THE definitive answer to the triangle hole illusion

[billysk8r]

I have read all the emailed explanations to this riddle, and I felt like clarifying who was indeed right. The final answer is that yes, the triangle is indeed a quadrilateral, not a triangle. And YES, that less than 2 degree difference does in fact make up for an ENTIRE square of space. Let me explain quantitatively in order to prove it:

Okay, so both shapes, before and after, are FAKE 5x13 triangles. They are both comprised of the same shapes: the 2x5 green tri, the 3x8 red tri, and the two remaining polygons which aren't the focus of this explanation.

So anyway, the first shape is a completely filled quadrilateral that looks a lot like a 5x13 triangle. Now that we know it's not, let's figure out the difference in areas of a true 5x13 and this fake. The point of interest here is the hypotenuse, since the "legs" are "true legs" anyway.

Let's think of how a true 5x13 triangle's hypotenuse should be: to make things simple, let's make a representation of a true 5x13 triangle's hypotenuse with a line with equation: y = (5/13)x (domain 0 to 13)
Now let's compare the fake hypotenuse of the first triangle to this.

We can think of this fake hypotenuse as two line segments. the first being y = (3/8)x (domain 0 to 8) and the second being y = (2/5)x - (1/5) (domain 8 to 13)
SO, as the picture illustrates (if you look closely enough), the first triangle is concave, which means the area of the fake triangle is less than that of a true 5x13. Let's calculate exactly how much less area it has. We'll use a little calculus to find the difference in area under two functions.
We'll need the difference of the integrals from 0 to 8 of the first fake hypotenuse segment and the corresponding real hypotenuse section, and the difference of the integrals from 8 to 13 of the second fake hypotenuse segment and the corresponding real hypotenuse section:

---First Segment---
integral from 0 to 8 of (5/13)x minus integral from 0 to 8 of (3/8)x = 8/26
integral from 8 to 13 of (5/13)x minus integral from 0 to 8 of [(2/5)x - (1/5)] = 5/26
sum of integrals = 13/26 = 1/2 = difference in area from true 5x3 13 hypotenuse to fake line segments.


So, now we know the "before" triangle has 0.5 units less area than a true 5x13 triangle.

If we compare the "after" triangle to a true 5x13 using the same process, we will see that the after triangle has 0.5 more units than a true 5x13 triangle (That is, assuming the hole is filled)

Therefore, through the substitution property of equality
areabefore = truearea - 0.5
areaafter = truearea + 0.5
areaafter - areabefore = 1
it is in fact apparent that the Area of the 5x13 fake "before" triangle is exactly 1 unit larger than the Area of the 5x13 fake "after triangle.

Since the area that the "puzzle pieces" take up on the graph paper is a constant, the 1 unit difference between the two pseduoshapes is clearly shown by a "missing piece"

[JacenDrive]

as i've heard before, there's only one real way to respond faithfully to something like this:

Show us your boobies!

[Ryan2003]

I know calculus but i dont know what the hell this triangle is you're talking about.... iow... I'm lost

[dbs_4eva]

?.....................???........................? ??? :confused: :confused: :confused: :confused: :confused: :confused: ..............................?.................?? ?.....................?????

[Nope4810]

The triangle that is supposedly bigger using the same shapes. Yeah, the calculus thing... I didn't read any of that, but my calculus teacher sucks, so I'm going to assume I wouldn't understand it even if I did.

[paINTbaLLA472]

can u please speak english? i didnt get 1 word..

[Trickee]

umm......great first post?? :wtf: :confused:

[ZX-6RR]

Oh, I get it... :insane:

[rifraft]

What our friend is really trying to tell us is that a hole in a circle makes a whole hell of a lot more sense.

[Goatman]

Does this mean I am going to die a slow and painful death?

[Nope4810]

Stop, collaborate and listen. Vanilla has two "l"s.

[absentminded562]

i am pretty sure this guy diddnt write it himself. Ive seen the same document elsewhere, and i doubt that anyone with "sk8r in their sn is going to come up with such an organized solution.

[Crackerjak]

Uhhh do you people not understand his fomulae or how there is hole? I can explain without calculus... we all know (or should know) area of a right triangle is base x height divided by 2. So theoretically the area is 13 x 5 = 65 then divided by 2 = 32.5. However, this isnt a right triangle because the hypotenuse (the longest line) is sloped inwards to make the area slightly less than 32.5 and the second one is sloped out make for slightly more area than 32.5. Now correct me if I'm wrong sk8er but the slight differences in area between the two slopes is exactly 1 square which is the "missing" square.
Wow that was wierd i had to change o u t w a r d s to out because it was censored...

[AntonioMainenti]

You people are imbeciles. The ones who say they've had calculus but can't follow it really leave me speechless. He's integrating a LINEAR FUNCTION OF ONE VARIABLE for Christ's sake. It's a good thing Crackerjak came along and saved the day for you morons. If you're still too stupid to visualize god damn line segments (not that you need to since the image of the "triangles" is given) you could always just write out on paper what he's doing and quit whining about elementary math being too difficult for you.

[RckWorld]

:icon_rofl *Cough* Geek *Cough* :icon_rofl

[BlinkingViki]

Well...I get it o_o quite simply, the two red triangles in that illusion are not the same, their difference being equal to one unit. There's no need for all that math, however. :/ its really...quite simple. Adding all that math just makes you sound like a dumbass. Nice try though.

[AntonioMainenti]

"Well...I get it o_o quite simply, the two red triangles in that illusion are not the same, their difference being equal to one unit. There's no need for all that math, however. :/ its really...quite simple. Adding all that math just makes you sound like a dumbass. Nice try though."

- BlinkingViki.

Please, for the love of God, tell me that was a joke. That was by far the stupidest thing I've ever heard in my life.

P.S. RckWorld, go f*** yourself. What was it that made you think im a geek? The ability to comprehend basic high school math?

[Idiot_Savante]

... Really ...

[BlinkingViki]

Please, for the love of God, tell me that was a joke. That was by far the stupidest thing I've ever heard in my life.

P.S. RckWorld, go f*** yourself. What was it that made you think im a geek? The ability to comprehend basic high school math?

Nice job, jackass. You've succeeded in making a complete ass of yourself. Try at least to be human, you egotistical pig-dog. Furthermore, my statement was completey coherent, as opposed to the crap you pulled out of nowhere.

Your blatant disregard of all those who don't think along the same lines as you disgusts me. Learn to open up, and quit viewing yourself as a "genius." People who look down upon others will in turn be looked down upon.

In addition, the only reason RckWorld insulted you was that you came in here bitching and moaning at people for not grasping a concept you deemed to be "basic high school math." I don't know where you went to school, but where I'm from, Calculus is not a requirement, especially since it is a math that has only a few applications to the living world. Most people won't need calculus in their adult years.

In conclusion, I have four letters for you: S...T...F...U! :thud:

[AntonioMainenti]

Nice job, jackass. You've succeeded in making a complete ass of yourself. Try at least to be human, you egotistical pig-dog. Furthermore, my statement was completey coherent, as opposed to the crap you pulled out of nowhere.You tell me to try at least to be human. Meanwhile you told billysk8r he sounded like a "dumbass" for "adding all that math." This is especially ironic due to the fact that what you said was so unbelievably wrong (no matter how coherent it was) as to become humorous (I would love to hear you argue that two triangles with the same side lengths, shape, etc. somehow differ in area by 1).Your blatant disregard of all those who don't think along the same lines as you disgusts me. Learn to open up, and quit viewing yourself as a "genius." People who look down upon others will in turn be looked down upon.You're probably right about this; I should clarify that those who have no idea what an integral is can obviously not be blamed for not understanding the explanation. But I don't view myself as a genius; my whole point is that anyone who knows any calculus whatsoever should find this easy to follow. You don't need to be a "genius." It makes you wonder how these people passed calculus in the first place if they can't do integrals over straight lines. In addition, the only reason RckWorld insulted you was that you came in here bitching and moaning at people for not grasping a concept you deemed to be "basic high school math." I don't know where you went to school, but where I'm from, Calculus is not a requirement, especially since it is a math that has only a few applications to the living world. Most people won't need calculus in their adult years.Again, you're right about that but I'm not complaining that they couldnt grasp the concept of integrals. It took Newton and Liebniz to do that. I'm just amazed by those who know integrals but couldnt figure this out even with the solution right in front of their faces. Also, as a side note, calculus is probably the most applicable branch of mathematics to real-world problems. It is found everywhere in science in engineering, far from having only a few applications.

Sorry for what I said though; it didn't occur to me until just now that you're a girl. :(

[billysk8r]

whoah... what an interesting set of replies! umm.... Well, I can only offer my word that my post is original, meaning that I didn't copy from another source. That of course doesn't eliminate the possibility that many people out there in the world already took the approach that I did, and that they just never bothered to post it here on the ebaumsworld forums. I just wanted to show an exact numerical representation of what the eyes are being tricked into seeing by the clever puzzle.

[Melonhead1]

GO TO FORUMS THAT GIVE A FUCK ABOUT YOUR GEEKNESS :censor: :censor: :censor: :censor: :banghead: :banghead: :mad: :mad: :









!!!!! Praise Melons :bow: :bow:

[Sk8erKid143]

holy crap calm down

[ac/dc]

sorry i dont speak arabian

[theViBe]

haha, its funny when nerds fight...
like on tv, a bunch of 'smart' gansters have a battle of wits...funny as hell


ive seen this math stuff, but...meh

[Naz]

Ugh.. question:

Who the hell cares? Calm down and eat your doughnuts, nerds. :rolleyes:

[AntonioMainenti]

Well, Naz, apparently you care enough to post in the thread.

I don't understand what's making some people so worked up about this thread. It's always annoying when people make posts like yours. If you don't care about his explanation of the triangle illusion, don't read the god damn thread. If you don't want to read the "nerd fight" then go somewhere else. I don't even see any fights in the thread; no one disputes his solution. Anyway, it's actually much more annoying to read worthless posts about how stupid a thread is.

[DeftonesBoy]

This is getting so old, can't we all just get along?

[RckWorld]

"Well...I get it o_o quite simply, the two red triangles in that illusion are not the same, their difference being equal to one unit. There's no need for all that math, however. :/ its really...quite simple. Adding all that math just makes you sound like a dumbass. Nice try though."

- BlinkingViki.

Please, for the love of God, tell me that was a joke. That was by far the stupidest thing I've ever heard in my life.

P.S. RckWorld, go f*** yourself. What was it that made you think im a geek? The ability to comprehend basic high school math?


EbaumsWorld Forums = Humor

We dont care about the size or anything just that it gives us a laugh or a headache. Yes I understand what you wrote but I think it was dumb to go through the trouble when most of us dont care. :lol:

Sorry if I offended you

[Trippin]

man it is so easy, it isnt rocket science, it is just that one of the triangles get bigger to cover the hole

[Eab]

I get your reasoning, makes sence.

The trick is the two smaller triangles dont have the same slope on the hypotanous.

[AntonioMainenti]

man it is so easy, it isnt rocket science, it is just that one of the triangles get bigger to cover the holeI can't believe my eyes. You're the second person to say that and it's STILL completely wrong. This thread has just gone to show how stupid some people on this forum are. After the first post on the thread there should have been several replies along the lines of "nice explanation." and then that's it. Instead we have people saying they had no clue what he's talking about and a few who say the parts that make up the "triangles" are somehow different. It really makes you wonder...

[moldy_hair]

Once, I was staring at the picture for a good five minutes, and I figured out how it worked. I couldn't explain it, but I somehow comprehended it.


You, AntonioMainenti, are an asshole. plain and simple. Some people not understanding calculus as well as you do gives you no right to insult and call names. How childish anyways. You're talking about how stupid some people are on the forum, well in this thread, you're the dumbass. Congratulations!

You've spent too much time studying math and not enough time learning how to keep from pissing people off. You keep up this attitude and you won't get far in life. People will always hate you and talk behind your back.

So go on, try to defend yourself. I guarentee everyone else will agree with me.

[AntonioMainenti]

All I'm saying is if you can't understand this simple problem don't post telling everyone how confused you are. I posted in this thread mostly because of the people who said they had calculus but still couldn't follow it. I was just taken aback by that. The two people who said the parts were different though... they were asking for it. You're right though, everyone probably will agree with you (since you're probably right).

[bwdelong031490]

Calculus is bullshit... :thud:

[Pacific]

and they say you never will use your calculus when you get out of high school.... :lol:

[AntonioMainenti]

Calculus is bullshit... :thud:Hmm.. interesting point... you may be on to something there... dumbass.and they say you never will use your calculus when you get out of high school.... :lol: :D

[bullgod7777]

can someone please enlighten me on what the fuck you guys a talking about, i have yet to see a damn triangel so can someone send me a link or somthin

[AntonioMainenti]

Good point.

http://www.ebaumsworld.com/trigrid.html