#right triangles

[storymaker14]

Bizzaro Right Triangles

Okay, we all know that, if you have a right triangle with sides a and b and hypotenuse c, that means a^2 + b^2 = c^2, right?

So you can have 3-4-5 right triangles, and 5-12-13, and 7-24-25, etc.

And technically, a 1-i-0 right triangle follows this pattern, too, which has made the rounds in the math meme community.

But I found something better and weirder. I found a whole family of better and weirder.

I'm gonna skip a bit of the beginning and start here:

cos x = (e^ix + e^-ix) / 2

That follows from Euler's formula; if you need a walkthrough, I can provide, but I want to start with the above. So then, it follows that:

cos (i*lnx) = (x + 1/x)/2 = (x^2 + 1)/2x

Which means that an angle of (i*lnx) radians in a right triangle will have an adjacent side of (x^2 + 1) and a hypotenuse of 2x. By Pythagoras, then, the opposite side must be (x^2 - 1)*i.

But please note: if x is odd, then the lengths of all three sides will be even, and thus can be divided by 2.

Which means -- are you holding onto your hats and shoes?!? -- means that the proportions of the right triangle with angle i*ln(3) radians... are 5-4i-3.

And for i*ln(5) radians, they are 13-12i-5.

And so forth, with all the familiar Pythagorean triplets sqrt(2n+1)-n-(n+1) showing up, just with one side imaginary and the hypotenuse and remaining side swapped -- so, (n+1)-n*i-sqrt(2n+1). They still fulfill Pythagoras, every single one.

Which I think is, pardon my directness, fucking terrific. But just as a little bonus, please note that this means the triangle with angle i*ln(2) radians, the proportions are 5-3i-4, which is just delightful IMHO.

[ravewing]

ever since i learned what a hypotenuse was i have always walked the hypotenuse instead of in a right angle and i think its saved me like a million steps

[math-journal2]

<— Unit 31 — Unit 32 — Unit 33 —>

Unit 32: Trig Basics

Part 1 —>

Related: Trig Review

. . . . .

<— Week 1: Page 5 —>

Standard Notation

Start: left most angle, go clockwise starting with alpha, then beta, last gamma

. . .

Solving with Ratios

Good to know

Missing an Angle?

For Exact Values

Weapon for Word Problem

[factomaniac]

Math Madness #3: The Pythagorean Theorem Wasn’t Discovered by Pythagoras, and Other Facts About

a² + b² = c²

Thursday, 13 June 2024, Chicagoland, IL

Mesopotamian mathematics historians have established that, despite the name it is known by today, the Pythagorean Theorem was widely known in the Old Babylonian Empire existing from the 20th to the 16th Centuries BC, over 1,000 years before Pythagoras of Samos, the theorem’s namesake, was even born around the middle of the 6th Century BC in either the Greek island of Samos, or the city of Tyre in present-day Lebanon (Wikipedia “Pythogorean Theorem” 2024, Neugebauer 1969, Friberg 1981, Hørup 1998, Robson 2008, Wikipedia “Pythagoras” 2024, Porphyry 2014).

The Pythagorean Theorem asserts that in any right triangle, a triangle containing one right angle (an angle of 90°–ninety degrees–or π/2 radians), the area of a square where each side is the length of the triangle’s hypotenuse (the side of the triangle opposite the right angle) is equal to the sum of the areas of two squares where each side of each square is the length of each of the other sides of the triangle, which are perpendicular to one another.

By the way, the word hypotenuse comes from French, by way of Latin hypotenusa, ultimately originating from the Ancient Greek ἡ ὑποτείνουσα (“hē hypoteínousa”), meaning “the [side] stretched below [the right angle]” because, in Ancient Greek times, right triangles tended to be drawn with the right angle pointing upward, therefore, the hypotenuse appeared to be stretched below the other two sides forming the right angle (Wikipedia “Hypotenuse” 2024, Liddle, Scott, Jones 1940/Perseus Project 2024, Online Etymology Dictionary “Hypotenuse” etymonline.com 2024).

Although typically this theorem is typically stated as something like “the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides,” and written algebraically as a² + b² = c² today, in ancient times, it was originally construed geometrically with actual squares on the sides of the right triangle. Remarkably, thousands of geometric and algebraic proofs of the Pythagorean Theorem exist. One very simple modern geometric proof is below.

Besides serving visually as a proof of the Pythagorean Theorem, the above illustration also makes a good jumping off point for an algebraic proof of it as well. Referring to the second frame of the illustration with the c² square in the center, notice that the sides of the entire frame each measure a + b. Therefore, its area can be found by squaring this expression: (a + b)². Also notice how the area of the frame can be found by adding the area of the c² square itself to the areas of each of the four triangles, ½ab: c² + 4(½ab). The Pythagorean Theorem then pops out just by writing an equation that sets these two expressions as equal to one another.

