Breaking laws of mathmatics

[Acanthus]

is this an optical illusion of some kind... i looked at this for a half hour and am very perplexed.

http://www.saunalahti.fi/~jylppy69/bizare.gif

if you can explain this to me PLEASE do.

 

i would like to know as well. I shall have my math and science friends figure this one out.

 

[Mingon]

simple answer is that the top combined triangle does not have a straight line along the Hypotenuse. the bottom combined triangle does.

 

[jbond04]

It's just an optical illusion....the slopes of the two triangles are different. If you were to overlay the 2 triangles, you would notice that the one with the "hole" has more area on the slanted "edge". The "edge" is actually made up of two lines, not one. It's not even a triangle.

 

[Acanthus]

yes i see that, but the surface area of the 2 triangles is still identical, yet one has the "gap" of 1 unit, even if it isnt a true triangle it shouldnt work

My brain hurts

 

[f95toli]

No, the surfaces are not identical. It is just an optical illusion.
I guess the explaination is related to how our brain percieves joints between a red and a green piece.
There is no math-mystery here.

 

[silverpig]

Originally posted by: Acanthus
yes i see that, but the surface area of the 2 triangles is still identical, yet one has the "gap" of 1 unit, even if it isnt a true triangle it shouldnt work

My brain hurts

That missing unit is made up in the top triangle by the line not being straight. Hold up a ruler along the top edge, blocking out the space above the triangle, and you'll see that there's a little bit of space in there.

 

[sao123]

Here is the mathematical secret...

Ignore the two triangles. The answer is in the Green/Yellow Rectangle.


Law 1 - All shapes of a given area A, do not have the same perimeter P.
The green shape, has a perimeter of 14 and an area of 8.
The yellow shape, has a perimeter of 14 and an area of 7.

The important figure here is the net perimeter...
in other words, how much of the perimeter is shared and overlapping, and how much is not.
net perimeter = total perimeter of both shapes (28) - the shared overlapping perimeter

In the top, the total area is still 15, but the net perimeter is 16.
In the bottom, the total area is still 15, but the net perimeter is 22.
The combination of the area and net perimeter, determine the gap appearing in the bottom shape.

hope this explaines it for you.

 

[Antza]

I can't really see what the perimeters have to do with this; it's just an illusion. The heights of the components just are not what they seem to be. The yellow block has a calculated height of 2,07 instead of two in the upper triangle and 1,923 in the lower triangle. (By standard geometry). This means that the red and green triangles do not have same angles...

 

[blahblah99]

Yup, the lower triangle has more area than the top triangle due to the bulge in the hypotenuese.

Ie - hyp of lower triangle is convex'd, and hyp of upper triangle is concave.

 

[woolmilk]

Ie - hyp of lower triangle is convex'd, and hyp of upper triangle is co

Yup, funny picture .
While the components are rectangular and triangular, the big shapes formed with them are not triangles at all.

 

[RossGr]

The 2 triangles are not similar they have different apex angles, so the two large "triangles" are not trangles but are really quadrlaterals.

 

[hevnsnt]

lol wtf.

 

[kylebisme]

the overall shapes are not triangles at all, one is an irreigular rectangle an the other an irregular octigon.

the last real illusion before the suprise in this thing here is one of my favorates: warning, not for people with heart conditions and the like.

 

[hevnsnt]

lol..

 

[Jeff7]

Originally posted by: TheSnowman
the overall shapes are not triangles at all, one is an irreigular rectangle an the other an irregular octigon.

the last real illusion before the suprise in this thing here is one of my favorates: warning, not for people with heart conditions and the like.

 

[Antza]

Wow! Well done site othwewise also...

 

[Peter]

Simple. Red triangle has a 3:8 ratio, while the dark green one is 2:5. This is not the same incline angle, thus what looks like the hypotenusis isn't a straight line, meaning that the entire thing isn't a triangle.

Since 3:8 is less than 2:5, the upper figure is a slightly concave shape while the lower one is slightly convex. This long thin difference amounts to exactly one square.

Or, mathematically,

Upper figure: area covered by the red, dgreen, green and orange pieces (in that order) is:
(3x8)/2 + (2*5)/2 + 8 + 7 = 32

The area covered by the whole thing if it WERE a triangle would be:
(5x13)/2 = 32.5

There's half of your "missing" square already. It vanishes in the unnoticeably slight inward "bend" of the not-really-a-hypotenusis.

The lower figure arranges the two triangle pieces in reverse order, which makes the "bend" point outward not inward. This adds another half square of missing surface (because we have the half one that is inherently missing from the surface of the pieces, and now have another half one because the arrangement partially is outside the imagined triangle shape).

 

[Peter]

Originally posted by: Antza
I can't really see what the perimeters have to do with this; it's just an illusion. The heights of the components just are not what they seem to be. The yellow block has a calculated height of 2,07 instead of two in the upper triangle and 1,923 in the lower triangle. (By standard geometry). This means that the red and green triangles do not have same angles...

The trick is NOT in the lack of precision in the drawing. See above.

 

[Ipno]

I sent this to a friend of mine who became near violent trying to figure it out. hehe.

So I made this drawing for him to explain it better.

Sometimes the trick in seeing something is to look at whats not there.

 

[sao123]

This is a true high level math problem combining Algebra, