Superhero Triangles - Numberphile

Heronian triangles and other fascinating things, featuring Dr James Grime.
Get your Superhero Triangle T-Shirt and other stuff: bit.ly/Super_Tri
More links & stuff in full description below ↓↓↓
More videos with Dr Grime: bit.ly/grimevideos
Dr James Grime: www.singingbanana.com
PAPERS: A unique pair of triangles - arxiv.org/abs/1809.09936
Heronian Triangles Whose Areas Are Integer Multiples of Their Perimeters - forumgeom.fau.edu/FG2007volume...
Discuss this video on Brady's subreddit: redd.it/ev4o6o
The Triangle Highway: • Triangles have a Magic...
Too Many Triangles: • Too Many Triangles - N...
All Triangles are Equilateral: • All Triangles are Equi...
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoundation.org/outr...
And support from Math For America - www.mathforamerica.org/
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[@Me0fCourse]

Recreating the horrible misshapen triangles on the animation made me burst out laughing. Sometimes, it's the little details that makes a difference.

[@Dabbleatory]

He really Parker Squared them.

[@TheGanamaster]

HEY, they are the lovely ones, don´t mock them hahahaha

[@magicalpencil]

I was laughing so hard when the numbers actually appeared next to the janky triangles

[@laurihei]

So we got the Parker Square; is this the beginning of the Grime Triangle? A metaphor for being so amazed by maths that you wholeheartedly ignore the quality of real-world applications.

[@ffggddss]

Hey! Those are jazz triangles, man!! They're totally cool. Fred

[@emirspahic2886]

Seeing James in the Thumbnail gives me a feeling of Nostalgia of 10 Years ago and the beginnings of NP, ahhh good times...

[@michaelhird432]

Polynomial vs numberphile - greatest unsolved problem in computing

[@LowellMorgan]

The man was genetically engineered to make math videos.

[@DivyaniSharma93]

Seeing James just made my day! Numberphile is never really Numberphile without James.

[@XanderFenikkusu]

There is a P vs NP joke hiding somewhere around here...

[@galgrunfeld9954]

@@LowellMorgan my ex was attracted to him. While we were together. xD

[@BKITU]

My wife died two weeks ago. She was a huge "Hitchhiker's Guide" fan, and used 42 in some way whenever possible. She would have LOVED this video. That she also had a bit of a crush on Dr. Grime wouldn't have hurt. I've purchased the 42 Superhero Triangle shirt, to commemorate my own beloved superhero. Thank you for this.

[@kabochaVA]

Sorry for your loss...

[@whatever946]

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 43,44,45,46,47,48,49,50,51,52,53,54

[@danedickerson]

That’s beautiful

[@PC_Simo]

My condolences.

[@Triantalex]

??

[@pinicius]

"Parker Super Triangle" hahahahah Yeah, that made me laugh

[@siddharthagrawal8300]

Was gonna say the same lol

[@xpqr12345]

What we need now is Matt and James cooperating on something, so we can have Parker-Grime [something]s. That could go down in the history books!

[@98danielray]

the ch community says hello

[@E1craZ4life]

xpqr12345 They were already together in a video about dividing by zero.

[@xpqr12345]

@@E1craZ4life They have done at least one video on Matt Parker's channel, with one of them proving something (I can't remember what) with geometry, and the other proving the same thing with algebra. But I would enjoy them coming up with something together, reminiscent of the Parker Square, or the Grime Triangles, so it can be called the Parker-Grime [what-ever-it-is]. Or Grime-Parker [thingamajig].

[@avikdas4055]

I'm glad to have James Grime back. Wonderful surprise Numberphile.

[@carlosgaspar8447]

james is no square.

[@ericlol1337]

why? was he ever gone or something?

[@Triantalex]

??.

[@Thror251]

"These triangles are really interesting - you can see where my printer ran out of ink"

[@1CO1519]

LOL True story!

[@RobotProctor]

5:49 this is a 3-4-5 triangle sized up. It's a right triangle as well!

[@cockneyse]

Yes I saw that, and yes he says it's a cheat... But because it can be made from two RA triangles when actually it's one itself. Not the same thing at all

[@stevieinselby]

I'm glad someone else has spotted that too, I thought I was going mad!

[@Azide_zx]

he probably drew up something quickly to illustrate his point without noticing it was a poor example

[@RZBredMaker]

13:43 This lovely drawn triangles should be called Grime triangles.

[@supermarc]

Usually mathematicians get recognized for their accomplishments, but on Numberphile for their mistakes!

[@ob3vious]

@@supermarc *cough* Parker Square *cough*

[@munjee2]

@@ob3vious not to mention the parker circles

[@Koisheep]

when a numberphile guest poorly draws something we shall call it a Grimes drawing, it's only fair to Matt

[@Wecoc1]

I was surprised to see there were not shirts and mugs with those

[@4ltrz555]

James Prime is back!

[@oz_jones]

James Prime > Optimus Prime

[@Bardigrade]

i think you clicked on the wrong video

[@4ltrz555]

@@Bardigrade what?

[@4ltrz555]

@@Bardigrade oh, James Prime is another KZbinr

[@onebronx]

Anti-hero triangle: the area is equal to the perimeter written backwards

[@alexandertownsend3291]

That is a cool idea. What would you call a triagle with area that the reciprocal of the perimeter for example, a=2/5 and p=5/2?

[@denyraw]

@@alexandertownsend3291 Villain

[@juanpgomez1299]

We've got ourselves a complete triangle cinematic universe

[@missphase8127]

less cool since it becomes base-10 dependent.

[@onebronx]

@@missphase8127 that's why it is not a true hero :)

[@floewqua]

This guy is a living legend

[@forestpepper3621]

As stated at 12:50 in video, for a given q, there are a finite number of Hero triangles with Area=q*Perimeter. An intuitive way to think about this is to note that as you you scale up the size of an object, without changing its shape, its perimeter grows in linear proportion to the scaling factor, whereas its area grows proportional to the square of the scaling factor. This suggests that, for a fixed value of q, if you increase the required area sufficiently, there will be no Hero triangles satisfying Area=q*Perimeter for that particular value of q.

[@idjles]

Can we have a t-shirt of Grime’s Lovely Triangles? 13:49

[@numberphile]

Your wish is my command: bit.ly/Griangle

[@DocBree13]

Numberphile thank you! 😁

[@whatever946]

NOPE

[@UltraLuigi2401]

5:51 that is a right triangle

[@drewduncan5774]

"Now I've given you permission to put clickbait thumbnails on this video." Numberphile goes self-aware.

[@laurendoe168]

15, 20, 25 triangle is also a cheat because it is a scaled up 3, 4, 5.

[@gavin5410]

I think scaling up will change the proportions since we're also tracking into account area: perimeter is in 1 dimension and area is in 2 dimensions. If you take a triangle and double all the sides, the perimeter doubles, but the area quadruples.

[@laurendoe168]

@@gavin5410 yes, the 15, 20, and 25 is not a superhero either (but the 6, 8, 10 is)

[@jeffreyblack666]

@@gavin5410 That is for his "Superhero" triangles. For the Hero triangles they just both need to be an integer. Scaling up a triangle with an integer length and area by an integer multiple will result in both still being integers.

[@dear-future-ai]

Liking for Hero’s formula, holy heck that would’ve made geometry so much easier

[@hazelgalban3566]

Glad that James Grime is back. Nostalgia brought back. BTW love your channel

[@patrickwienhoft7987]

9:10 it's quite easy to work out by hand as well since it's the square root of a product and it's an integer. 32 = 2^5 27 = 3^3 which means 32x27x3x2 = 2^6 x 3^4 and then the square root is simply 2^3 x 3^2 = 8x9 = 72

[@adityak1231]

0:46 James we have got you on the thumbnail!!That is the best clickbait though this one isn't

[@titleloanman]

I love how you troll the presenters so ruthlessly 😂

[@Qermaq]

13:38 Grimes triangles, akin to Parker circles? When’s the tshirt coming out?

[@ykl1277]

Qermaq's Shape. It is the Parker Square

[@ronzang]

Welcome back James!

[@dustytompkins9234]

15-20-25 triangle: why drop a verticle to "cheat"? It's a 3-4-5 scaled up by a factor of 5, no?

[@itioticginger9520]

13:43 is my new favorite thing Brady has ever done

[@carlhedin7279]

12:47 the left triangle is an impossible triangle. The third side cannot be longer than the sum of the two first.

[@lukesteeves1291]

Guessing that the left side is a 2-digit number with the left digit cut off by mistake.

[@ar_xiv]

Yeah looks like 75

[@ngyipin1]

Seems like the second longest side is 72, the 7 got cut off the screen

[@Bayerwaldler]

@@ar_xiv Yeah, that works: Perimeter 162, Area 324, ratio 2

[@total_dk6517]

If the cut-off number is 72, how come the number shown is 5?

[@VyTran-tk8jq]

more videos of James Grime please! I really like the way he explains math. He makes it very interesting!

[@jarbies004]

This man's voice us so nice to listen to. And he's SO WHOLESOME. I STAN James

[@tomharner83]

@10:30 how did the "s" become "x+y+z"? If "(s-a) = x", "(s-b) = y", and "(s-c) = z", then "s = x*y*z/4".... Why are the terms being added together?

[@HediMaatoug]

Same remark, I didnt't get that part!

[@user-hc4wi9ks9r]

Well, s is the semi-perimeter. Too lazy to re-watch the video, but going backwards: x+y+z = (s-a)+(s-b)+(s-c) = 3s-a-b-c = 3s-2s = s

[@HediMaatoug]

@@user-hc4wi9ks9r Thanx!

[@madacol]

@@user-hc4wi9ks9r Thank you!, I forgot **S** was *half* the perimeter

[@lynk5902]

What I find so astounding, is that with the infinity of whole numbers, all the super hero triangles use such small numbers. Yes I realize that the fact that the perimeter will scale linearly while the area scales to the square so one could intuit that they all should be relatively small, but the fact that even the areas and perimeters are still only two digits (at least in base 10) I still find surprising.

[@hybmnzz2658]

Many number theory problems where one side of an equation is linear and the other side is quadratic usually has its tipping point within a dozen. In that sense, base 10 is quite large. I guess the cool thing is that equating area and perimeter makes such a small looking equation as 4(x+y+z)=xyz as opposed to 1000(x+y+z)=xyz. Heron's formula is the star here.

[@professorsogol5824]

Something I would like to see you do some day: Take one of these ancient mathematical truths and do the arithmetic using the ancient system of numerals. Some possible ideas: Calculate pi using Roman numbers. (Roman numerals were used for over a thousand years. How does one add, subtract, multiply, divide, take square roots with Roman numbers? ) As you know, the Pythagorean theorem predates Pythagoras. How did those earlier mathematicians find the length of the hypotenuse using the numerals available to them?

[@jamesfilosa6277]

16:01 - oooh the banter..

[@kingoffire9373]

Was feeling down this morning, thought to myself I need to see some numberphile with James. Went to youtube and immediately a new video, love you guys you are my comfort channel for the past 5 years xD

[@saikatkarmakar9370]

Yeah!!!! Brady does read and takes heed of the comments. Nice to see Dr.James.

[@JAzzWoods-ik4vv]

"Putting captain america on one side in the tumbnail" *puts James Grime*

[@woufff_]

Happy to see James Grime back, I just love the guy

[@felixcroc]

I love the Graham's Number paper framed in the background

[@bjmermelstein1911]

Looks to me like he had it signed.

[@datarioplays]

where? time frame?

[@mienzillaz]

i love this joke continuation about "well" drawn triangles..

[@OlafDoschke]

15:24 - so now Parker geometry has taken on a life of its own.

[@davidgillies620]

I found an intriguing connection between nearly-equilateral Heronian triangles, the Pell numbers, and sequences of consecutive integers whose standard deviation is an integer, but a KZbin comment is too small to contain the proof.

[@emeryboehnke4259]

Wow, I love these pretty easy to follow proofs.

[@Bayar-oe7rj]

It is quite nice to see James again! I rushed when I saw him and 6 hours ago

[@neuron05]

I am so happy Grimes is still here :) I have to find his banans channel again.

[@mojann1]

Yaaas. Grime is back! Just what I needed today!

[@phillipreay]

Really great topic. I especially enjoyed how you finished on the big isosceles.. had the intuitive feeling there would be near misses to integers but you moved on...but looped back! Now I’m curious about hero rhomboids made of these composite pairs..

[@saulcaneloalvarez1116]

Yay, Mr.James is back!!!

[@scienceandmusicmix]

Was waiting the whole video for mention of a Parker triangle and you did not disappoint!

[@rosiefay7283]

12:49 "If you want your area to be a multiple of the perimeter" A nice thing about all triangles (not just Heronian ones) is that A=rs where r is the inradius.

[@leo17921]

wtf is the inradius

[@rodrigomarinho1807]

@@leo17921 the radius of the inner circle. For right triangles you can use the formula b + c - a, let a be the hypotenuse and b, c the catheti

[@anindyabiswas1551]

Sir Grime, I have waited a long time for you! ❤️❤️❤️

[@molamola8305]

That semi-perimeter formula was taught in my school in 8th grade in India.

[@storyofsharplyon]

loved this episode, good work

[@daapf8232]

Parker-Squares and Grime-Triangles... which polygon will be next?

[@Tombsar]

Following the progression, we need a 2D shape with two sides. Annulus?

[@daapf8232]

@@Tombsar Interesting idea. Now there's only the need for an eponym the annulus will be named after.

[@clockworkkirlia7475]

A Zvezda Circle (a straight line with "Ptolemy>Pythagoras" graffitied over it)

[@3ckitani]

13:43 I was going to comment about the other mathematician, but it's James Grime, so...

[@jacobjoshi4426]

Reminds me of this (though not as amazing) A square with all sides of 4, has area 16, and perimiter 16.

[@quinn7894]

A circle of radius 2 has perimeter 2*tau and area 2*tau.

[@mustafaemrebasaran7701]

A James Grime video! Hallelujah!

[@KedarOthort]

"Parker super triangle" I literally laughed out loud

[@PC_Simo]

Funnily enough, James’s drawings of the ”P = 70; A = 210” -pair would slot together perfectly, like pieces of a jigsaw 😅.

[@XanderFenikkusu]

My favorite presenter is back. Welcome back, James!

[@vijaytripathi7675]

I still don't know but this channel give me an insight of how amazing and elegant mathematics really is ...thank you numberphile... love from India...

[@jesse8167]

*sees James* Never clicked on a video so fast

[@cerwe8861]

Yeah, he is the best Clickbait!

[@aidanmcdonagh7861]

@FullTimeSlacker so? 😂

[@DocBree13]

same!

[@SlingerDomb]

As soon as I saw Dr.James, I click so so fast .

[@nonpopscience3291]

really great video, loved it, very elegant

[@murozond8473]

I could watch James Grime talk about numbers all day, bring him back soon pls!

[@cerwe8861]

YAAAY JAMES!!!!! MORE OF HIM!!!

[@wompastompa3692]

5:45 That's already a Pythagorean triangle. 5 * 3-4-5.

[@okuno54]

13:41 My compliments to the animator!

[@HassanAhmed-bs5fn]

James always does the best videos

[@ItachiUchiha-ns1il]

HE’S BACK

[@dougg1075]

Love this guy. I don’t even like math but love this channel.

[@MysterX79]

In school if I said something like: "The length is 5", I got asked "What 5? Five eggs? Five apples?" we were always demanded to name the unit. Even if the actual unit was not needed, we were demanded to call it. Lets say "30 length units" or "30 length units to the square" and so on. And honestly this is so much inside me, I was cringing alot in this video. "It has the same perimeter as it has area" and such things simply feel very wrong from the beginning. - Of course I like the video, it was very cool as always! :-)

[@clockworkkirlia7475]

Yay! Back watching Numberphile, and even more interesting things about triangles! You'll get me to like Pythagoras someday (though I content myself with knowing that Ptolemy is, obviously, superior).

[@cineblazer]

The 15-20-25 triangle was a right triangle to begin with!

[@adityak1231]

Finally , James Grime again(not that others aren't acceptable but James is unbeatable!)!! Clicked instantly

[@8Clips]

Oh this is phenomenonal, easy enough to understand for almost anyone and a great introduction to the concept of Mathematical proof for people who haven't studied it.

[@fwiffo]

After watching this video, I had a dream where The Moody Blues were on Sesame Street performing a song about perfect numbers with Cookie Monster. The song included some sort of nonsensical visual proof in high dimensions about their properties. It did not sound much like a Moody Blues song, and Cookie Monster sounded like he had a cold, so he was a particularly unconvincing lead when they did "Nights in White Satin."

[@SamiSioux]

I could listen to Dr James Grime talk Geometry all day. 😍

[@radonato]

Non-Euclidian Superhero triangles? Semi-SuperHero cones? (Surface area is ½ volume?) Lovely to see and hear you, James!

[@The_Omegaman]

James has more math enthusiasm than anyone is enthusiastic about anything. Great vid.

[@albinoagellar268]

James on thumbnail - instant click

[@ReyMysterioX]

The framed brown paper about Graham's Number behind James, written and signed by Ron Graham himself, really is something like the holy grail of Numberphile!

[@nitfumble]

I see James, I press the like button!

[@numberphile]

I see people liking James, I like their comment!

[@NonTwinBrothers]

Always enjoying these animations! Haha

[@hansgrettle8240]

*sees thumbnail* 24 *was* a superhero

[@WilfordBrimleyGaming]

I wonder if that was intentional.

