| Author | Message | ||
     M1combat |
I guess I didn't get far enough into Trig or Geometry or something...  | ||
| Oz666 |
It's an optical delusion, dude. Upper "hypotenuse" is concave, lower is convex.  Oz | ||
| M1combat |
But they're both the same size triangle... One seemingly covers more area though. | ||
| M1combat |
So, what you're saying is that because the lower hypotenuse is convex, it allows the 2 square height on the right side of the fifth square, where if it were a straight line it would not... I put a piece of paper up to the lines and the top line looks pretty straight, but the bottom does look a little convex... So, if I were to build these pieces straight and square, I would not be able to reproduce this result? | ||
| Crusty |
OZ is right. The hypotenuse or top line of the triangles is not straight. The top triangle has a top "line" that's bowed down, while the second triangle's is bowed up. They're not straight lines. | ||
| Mr_grumpy |
Damfino  | ||
| Blake |
Thus they are not "triangles." | ||
| Danny |
Don't know if this answers your question or not, but... In figure 1, light green and Orange occupy an area 5x3 or 15 squares. In figure 2, they are 8x2 or 16 squares, hence the one "missing" square. Now that I'm home, gimme a little more time on the rest. Danny | ||
| M1combat |
Got it  . Thanks guys. My trusty tech bench co-hort kept telling me that I was too stupid and un-educated to know how to explain that. I haven't told him the explanation, but he still insists that there is a perfectly good mathematical explanation (which I understand that there is, but almost certainly not one that he knows). Of course, he won't explain it to me... He just likes to think he's smarter than me. At least I'm not arrogant enough to assume I know an answer and then tell someone else they are un-educated because they don't know... Anyway. He's still an •••••••, and I know to look closer next time . I noticed the discrepancy WRT the height of the hypotenuse of each "triangle" but wrote it off as an imperfect drawing... Thanks again. | ||
| Danny |
If green trangle and red triangle are actally triangles, they have different angles to the left of the right angle. red is 20.55 degrees green is 21.8 degrees If figure 1 were a true triangle is should have an angle of 21.04 degrees... it does not. Oz and Blake are correct. | ||
| Dasbuell |
You guys just made my head hurt!!! I was trying to think... ...and nothing happened!!! (formerly dasxb9s) | ||
| Glitch |
But did you ever stop to think...and forget to start again? | ||
| Tucsonxb9s |
Well, I don't think it's a convex or concave thing. As stubborn as I am, I had to cut a triangle out of paper as shown. When the pieces are restacked the hole appears...I don't understand...but seriously...cut up a triangle and see for yourself!! Dang...it's not like a gotta a ton of work to do and here I am cuttin up paper!!!! Thanks M1! | ||
| M1combat |
No problem... I think it is the case though. If you take a look at the hypotenuse of the upper and lower "triangles" you will notice that they don't intersect the grid lines at the same places. Start at the bottom left point and go five squares to the right and two up. The upper triangle is shorter at that spot. They are correct is stating that the partitions are used in the upper and lower shapes are the same. Look at the 5,2 spot on the red triangle in both shapes... the two triangles (as danny stated) so not have the same angles. That's where you get the height of 2 at the fifth grid from the left with the green triangle. Theoretically you would be able to exchange the left portion of the red triangle with the green triangle but you can't because each hypotenuse is a different angle. If it were NOT an illusion, you would be able to divide the two sides lengths and come up with the same ratio... I haven't tried it (I suspect that Danny has) and I think you would come up with a different ratio. Apply ?vector math? (probably just trig) and you would probably get the angles that Danny mentioned. | ||
| Bluelightning |
I guess none of you ever played tetris as a kid. The angles are the same, the 2 triangles have the same angles (at the long point) the only difference is that the red triangle is physically larger, and to make up the difference, the yellow piece has been shifted over and lowered with the green triangle placed at the end. | ||
| Tucsonxb9s |
M1....quit hypothesizing and cut up a triangle! It's not an optical illusion. | ||
| M1combat |
They do not have the same angles... The L/H ratio of the red triangle is 2-2/3 to 1. The L/H ratio of the grn triangle is 2-1/2 to 1. | ||
| Tucsonxb9s |
http://www.ebaumsworld.com/holeexp.html From the site with the puzzle. I think no one knows what the heck is going on. | ||
| M2me |
I don't get it. I printed out the image and did just as Tucsonxb9s said and cut out the triangles. It's not an optical illusion! The triangle pieces are interchangeable between the top arrangement and the lower. There is no way I could of cut a convex or concave side exactly right. | ||
| M1combat |
