equilaterals

shoutout to @equilateral-asshat for showing me this post and giving me the laugh of a lifetime

From Dan’s Rush AMA
  • He’d like to possibly cover Closer to the Heart by Rush in the future
  • His favorite concert of Rush’s he’s seen (out of 11) is the R30 tour in 2004
  • His favorite thing he’s seen them play live is Xanadu
  • He doesn’t plan on trying to start playing an instrument (aside from the bass which he has said he’s mediocre at) because the band he’s got now has such awesome musicians that he wants to focus on being the best vocalist he can be
  • His favorite Skyhill song is Run with the Hunted, and his least favorite is Only One
  • His favorite Rush song to play on the bass is Fountain of Lameneth (but it’s hard)
  • It’s pretty likely that there will eventually be a new Skyhill song or album
  • He first saw Rush in 1994 in NJ at the Brendan Byrne Arena. He’s seen them on every tour since.
  • He’s hoping that next year they’ll be able to tour in the UK, and he’d like to be able to tour in Australia too at some point
  • What influenced him most from Rush is their chord choices, their worth ethic, and their integrity.
  • If he could perform vocals on one song with Rush, it would probably be Time Stand Still
  • His best Rush-related experience was zoning out by himself as a kid and letting their music take his imagination places
  • His favorite Rush albums are A Farewell to Kings, Hemispheres, Caress of Steel, and 2112. His least favorite (though he still loves them!) are Hold your Fire, Counterparts, and Snakes and Arrows
  • The most direct way that Rush has affected him is that they helped inspire him to commit his life to music.
  • His top 3 songs are Xanadu, Hemispheres, and Fountain of Lameneth
  • He started listening to Rush at 13 and he became obsessed with them. He sang their music to himself all day during school.
  • When asked how many Rush shirts he has, his answer was “Too many!!”
  • A Farewell to Kings is his alltime favorite Rush album
  • What initially caused him to enjoy Rush was that it was an awesome sound that he’d never really heard before and lyrics that appealed to his nerdy Lord of the Rings loving self.
  • His favorite >10 minute Rush track is (possibly) Circumstances
  • “ I’m more Jewish by culture than by religion. I certainly believe that there’s a lot to life and the universe that we don’t understand, and I respect anyone who finds their answers in religion, but it’s never been a big part of my life personally.”
  •  His favorite Rush show to watch is one from Rio
  • The song he’s most proud of is 6969 because it’s a comedy prog song, which is something he’s always wanted to do.
  • His favorite part of 2112 is the overture
  • Hemispheres will always remind him of his freshman year of high school. He listened to it constantly.
  • Both him and Arin write the lyrics to Starbomb songs
  • He feels an immediate kinship with other Rush fans. “We’re like a world-wide secret club!”
  • The rush album he feels is most underrated is Presto
  • The first Rush song he listened to was Roll the Bones
  • He would be interested in giving talks about music/comedy/career stuff at colleges or high schools if there was an audience for it.
  • He got into Rush at age 12 because the guy that worked at his local comic book shop (named Keith) introduced him to them. “God bless that man.” Before he discovered them he listened to a lot of Def Leppard and Nirvana
  • Q:  if you were to start your career again today, where would you start? Youtube / twitch / podcasts?
    A:  I don’t think I’d do anything differently because even the years of “failure” taught me a lot and allowed me time to improve. Once that spotlight is on you, you’d better be ready for it. I’d say start in the medium that inspires you individually the most and work as hard as you can on it from there.
  • Q:  how do you think the future of NSP has changed now that you guys have topped the motherfucking charts with Under the Covers?? You guys fucking rule and I’m so happy for you and your multitudes of success!
    A:  Thank you! We were psyched that our recent albums have charted so well, but I don’t think we’ll do anything differently. We’ll just keep doing what feels right to the best of our ability!
  • 6969 came together quickly when writing it. The video only took so long because it was such a huge jump in ambition and it has a learning curve. “If we do something like that again, things should go a lot more smoothly!”
  • The Pass has helped him through some tough times.
  • (I believe) that the lyrics he’d get as a tattoo are “ All of us get lost in the darkness Dreamers learn to steer by the stars All of us do time in the gutter Dreamers learn to look at the cars!”
  • Q: What would you say is the best dish at Olive Garden?
    A: The dish where you get up, walk across the street, and eat at Outback Steakhouse
  • He meditates in silence before a show because he’d get overhyped if he listened to music.
  • Doesn’t have a favorite member of Rush, “it’s an equilateral triangle”
  • Says Moving Pictures is usually a safe bet to start off with introducing someone to Rush
  • He picked Madrigal and Subdivisions for the first UTC because he loves them and they’re easy to sing. “I don’t know if I could hit some of those crazy high notes Geddy rocked in the 70s”
  • He thinks Witch Hunt would be a very good song to cover.  Such a great song and so relevant right now! “Quick to judge, quick to anger, slow to understand”
  • His favorite song to try and fail at on bass is YYZ
  • “ I don’t think there’s any conscious decision to have Rush influence my music, but I’ve spent so many thousands of hours listening to them, it’s inevitable that it just comes out naturally”
  •  His favorite album to work on in his career is what he’s doing now (UTC2 and Cool Patrol) because “we’ve grown so much and finally have a full band!”
  • He doesn’t mind when fans are super nervous to meet him. He was nervous when he met Rush, too, “so it’s just the circle of life!”
  • He chose bass because it was different and fun to find submelodies.
  • Q:  Hey Dan! I’m looking forward to Under the Covers 2. I started writing comedy music and NSP is one of my main inspirations. What do you wish you knew about writing comedy music earlier on in your career?
    A:  Nothing in particular really. You learn and you go and there’s no substitute for just doing something over a long period of time. Of course there are things I would change about our early work, but it was the best we could do at the time so I have to be cool with it.
  • “I listened to so much of them [Rush] growing up that it’s in my blood and unconsciously shapes the way I approach music”
  • His second favorite album flops between Hemispheres, Caress of Steel, and 2112 on any given day
  • His favorite story/message told in a Rush song→ Story is Fountain of Lameneth because it’s a beautiful summary of life. Message is Spirit of Radio where they talk about the importance of integrity in art.
  • His biggest influences are Pink Floyd, Led Zeppelin, Genesis, Nirvana, Spacehog, and Gentle Giant (to name a few)
  • He does his best on his Rickenbacker bass but it’s just for fun. “My band already has a bass player that’s better than I’ll ever be!”

“Thanks everyone! I have to go see my family now otherwise I’d stay longer. Sorry I couldn’t get to everyone’s questions but I had a lot of fun. I’d be happy to do this again in the future!
You being fans of mine means the world to me and I promise to keep making comedy and music for a long time! Thanks to you, the moderators who set this up, and of course Rush for being an awesome and inspirational force!”

So yeah reminder that not all relationships in a polycule need to be equal and equivalent! Just because you love somebody doesn’t mean you automatically love all their significant others.  

I love my girlfriend, I’m friends with her wife, and I appreciate the humanity of her husband, but he shares an unfortunate physical resemblance to someone who hurt me so while I appreciate that he is a Good Dude who lends me books and looks after the woman I love, I have a hard time feeling super chummy with him. And he’s cool with that!  We share the metamour bond of “you make someone I love very happy, and that makes you cool” but like… no way am I about to be as physically or emotionally close with him as I am with my girlfriend.

I know fandom loves their equilateral triangle OT3s, but if you don’t feel like that’s how it would go? It doesn’t need to go that way.

Seems like Ishida is associating Hide with equilibrium through the formation of an equilateral triangle with three circles

I wrote a post a few days ago talking about how the name Scarecrow could be alluding to Hide cooperating with both species, being perhaps the triangle of the image an allusion to this fact.

The meaning to which the equilateral triangle is associated, is also associated with number 3 and the tarot card The Empress:

“It represents the synthesis and harmony between spirit and mind, balance and emotion”

On the other hand, a triangle whose tip looks at the sky represents the masculine gender.

Therefore, both the triangle and the number 3 that forms it, are symbol of balance, which could be alluding to the actions and intentions of Hide during the first manga and :re.

anonymous asked:

hey, congrats on the gre score! any tips you can give us on studying since you did so well? i'm finding memorizing words rly hard. tips for each section please! & how was the timing? rushed?

Thank you very much! And sure, I’ll do my best, although I think studying depends wildly from person to person–for example, many of the grad schools I’m applying for explicitly state they don’t consider your math score, so I didn’t spend much time prepping for that. Timing is also quite subjective, because while I have time to check my verbal answers, I always have to guess on 2+ quantitative problems because I’m not that quick with sums.

First, invest in a practice book. I highly recommend Manhattan Prep’s 5lb Book of GRE Practice Problems, which is only $12 on Amazon right now. Not only is it as ridiculously expansive as the name suggests, it includes a diagnostic test, 9+ fully outlined and explained essays, maybe one thousand vocab words, and a solution explanation for each math problem, broken down by type (ie probability, triangles, sets, etc). In addition, a quick google search will pull up free online practice tests (three from ETS, the actual GRE-makers) which mimic the computerized test exactly–and they’re free, so why not at least power through a few sections?

Now, more specifically:

The Written: No one wants to spend 30 minutes writing a practice essay, but at least try to do a few full, typed essays from practice prompts (preferably from a source that also includes full 4, 5, and 6 essays to measure yourself again). Type it somewhere without spellcheck, because you don’t want to be like me and realize during the exam that you can’t remember if millennial has two Ns. Because I feel confident in my writing skills, I usually only outlined the essays for my practice test, but this allowed me to compare my line of reasoning to the examples given. Standardized testing demands a very particular type of logic. If you want to be especially thorough, you could mark up an example 3 or 4 essay with thoughts on what could have made it a 5 or 6–by learning to efficiently recognize others’ shortcomings, you may be better equipped to see your own.

The Verbal: Flashcards. All day, every day, until you despise the English language (but also start to recognize your expanded vocabulary in the wild–just the other day I heard both laconic and taciturn on Buffy). Manhattan prep has online flashcards, but I went ahead and made a huge set on Quizlet (because I love making flashcards). Quizlet allows you to star words you have a lot of trouble on. I found that 200+ words quickly dropped from my list, and I began to recognize others showing up repeatedly on practice tests, which helped me gauge what was really important to know. (Flue? Probably not going to come up. Quixotic? Most definitely.) On any of the passage summary readings that sound like trick questions, I write out what the question/answer is saying in my own words, along with any unspoken assumptions. This saved 5+ points on the test, because a lot of the questions are purposefully written with assumptions that logically follow–but if the question doesn’t ask you to make a conclusion, stay to what the text says to the letter. It’s not testing your ability to be a rational, practical thinker, it’s testing your ability to jump through its evil, evil word-hoops.

The Quantitative: Aside from a basic college algebra review, I haven’t taken a math class in almost six years. The math section is bittersweet: While it does rely more heavily on reasoning than on equations (and unlike math, reasoning is familiar to my day-to-day life), it still expects you to memorize obscure equations. And it forces you to use a tiny on-screen calculator with minimal functions. When the SAT is vastly kinder, you know they’re just screwing with you. (Have I mentioned how expensive this test is? Honestly, that price is a large part of the reason I was determined to get it right the first time. But I digress.) To supplement my prep books, I had a friend tutor me in concepts I’d totally forgotten, and I made a cheat sheet of formulas that the GRE excepts you to know. (Not a literal cheat sheet, GRE police–I know you’re watching me.) Some of those are as follows:

  • Quadratic equation
  • Slope of the line
  • Areas of equilateral triangles + assorted polynomials
  • Area of part of a circle
  • Standard deviation principles
  • THE GODDAMN COMBINATIONS EQUATION
  • Regular/compound interest

That’s not comprehensive, but it’s a start. Tailor it to your own needs, and decide how important math is to you/your top grad schools.

This is the part where I repeat all the cliched stuff about going in well-rested, remaining calm even when the timer flashes the 5 minute mark, and remembering you can retake it in a few weeks. Really, it’s important to remember that this is a test designed to measure skills you don’t actually need to be a smart person. Decide ahead of time the minimum scores you’ll send (check data for your intended schools/programs and national percentiles). And if you have any more GRE/academic questions, I’m absolutely open to support you as best I can!

Problem of calissons

Couple years back I met a nice math problem, which is mainly know as the problem of calissons. Given a regular hexagon with side of length n. You can fill it with rhombuses, where every rhombus is made by gluing together two equilateral triangles. Here is an example:   

As you can see the rhombuses have three different orientations. 

Theorem 1: The number of rhombuses with a given orientation is one third of the number of the rhombuses.  

It is not hard to see that we will always use $3n^2$ rhombuses to fill the hexagon. So the theorem says that we will always have $n^2$ rhombuses of each orientation. 

Proof No. 1 

This problem became known it has a nice solution. It is a “Proof Without Words”. Just look at the following picture.  

We colored the rhombuses according to their orientation. If you try to look at them in 3D, you will see some cubes, and it is easy to feel the truth of the theorem. This proof is not too rigorous though.

When I was researching this topic I found two really interesting notes on the topic from Dijkstra (who is known for Dijkstra’s algorithm). In these notes he shows that this proof obscures a more general theorem, where we need an other proof.   

If we have a figure that is covered by rhombuses with $a$ of them in one orientation, $b$ of them in an other orientation and $c$ of them in the third orientation, then we will say that $(a,b,c)$ is the frequency of the covering. So Theorem 1 says that no mater how we cover a hexagon, the frequency is the same.    

Theorem 2: If we can cover any kind of figure with the rhombuses, then the frequency is independent of the covering.

For example we the following figure always needs 8 rhombuses of each orientation:

Proof of Theorem 2.

I will show of the proof of the second theorem using the original hexagon. First divide the figure into triangles. Color the triangles black and white such that neighbors have different colors. Assume we have a covering. This way every rhombus covers one black and one white triangle. In each rhombus we can draw an arrow from the middle of the black triangle to the middle of the white triangle. For example this is what we get from the covering above. 

  And now we must use a bit more advanced math. Consider the arrows as vectors!  We will consider the sum of these vectors in two different way. 

We have three different vector, one for each orientation, denote them by $v_1,v_2$ and $v_3$. Let $(a,b,c)$ be the frequency of the covering. This means we have $a$ of the $v_1$ vector, $v$ of the $v_3$ vector and $c$ of the $v_3$ vector. 

We can now calculate the sum of these vectors: $av_1+bv_2+cv_3$. 

 We need an other way to consider this sum. The important thing here is to show that the sum does not depend on the covering. Each vector goes from a black triangle to a white triangle. This means that we can write each vector as a difference of two vectors, one pointing from zero to the center of the white triangle minus one pointing to the center of the black triangle. If we consider the sum of all vectors now, we can write everything as a difference. So the sum of the vector is just the sum of vectors pointing to the white triangles minus the sum of vectors pointing to the black triangles. This quantity does not depend on the covering, just on the figure. We will call this vector $v_{sum}$.

So we have  $av_1+bv_2+cv_3=v_{sum}$. Also $a+b+c$ is a fixed number since the number of all rhombuses can be calculated from the area of the figure. 

Since we are in two dimensions, the equation  $av_1+bv_2+cv_3=v_{sum}$ is actually two equations, one for the x-coordinates and one for the y-coordinates. 

So all together we have three equations for $a,b$ and $c$ that has to be true independently of the covering. From linear algebra we know that three equation uniquely determines three variables. So $(a,b,c)$ is independent of the covering. QED 

Proof  No. 2.

Once I posted the original problem in a math competition. Most of the students gave the following solution. We can draw $n$ paths from one side of the hexagon to the opposite site using two orientations. 


In each path we must have $2n$ rhombuses. All rhombuses outside the paths belong to the third orientation. So we have $3n^2-2n\cdot n=n^2$ rhombuses in the third orientation. By symmetric arguments we have $n^2$ rhombuses in each orientation.         

Oh and my logo was inspired by all of this :)    

Okay so I spend so much time reading One Direction fanfic that I feel the need to organize all my best readings in this post and let here for reference. This probably will be so f*cking long cause I read most of the ships. So here we go.