(a + b)² = c² + 4(½ ab) → a² + 2ab + b² = c² + 2ab → a² + b² = c²

By the way, the equation of the Pythagorean Theorem can also be rearranged to find the length of one of the sides of a right triangle while knowing the length of the other two sides:

∵ a² + b² = c², c = √(a² + b²), a = √(c² - b²), b = √(c² - a²) (∵ symbolizes "because" and √(x) is "the square root of x").

Originally, Euclid proved this theorem by taking a right triangle with the right angle pointed upward and the hypotenuse as the base at the bottom and then, as I mentioned above, making squares off of all three sides of the triangle whose sides are the length of the respective triangle sides they are formed from.

Although I’m presenting what is essentially Euclid’s proof of the Pythagorean Theorem, I’m going to put my own spin on it in that I’m going to refer to the points, lines, and shapes in a way very different from how it is done in classical geometry. Because one thing I recall that I found confusing when learning classical geometry in primary and secondary school was the reference to shapes by the points that form their corners, such as referring to a shape as Triangle ABC—or worse △ABC—because it contains the points A, B, and C, or saying that one shape is congruent to another because they contain proportional sides that meet at points forming the same angles. I’ve always found the semantics and vocabulary of classical geometry baffling compared to, say, algebra or even calculus. Therefore, points will have miniscule Greek letters, the lengths of lines will be referred to using miniscule Roman letters, and the shapes are depicted with Roman numerals. (Please note that I will occasionally use traditional notation as well as my own notation in order to avoid confusion.)

First, we have Triangle I in dark grey (traditionally defined by points ϙ [qoppa, pronounced “COPE-ah”, an archaic Greek letter now only used as a numeral], α, and β, or symbolically as △ϙαβ) with sides of lengths a, b, and c, a is the length of the side of the triangle on the right side of the right angle, b is the length of the side on the right angle’s left side, and c is the length of the hypotenuse, in the style of the ancient Greeks, on the bottom of the triangle forming its base. Next, there are the squares in white (Squares II [□ϙηιβ], III [□αφγϙ], and IV [□βαδε]—see the uniting brace on the bottom of Square IV) formed from each of Triangle I’s sides, all of whose sides, by definition, are all the same exact length, and the same length as each of the sides of Triangle I from which they are formed.

Further, taking point ϙ, let’s draw a line segment straight down through Triangle I through point κ and all the way down through Square IV (□βαδε) to point λ, dividing the square into two rectangles, Rectangles V (▯καδλ) and VI (▯βκλε). Now, also in lighter grey, we have two larger non-right triangles, Triangles VII (△φαβ) and VIII (△ϙαδ), by drawing line segments from point φ to point β (line segment φβ, or φβ) and from point ϙ to point δ (ϙδ). Finally, also in lighter grey, we draw line segments from point α to point ι (αι) and from point ϙ to point ε (ϙε) to create another pair of smaller non-right triangles, Triangles IX (△αβι) and X (△ϙβε). (Usually, a line is placed above the endpoint letters to signify a line segment, but that type of font is not easily available on Tumblr, so I used crossout print instead.)

Note that the upper left corner of Square IV (angle βαδ, or ∠βαδ) and the lower left corner of Square III (∠φαϙ) both meet at point α and because they are both corners of squares are both right angles. So because Triangles VII and VIII have a common corner at point α, their angles both measure 90º (π/2 radians) plus the angle of the corner of Triangle I at point α (∠ϙαβ) and, therefore, both angles of the corners of Triangles VII (∠φαβ) and VIII (∠ϙαδ) that meet at point α are equal.

Because the side on the left of Triangle VII (φα) and the side on the top side of Triangle VIII (αϙ) both also lie on sides of Square III and both are of length b, and the bottom side of Triangle VII and the left side of Triangle VIII lie on sides of Square IV and both measure length c, and the angles of both triangles that meet at point α are equal, both Triangles VII and VIII are congruent to one another in that all their sides and all their angles are equal, and, therefore, we can assign the letter d to the length of the diagonal sides of both triangles.

Because the line starting at point ϙ going down to point κ and continuing down to point λ (ϙκλ) is a straight line parallel to left side of Square IV (αδ), that is (ϙκλ ‖ αδ), which is the same as the left side of Triangle VIII, measuring length c, and the top side of Rectangle V (αϙ) is parallel to its bottom side (δλ, or αϙ ‖ δλ), both each measuring length f, which is the height of Triangle VIII, therefore its area (½ cf) is half the area of Rectangle V (cf). By the way, if you would like to know the derivation of the triangle area formula, see my previous blog entry.

Further, since the line segment starting at point γ, on the right side of Square III, going to point ϙ and continuing to point β (γϙβ) is a straight line segment parallel to the left side of Square III (φα, or γϙβ ‖ φα), Triangle VII with a base of b and a height of c is half the area of Square III (½ bc = ½ b²), and is therefore, Triangle VIII with a base of c and a height of f is also half the area of Square III (½ cf = ½ b²), which is also half the area of Rectangle V (½ cf = ½ b²). Therefore, the areas of Square III and Rectangle V are equal (b² = cf).

Finally, since the line segment starting at point α, on the left side of Triangle I, going to point ϙ and continuing to point η, on the top side of Square II is a straight line segment parallel to the bottom of Square II (αϙη ‖ βι), Triangle IX has base and height a and, therefore, area ½ a², and the line segment starting at point ϙ, going through point κ and ending at point λ is a straight line segment parallel to the right side of Rectangle VI, Triangle X has base c and height g and, therefore, area ½ cg, and both Triangles IX and X are congruent to each other and, therefore, their areas are equal (½ a² = ½ cg) and, therefore, Triangle X is half of the area of Rectangle VI, and, therefore, both Square II and Rectangle VI have the same area (a² = cg). Therefore, Square IV has the same area as the sum of the areas of Squares II and III (a² + b² = c²). QED

But, as I said, there are thousands of proofs of the Pythagorean Theorem. Next time, we will cover one such proof recently discovered by two high school girls.

Works Cited:

Friberg, Jöran. "Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean Triples, and the Babylonian Triangle Parameter Equations." Historia Mathematica 8 (1981): 277-318.

Høyrup, Jens. "Pythagorean 'Rule' and 'Theorem' – Mirror of the Relation Between Babylonian and Greek Mathematics." Homepage of Jens Høyrup of Roskilde University. Jens Høyrup. 1999. http://akira.ruc.dk/~jensh/Publications/Pythrule.pdf (accessed 2024).

Neugebauer, Otto. The exact sciences in antiquity (2nd ed.). Mineola, NY: Courier Dover Publications., 1969.

Wikipedia "Pythagoras" http://en.wikipedia.org/wiki/Pythagoras , (accessed 2024).

Wikipedia "Pythagorean Theorem" http://en.wikipedia.org/wiki/Pythagorean_theorem , (accessed 2024).

Coming soon: my new blog about Philosophy, Mythology, Religion, and Politics, also in Spanish and English. It's the Aristotelian, and it's available at the-aristotelian.tumblr.com

[wingedpikmin]

Helping flood people's dashes with transition timelines!

Left: 4 years before HRT

Right: 3 years after HRT

Transition saved her miserable ass. Love and support transwomen today.

[littlemoonflowr]

is this anything

[h4unted-d4rling]

I would be that one person in the town during Weirdmageddon where I’m like “is it just me or is that Bill Cipher guy kinda hot?” And everyone would look at me like

[candycatfalls]

just a silly guy! who's definitely not wanted for mass murder! so stop asking!

[poorly-drawn-mdzs]

ghost horses

GHORSES

[captainyaps]

I never really shipped Odydio until last night 12am thinking about how Athena favoured Odysseus for his wits while for Diomedes his strength (though the two did not lack in the other quality, it just wasn’t their standout thing) and so together they are basically Athena Incarnate. Like, the unstoppable war machine Athena. All they need is someone who’s good at weaving and- oh would look at who it is! PENELOPE. ODYSSEUS’S WIFE. PENELOPE.

So yeah. Odydiolope (or some other better ship name) is literally Athena incarnate, the Athena trinity, the holy-god-fucking-shit-we’re-so-screwed duo on the battlefield, and the holy-god-fucking-shit-we’re-so-screwed girlboss trio at like… any other domestic setting.

[iheartbookbran]

My favorite part about the Debling plot was Penelope and Cressida doing the most over there top cat-fighting in order to win him over, not because they genuinely liked the guy, but because they wanted to escape their families and also lowkey just hate each other.

[missinglaterals]

The epicness of 'just some guy' fashion

(She/Her)

[darcyolsson]

avery explicitly using the word 'throuple' to refer to the three of them, ody3 hot tub cuddling sesh, captain massey referring to said cuddling as '...whatever this is', a joke about the threesome, a 'you can love two people at once' speech delivered by quinn fabray playing a hallucinated dead wife (also in a hot tub), avery sneaking both max and tristan into the operating room for her surgery... i can't quite believe it but a throuple endgame on doctor odyssey is actually fully within reach

[urthefartmike]

anyone who says byler is portraying unrequited love has never written or read an actual unrequited love trope in their life because this is NOT how you do that!

it would not drag out for this long, it would not be made such a big point of, and mike wheeler would not be taking glances at will’s lips every interaction.

let’s be real now guys.