The L/H ratio of the red triangle is 2-2/3 to 1. The L/H ratio of the grn triangle is 2-1/2 to 1. The angles ARE different. Those two triangles are NOT the same shape. On the bottom triangle... Look at the green one. From the left point, go five to the right and two up (grid lines). The green triangle comes out to the 5,2 grid line. Do the same for the red triangle... It comes out BELOW the 5,2 grid intersection. Extrapolate (or interpolate??) where the height of the green triangle would be if extended to the length of the red one... It will come out 4% higher, or at the 5.2 grid line instead of the 5 grid line like the red triangle does. The hypotenuse of the green triangle is steeper. That allows you to get the height of two where the green, gold and red pieces meet. You can not interchange the green triangle with the left point (can't remember the name of that one) of the red triangle. It's only a degree or two off, so you won't see the convex/concave situation. If you machined the pieces from metal and placed a straight edge you would find that the hypotenuse of the triangles (the two made up of all the parts) is NOT straight. Therefore, they are not triangles. The hypotenuse of the green triangle is 36 degrees. the hypotenuse of the red triangle is 33.75 degrees. | ||
| M1combat |
"The triangle pieces are interchangeable between the top arrangement and the lower. There is no way I could of cut a convex or concave side exactly right." Probably true - BUT... You DID cut an angle of 36 degrees for the green triangle (the angle on the left, 54 degrees for the top angle), and 33.75 degrees on the red triangle (56.25 for the top angle of the red triangle). | ||
| Tucsonxb9s |
You're gonna drive yourself nuts!  It is what it is....don't ask why! | ||
| Uwgriz |
M1 is right. In reality, you're not looking at a triangle, it's actually a four sided shape. | ||
| Newblaster |
You're just overthinking. Trust me, I'm an expert on overthinking. Okay, mostly in social situations, but still... Remember the first lesson in your geometry classes. "When something is Given information, it should always be treated as true." "Given" in this case is that the shape in question is a triangle. Sure, if you have a good straigtedge and a protractor, you may be able to prove it's a four-sided polygon with one very very slight angle, but that's immaterial to the question asked. Why is the hole there? Simple, there's a triangle that's 8 x 3, one that's 5 x 2, and two blocks, one of which is a 2 x 2 with a 3 square overhang, the other is 3 x 2 with a 2 square overhang. Okay... Switching the triangles... The one-square difference in height is taken away by putting the six-sided blocks on the same horizontal axis (instead of interlocking them.) Together they only are 7 squares long, but since they no longer interlock, there is a space (3-square overhang meets 2-square overhang.) So... It may or may not be a triangle, but it isn't an optical illusion, and that's why cutting up the paper works... 2 more cents in an ever deepening pile... (Message edited by newblaster on August 12, 2004) | ||
| Uwgriz |
It's not a given that it's a triangle. You don't need a straight edge or protractor either, you do exactly what M1 did. 5/2 = 2 1/2 8/3 = 2 1/3. Translation - they are different angles and therefore it is actually a four sided shape and that's what makes this work. If you can produce the same phenomenon using triangles with the same angles, I'll ship everyone who's posted above this post a case of beer. | ||
| M1combat |
ME too, including Uwgriz... I'm kind of looking forward to that too, I'll get 16 cases of Beer... If I can make the request, I'll take Bass Ale. Just for the record... Take a piece of paper and line it up with the top line (hypotenuse) of the green triangle on the bottom "triangle". It'll come out .2 squares above the top of the red triangle. It's just enough of a difference that your eyes don't pick up on it because of the straight grid lines. You can also look at the top of the fifth grid square of the green and red triangles and see that they are not the same height at that point. | ||
| Uwgriz |
Let me explain this more simply. The area of a right triangle is multiply the two shortest side together and divide by 2 (it's half of a rectangle). Top "triangle" => (13x5)/2=32.5 Bottom "triangle" => (13x5)/2=32.5 Now let's add the areas of the shapes: Green triangle => (5x2)/2=5 Red triangle => (8x3)/2=12 Green block = 8 Orange block = 7 Added up they equal 32 therefor the area of the top shape is 32. This is not equal to a the area of a right triangle with base of 13 and height of 5 (see above). Add the 1 unit of area in the bottom picture and you get 33. Also, not the right area. Translation, neither shape above is a triangle and that's the trick. Case closed. (Message edited by uwgriz on August 12, 2004) | ||
| Socoken |
wait, i thought the dark green one and the red one were triangles???????? | ||
| Newblaster |
Okay, my bad. It never calls it a triangle. Probably shows that I never enjoyed division or fractions, and avoid them whenever I can. So the triangular shape is an illusion. At least we figured out where that hole comes from... Socoken--The dark green one and the red one are triangles. It's just that the entire shape from assembling all the blocks isn't one. (Message edited by newblaster on August 12, 2004) | ||
| Uwgriz |
Socoken, they are, but the shapes that they help make up are not. |