Honorable Mentions: 

I’ll update this post later to add more I’m tired now LMAO

Triangle Of The Art

The triangle of  the art also referred to as a solomonic triangle, and a triangle of evocation is a equilateral triangular symbol, that usually also has words, or symbols of power written around it with a circle inside it, that is used as a tool in envoking entities. The traditional triangles of the arts had three names of God around the sides of the triangles, these names were Tetragrammaton, Primeumaton, and Anaphaxeton, and also had the name Michael split up into three parts (Mi - Cha - El) in the center spots between the triangle, and the circle. This symbol is placed outside the confines of a practitioner’s magick circle, and is a place for the practitioner to envoke an entity to. This is usually done did the act of putting a sigil in the center circle in order to call the entity there. Sometimes triangles of arts are made into scrying mirrors, so that the entity can be envoked into the black mirror. The triangle of the art is used in order to keep the entity in a confined place why you are communicating, and interacting with them. It is a way of keeping yourself safe while you are summoning entities, and will not give them free reign of the environment that they have been summoned into. The triangle of the art was first given to us in the Lesser key of Solomon, and has become a very popular part of a ceremonial magician’s practice when they are trying to envoke entities in order to work with them, and to allow them to aid them in their practices. They are one of the most important tools right next to the magickal circle that can be incredibly helpful when placed in the correct situations. It is always good to have an understanding of what the triangle of the art is, so if you are ever trying to envoke something you can have some distance between you, and what you are trying to summon.

anonymous asked:

Got any stuckony recs?

YES! Absolutely yes!

My apologies this took until today. I had a busy weekend and had to rest a lot, but I’m here now. :) Also, I don’t have my old bookmarks list (long story) so I’m having to go through and look for my favorites, old and new. Long process! (This took me over four hours, though I was doing a few other things as well. Still, I’m a bit of a slowpoke.)

See below the cut for some of my favorites, and don’t forget to leave kudos and comments (even just an “I liked this!”) for the authors, to let them know their hard work is appreciated!

(If someone knows an author’s Tumblr handle, let me know or tag them so that I can edit this rec list and tag them appropriately!)

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I’m never getting over this

So I keep thinking that I already over analyzed the entire poem of Bill by the Axolotol that I WONT be able to find something worth fussing about but nooo

So

Sixty degrees that come in threes 

Which basically establishes Bill as an Equilateral Triangle who, in Flatland society class hierarchy, is the equivalent to our fast food workers. So he’s practically not that special relative to the fact he’s often portrayed as overpowered. 

SAW his own dimension burn.

Now I’m incredibly BOTHERED by this. Saw. He just saw his own dimension burn. Why? The poem could’ve gone well with it saying “Made his own dimension burn,” implying that Bill actively participated in the destruction of his own world, but no. It’s SAW, meaning he just watched his dimension reduced to ashes. 

Misses home and can’t return 

This is interesting because it’s in the damn present tense which means that he still does miss his home and he yearns for the comforts it once held for him. 

Says he’s happy, he’s a liar

DEPRESSED BILL: CONFIRMED. I mean, why would he lie about being happy if he’s not depressed, amirite? He lies not only to himself about being fine, but to others, and maybe to us as well. I mean, he does have limited screen time so who knows what he’s doing behind the scenes.

I have a headcanon that Bill’s ploy to conquer Gravity Falls is just him committing suicide

Blame the arson for the fire

Now why would he blame the arson for the fire when he insisted that he himself liberated his dimension. it just doesn’t add up.

This poem has been driving me crazy. Send help

Moonlighting Part 1: TAZ Pre-Finale Relisten Recap

Okay gang I think I underestimated how long TAZ is.  Don’t get me wrong, I am still fully committed to this relisten and totally prepared to die on this hill, but I’m just saying that this is definitely a less hill more mountain situation. 