[oneroomjestershow]

when

[cultofcipher]

Bill Cipher thoughts (BoB Spoilers Ahead)

I'm really sitting on how Bill's displayed so much of himself indirectly in the BoB. How during the Love section he denies having exes, marking them out. How said exes show up SEVERAL times scratched out or are regarded with this bitterness of someone who did NOT do the breaking up part. Bill got dumped. Every time. And is desperately trying to bury his feelings.

And that's something I think the Book of Bill really highlights in a way. The fact that Bill has feelings. That deep down he's a broken triangle. It's all over the book's writing. Him pointing out how to use denial and rationalization and other bad coping mechanisms to basically ignore and lie to himself (and show us how to do it) and basically convince himself that he is as heartless as he tries to be. Him avoiding his exes. The tone he uses and the avoidance really giving the "I don't handle breakups well and I'm still petty about it". Him constantly telling himself that he's fine. He's not fine. Him crying over Ford leaving and getting wasted. Him being bitter about the henchmaniacs not calling. His regret over what happened to his world. His loneliness. GOD his loneliness. His self-hatred. His scathing remark about definitely NOT having some tragic backstory that humanizes him and how he's not an "I can fix him case". Calling himself a monster. His longing for home. The "Last one breathing". The "I tried to change the past". The "my hands shaking, as I realized I could never undo the". The "until there was no one left but me, covered in blood, alone in the universe". The goddamn "I don't want to die alone" Valentine's card. The last few pages. Just, the last few pages. That isolation, his pained "I'M FINE". The almost sad plea for someone to let him out.

Bill cares. He's fucked up, unstable, violent. But he does care about people he gets along with and he feels understand him. For every "I'm just playing the bit" and using people with nice gestures, I think a fraction of that is somewhat genuine. And he hates it. He hates his own vulnerability. He hates his lack of apathy. He's denying himself his own emotions constantly under so many layers of distractions, eldritch horrors, and repression. He can't think about home, about failure, about how every relationship he's ever had, platonically or otherwise, ended. And it wasn't on his terms.

Him talking about/to his mom when he's drunk. How his mom called him Billy as a kid. How his home life sounded simple. How Bill as an individual is anything BUT simple. And how his drunken state holds such fondness for that simplicity, yet it was suffocating. How he would've broken free eventually, inevitably, because he knew that's who he was. It's his nature. He was destined for more.

How it cost him everything.

How he's constantly chasing insanity like it's a drug. Like he needs the power trip to stay high. To not think too hard. To drown out his emotions and his self-reflections and everything he hates about himself.

How in Gravity Falls he still tried to get Ford to side with him after everything, cause that was his vulnerability showing, for the slightest glimpse of a moment. Cause he doesn't want to do it alone. Him reaching out to the reader in his book, because he doesn't want to do it alone. Can't do it alone. Even when he eventually betrays that person, I think him offering Ford that cushy spot alongside his henchmaniacs makes me think that yeah, Bill actually would've upheld his end of the deal.

He thinks he wants multiversal domination. He thinks Weirdmageddon is his Magnum Oppus. His purpose. But he's so lost. If he ever does get what he wants, he won't know what to do with himself. He'll be faced with the "Now what?". He'll hit the end of the road and realize how unsatisfying it is. How this isn't what he wanted.

How lonely it is to be God.

I think the Axolotl sees that in Bill. It's why he doesn't try to destroy him or attack him or anything. He sees that inner self of Bill. Sees him for what he really is. Someone who needs a LOT of therapy, a true, honest to goodness friend or partner in his life, and maybe a more sustainable life purpose or hobby. He has so much potential and in a way his pursuit of power, rather than being an actualization of his abilities, is a waste of them, because it gets him nowhere.

And he needs help, even if he doesn't think he does. He's a depressed alcoholic frat boy trying to drown his misery in a way that hurts and kills worlds. He's a girlfailure, a bisexual/pansexual disaster (he's at LEAST canonically bisexual or at MOST canonically pan cause this guy has dated both ways).

Bill's book is so incredibly amazing for what it is. All the lies, all the unrealiable narrator parts of Bill's facades and flaws and him being himself and all of his genuine thoughts and feelings bleeding through the lines and showing themselves but only in a way that you can really understand if you understand him and can tell when he's lying and when he's not. To see the real parts of him, and everything else. This book was perfect, and it was perfectly imperfectly him. This truly is Bill's book. It's so him in such a raw and genuine yet dishonest way. I'm gonna cherish this damn book forever.