[@nicholasbrown3197]

Around 0:47 he mentions something about clickbait titles. Totally thought this was a Kobe vid

[@JuanSanchez-qt1ue]

There is no coincidence the area of a hero triangle is 24, the laws of math knew what they were doing

[@nicholasbrown3197]

Gotta believe it’s a shout-out, I can see them asking James, “Hey can you do a video that includes the number 24?” “Why yes I can”

[@AAbattery444]

What happens to the area of a triangle as its "base" and "height" remain constant but length of its two other sides approaches infinity? Does the area really remain constant? Or does it increase ever so slightly?

[@Domancave]

This one is for Kobe. He’s a hero. Days after he died a video about 24 comes out. The world just works like that

[@tamarpeer261]

6:04 The 15-20-25 is already a right triangle, tho

[@matthewellisor5835]

I absolutely lost it at 13:40! Thanks for that.

[@unreal-the-ethan]

oh, i thought this was uploaded years ago. it was actually only uploaded 12 minutes ago.

[@noobofnoobs6165]

E

[@klaasbil8459]

5:57 this is a right-angle triangle (3-4-5 scaled up), so it's not only a cheat by dividing it in two right-angle triangles.

[@SpiderGirl42]

I want merchandise of "these really valid, lovely drawn triangles" (along with quote underneath lol)

[@ccppurcell]

I scrolled through most of the comments and didn't see anyone point this out: the 5,29,30 triangle can be obtained by taking a 5,5,6 triangle (Heronian) and a 6,25,29 triangle (also Heronian) and gluing them along the edge of length 6. So there should be some "prime" Heronian triangles that cannot be obtained this way...

[@sohambasak6382]

Super curious about the framed sheet of brown paper on the wall behind James. Edit: 9:44 looks like a signed sheet showing Graham's number!

[@kevinmartin7760]

At 5:52 the triangle is already a right triangle with pythagorean ratio sides (3:4:5); there is no need to split it up to reveal the "cheat."

[@sakkikoyumikishi]

"The way I was told to work out the area in school requires using Pythagoras occasionally. Let me instead show you this other method, which ALWAYS uses Pythagoras (or, actually, Ptolemy, who Pythagoras' theorem is a special case of) in order to avoid occasionally having to use Pythagoras." If that isn't the biggest troll ever then I don't know what is. 😂

[@standard_limbo]

James is the best on this channel

[@rosiefay7283]

14:13 Thank you, Prof. Grime for making a connection between this interesting phenomenon and something I'd already been studying. The latter is triples of integers {x_i, y_i, z_i}, with the same sum and the same product. Following your notation at 10:28, suppose that for various indices i we have Hero triangles {a_i, b_i, c_i} with the same perimeter a_i+b_i+c_i=2s and the same area A. Then define x_i=s-a_i, y_i=s-b_i, z_i=s-c_i. Then x_i+y_i+z_i=s as desired. Moreover, x_i y_i z_i=P=A^2/s. So if you can find triples with the same sum s and the same product P, then you have triangles with the same perimeter 2s and the same area sqrt(sP). And if sP is a square, then your triangles are Heronian.

[@rosiefay7283]

Here's a further nice feature. Define q=s/2, the quarter-perimeter. Then if we define d_i=a_i-q, e_i=b_i-q and f_i=c_i-q, then all of the (d_i, e_i, f_i) have the same sum q and the same cube-sum q^3-3P. Now suppose that you managed to find two such Hero triangles, and in one triangle the shortest side c_2 was q, the quarter-perimeter. Then f_2=0. (I'm using _1 and _2 to index the two triangles.) Now define sets L={d_1, e_1, f_1, -d_2, -e_2} and R the same but with all five signs flipped. Then L and R each have sum 0 and cube-sum 0 (by the previous paragraph). More trivially, L and R agree in square-sum and fourth-power-sum, because a number's even powers remain the same when you flip its sign. So L and R are a solution to the Tarry-Escott problem for powers up to the 4th. For example, triangles (18, 17, 5), (19, 11, 10) with perimeter 40, and 10=40/4 as required. Then L={8, 7, -5, -9, -1} and R={-8, -7, 5, 9, 1}.

[@jibster5903]

Would love to see a generalized proof for other shapes and possibly in higher dimensions

[@fiziwig]

So much fun to be found in the world of numbers!

[@Henrix1998]

James Grimes utilising superheros to become the most liked Numberphile person again!

[@harmidis]

James Grime.... my hero! He rocks the Math world!!!