  • Shit That Probably Isn’t Technically Canon Anymore But Remains Canon In My Soul:
    • Taako’s gogurt curse
    • Taako’s fucking key lime gogurt curse.
    • Listen.  Listen, I will fight you for this one.  I will fight you for the canonization of a key lime gogurt curse
  • Avi: “Oh, I have an uncle in Phandalin!”  Taako: “Correction, darling – you HAD an uncle in Phandalin.”
    • Avi: “Well he wasn’t my favorite uncle, or anything–”   Magnus: “Good news!”
  • Clint calls the grass on the moon ‘astroturf’ and Griffin completely fucking misses it
  • “Listen, if I don’t tell you there’s gogurt in your flask then you don’t tell me how I feel about my hind end.”  – Justin to Griffin, regarding his D&D character’s opinion of his own ass.
  • Griffin’s character voices rating – Johann:
    • Full disclosure: the epiphany that Johann is actually fantasy Eeyore just hit me like a fucking truck.
    • I’ve just ruined every single scene he’s in for myself.  Please send help.

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The Bookkeeper

Title: The Bookkeeper
Summary: The first time Randall sees the Cipher boys, it’s in his bookstore. 
Characters: Bill Cipher, OCs
Rating: K 

This one is for @phin-dicated, who requested some more of Bill and Liam. Hope you like it, and thanks a lot for your charity donation!

***

The first time Randall sees the Cipher boys, it’s in his bookstore. And, truth be told, he hears them before he sees them.

More accurately, he hears Bill. Liam is silent - Randall will understand soon that he often is when outside the safety of his home, and for good reason - but Bill talks enough for both, voice gratingly loud and spewing out words so fast it’s almost hard to follow him. As he’ll learn later, Bill has a lot to say and the utter certainty everybody should listen to it.

“C’mon, don’t be dumb! You want a book so we’re gonna get a book and then we’re buying candy!”

Randall, who has been taking advantage of a few quiet minutes without clients to take a better look at the inventory, looks up to see two Triangles walking in. One is just a child, no older than maybe five or six; the other looks like he could be twelve or thirteen at most, but it’s not his age Randall focuses on. Very much unlike the smaller Triangle, who is perfectly Equilateral like himself, the boy is very clearly Irregular, to a degree that makes a death bell toll in Randall’s mind.

He will not pass the Inspection.

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anonymous asked:

post/159918431327/more-team-gai-bc-im-deeply-attached-to-them whats up witht he nazi symbol on one of the characters?

I think i need to clear up some things because for some reason this isnt public knowledge? 

Taken from wikipedia:
The Swastika (also known outside the Indian subcontinent as the Hakenkreuz, gammadion cross, cross cramponnée, croix gammée, fylfot, or tetraskelion) (as a character 卐 or 卍) is an ancient religious symbol originating from the Indian subcontinent, that generally takes the form of an equilateral cross with four legs each bent at 90 degrees.[1][2] It is considered to be a sacred and auspicious symbol in Hinduism, Buddhism, and Jainism and dates back at least 11,000 years

Im also gonna put a link to the wikipage, so that you guys can be more informed: https://en.wikipedia.org/wiki/Swastika

Taken from http://www.japantimes.co.jp/community/2016/02/10/voices/educate-people-keep-manji-%E5%8D%8D-japans-maps/#.WQBR31PytE4

The symbol, as most of you likely already know, has been around for thousands of years, and out of those thousands of years, for only roughly 70 or 80 has it been associated with anything negative. It’s also oriented a bit differently. The Nazi version usually sits in a diamond shape, while the manji version is a square. Some will also say they face opposite directions, but that’s not altogether true: Japanese Buddhism has both a right-facing and left-facing manji.

and obviously you and some of the people who commented didnt read the manga so im gonna put a page from the manga right here. 

Now if your question is why would neji have that symbol on his forehead then maybe you should actually watch the show or read the comic. 

please make sure you are well informed and do your research first before saying anything regrettable, you’re just shaming yourself for not knowing a well known fact. 

If there are mathematicians in the animal kingdom, honey bees may top the list. Here is why. To store their honey, bees use a structure of hexagonal cells called a honeycomb. Why hexagons? Turns out bees did their homework and hexagons are actually the optimal cell shape.

