now, don’t get me wrong here. The Golden Ratio really is a very interesting number with a number of outstanding mathematical properties. This is why it saddens me to see so many people praising it for all the wrong (and wildly innacurate) reasons.
For instance, several spirals in nature are logarithmic spirals because they are the same independent of the scale. This sort of thing is bound to show up whenever you have exponential growth in a circular fashion, two phenomena that are extremely common in nature. In the end, logarithmic spirals are really just exponential functions in polar coordinates.
However, not all logarithmic spirals are Fibonacci spirals. In fact, what it is known as the Fibonacci or Golden spiral, derived from the famous construction using nested squares and golden rectangles (shown below), is a very gimmicky geometric construction that really shouldn’t be expected to show up in nature at all. Nature doesn’t work with squares and rectangles!
In the study of aesthetics, the Golden Ratio is often praised as being the most beautiful ratio for things, a dogma that gets passed around a lot in design circles. Several studies have shown no correlation between the Golden Ratio and a sense of beauty or aesthetics. (check links at the end of the post for more on this)
I could list most of these myths here, but I would just be repeating what has already been said by many others. So if you want to find out what’s true and what isn’t about the Golden Ratio, I recommend that you watch this talk by Keith Devlin or read this article by Donald Simanek. More links and resources can be found a the end of the post.
With that usual Golden Ratio crap out of the way, I can now finally talk about why this number is REALLY cool.
φ - The most irrational of all numbers
Irrational numbers are numbers that cannot be expressed as the ratio of two integers. Note that the keyword here is integers. This little important detail gets a lot of people confused, usually because of π.
While π is usually defined as the ratio between the circumference of a circle by its diameter, you cannot have both of those quantities being whole numbers, because π happens to be irrational. You can approximate an irrational number with rational approximations, such as 22/7 = 3.142857142857… or 3141592/1000000 = 3.141592, but no matter how large the two numbers of the ratio are, you’ll never find a ratio that is exactly π. The same is true for any other irrational number, φ included.
That animated infinite fraction you see at the top is an example of what we call an infinite continued fraction. Continued fractions are a powerful way to represent irrational numbers because they show you how good a rational approximation is: larger terms in the continued fraction mean you are adding smaller corrections, which tells you the approximation is good. Additionally, all irrational numbers have unique infinite continued fraction representations, a very useful property.
But since we know the larger terms mean “better approximations”, we can think of what would be the worst approximation ever for any number. This would be the infinite continued fraction where the terms are the smallest integer available: 1.
And, it turns out, this infinite continued fraction represents the number φ! This is what the animation is representing.
Think about that for a second. There are an infinite number of irrational numbers, and of all of them, φ is the absolute worst number to approximate using a ratio of two whole numbers. In a sense, φ can be said to be the “most irrational” of all irrational numbers!
This makes me wonder why we even call φ the “Golden Ratio” to begin with, as it is the one number that is as far from being a ratio as it is mathematically possible.
φ and Nature
This “super-irrationality” of φ can be pretty useful, and it is one of the reasons (if not the only one, other than those related to pentagonal symmetries) why approximations of φ show up in Nature, for real this time.
Imagine you have a periodic process, such as leaves growing on a plant stem. If one leaf grows directly on top of another, the leaf below will not be exposed to the Sun due to the shadow cast by the leaf above, so the leaf below will be pretty much useless.
Evolution would favor plants that add an offset between leaves, perhaps by having the stem twist as it grows. This would improve the amount of sunlight each leaf is exposed to, making the plant more efficient and giving it an evolutionary advantage.
However, if the amount of twist between consecutive leaves is a nice ratio of full turns, say 2/3, you would get an overlap between every 3rd leaf. So in this case, you don’t really want nice ratios. You want the leaves to be as spaced as possible, that is, you want the worst ratio you can think of.
As we already know, φ would be that ratio. However, φ cannot really exist out there in the real world, so approximations are as good as we can get.
And guess what? The rational approximations available for φ are the ratios between two consecutive Fibonacci numbers. But you probably knew that already.
This explains why Fibonacci numbers may show up in Nature. Whenever you have a periodic process that would benefit from being “as irregular as possible”, Fibonacci numbers are bound to show up as approximations for φ.
The “real” golden spiral
Let’s say you have a bunch of points that you want to distribute evenly on a disk, as efficiently as possible. This sort of problem shows up in Nature, like in the case of sunflower seeds.
The easiest way to do this, in terms of a set of basic rules, is by placing the points along a spiral, adding layer after layer of points.
But the BEST way to do it uses a very special spiral known as Fermat’s spiral, in which the radius is proportional to the square root of the angle, that is, r(θ) = k√θ, for some constant k.
Since the area of a disk grows with the square of its radius, this spiral has the property of “covering” equal amounts of area for the same amount of rotation.
If you pair this property with the irregular spacing mentioned previously, by picking points along this spiral in multiples of φ (in terms of full turns), you have a very simple rule to achieve the goal of distributing these points along the disk.
You can play around with this idea in the applet below. Apart from the sliders, you can also change the ratio using the left and right keys. Hold shift and/or control to increase the rate of change. You can also type in a fraction like 22/7 in the ratio text box and hit enter.
To be clear, the x and y coordinates of the n-th point will be: x = cos(2πkn)·r(n) and y = sin(2πkn)·r(n), where r(n) is the radius function (that is, the polar function for the chosen spiral) and k is the ratio being used to place the points around the spiral. Only the fractional part of k matters in this model.
You’ll see that most irrational numbers produce some pretty obvious patterns right away. φ and its reciprocal (in fact, the entire family of numbers sharing that same fractional part) are the only numbers that get everything as evenly spaced as possible, no matter how large the spiral is or how many points you use. In fact, even tiny variations from these ratios already ruin the whole pattern.
Picking different functions for the radius will reveal how Fermat’s spiral is special regarding the radial spacing between dots.
For fun, I also decided to plot lines connecting two consecutive points. You can get some pretty neat images with this, showing the patterns even more clearly. As expected, φ gets you the most messy and irregular of all images, as shown in the second image in this post. For comparison, I also included some other irrational numbers as ratios.
In three dimensions
Now imagine that instead of a disk, you wanted to distribute points uniformly on the surface of a sphere. This problem shows up every now and then, and it cannot be solved so easily. The usual algorithms to solve it involve physical simulations of repelling particles with friction. After a long simulation time, the system will achieve a somewhat decent equilibrium state. This method is particularly troublesome if we’re talking about thousands of points, as we’d have to simulate the interaction between every possible pair of points.
However, we can do better than that. A spiral similar as the one for the disk can be used to distribute points across the surface of a sphere, in a way that makes them relatively uniform.
So thanks to φ and its irrational properties, we can tackle a hard problem in a relatively straightforward and direct way. Pretty clever stuff!
Find out more of the truth about φ
Well, there’s a lot of other cool stuff I could say about φ, but this post is already pretty long as it is and the links below are full of more stuff.
- "Math Encounters — Fibonacci & the Golden Ratio Exposed" by Keith Devlin. Accessible to all ages. (I liked it, though from the comments, a lot of people seem to hate this lecture.)
- "Fibonacci Flim Flam" by Donald E. Simanek, which criticizes most of the myths about the golden ratio
- "The golden ratio and aesthetics" by Mario Livio, a skeptic look at the claims about phi in arts
- "The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number" by Mario Livio, if you are looking for all the real, no-nonsense mathematical coolness behind φ
- "Doodling in Math: Spirals, Fibonacci, and Being a Plant", ViHart’s series on Fibonacci numbers in nature.