The bees were all like, “We want to store a crap-ton of honey and our idea is to use a bunch of cells.” Their first idea was, naturally, to use circular cells, but circles do not fit nicely side-by-side. “We would lose some space in between the cells if we used a circular array of cells.” (That’s the bees talking.) One bee said, “How about square cells? They fit nicely side-by-side.” Another suggested using equilateral triangles. A particularly clever bee said, “Assume polygons of a fixed maximal width, then the more sides a polygon has, the larger the area—and the more the honey we can fit into the cell.” The bees deliberated: “Okay, so square cells will fit more honey than triangular cells, but what about pentagons, hexagons, octagons, or some other n-gons?” The more sides, the more area, but which ones fit nicely side-by-side? The bees quickly realized they would have to use math to solve this problem.

A bunch of polygons fitting nicely side-by-side is called a tessellation (see 3 examples above). Since bees have an affinity for aesthetics and want to keep things simple, they decided to stick to tessellations of regular polygons. So, the problem for the bees was to find the regular polygon with the greatest number of sides that would form a tessellation. The bees did a quick proof and found the hexagon (6-sided polygon) to be that shape.

Why hexagons? The idea the bees realized is that for the regular polygons to fit perfectly side-by-side the measure of the polygons’ interior angles must divide 360 degrees. Regular triangles work since 360/60=6 and squares work since 360/90=4 (notice in the tessellations above we see that 6 triangles fit perfectly around each vertex and 4 squares fit around each vertex). We can use the formula above to check interior angles of all regular n-gons and prove that the largest value of n such that 180x(n-2)/n divides 360 is n=6.

And there you have it. Bees really put a lot of thought into their honeycombs!

Axolotl’s Poem: Full Analysis

Well, this post took a long time to make. Going by the general theme of my blog, you would’ve thought that this would be one of the first posts I made. But better late than never.

Sixty degrees that come in threes

This is one of the more straightforward lines. It simply confirms that Bill is actually an equilateral triangle, not an isosceles as he was called a couple times in-show. This actually does have some significance, as when Bill was asked where he came from in his AMA, he replied “EDWIN ABBOTT ABBOTT HAD THE RIGHT IDEA!” Edwin Abbott Abbott was the author of Flatland, a book about a 2D world, with a shape hierarchy. The more sides you have, the higher up in society you are, so Isosceles triangles were lower class, while Equilaterals were middle class. 

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anonymous asked:

The first time Grog saw Pike, she was a circle. There was this halo around her, all golden and brighter than a bonfire. It hurt him to look at her, but she healed the rest of his hurt things pretty well so he didn’t mind. She kept being a circle, always coming back to them. He saw the wide arc of her maul in the air and the pendant around her neck. Sometimes, Grog thought she might be the little bit left over around when you went three times across. The last linking bit. (1/5)

The twins and Trinket, they were a triangle. The sort changed sometimes. For a while, they were obtuse. Vax and Vex worked together alright on the battlefield. Grog grew to learn he could rely on them to take the baddies down a peg before he got his hands on them, but Trinket was far off to the left, always close to Vex and never much help. Then they got tall and acute. Vex and Trinket crowding together while Vax got further and further away. Grog was sure they’d be equilateral someday. (2/5)

Keyleth was a rhombus. Solid on some ends. Certainly looked solid, but Grog was always just a little afraid she might topple over if her slant went too far. Her sizes and angles changed too much all the time. Early on, he’d worried about every animal he saw on the battlefield. There were so many shapes she could take. It made Grog’s head hurt, but he learned them one by one. Still, every squirrel was suspect. (3/5)

Scanlan started as a square, but he’d widened over time. The more songs, the more conversations, the more voices his friend brought out, the more sides Grog saw around him. He was a nonagon now, but Grog wondered sometimes if he was trying to get to a circle, same as Pike. He didn’t think it would work. Scanlan worked himself in and out of too many corners. (4/5)

Percy was a line. Straight on down from far away to whatever he chose to hit. Grog admired that in him. Sometimes he couldn’t see the beginning or the end of a line. Roads were one. Rivers were another, though he sometimes tried to put words to just what shape rivers were when they didn’t stay straight as he might prefer, but he was comforted knowing a line always landed somewhere. Pow. (5/5)

GROG AND SHAPES I’M!!!!!!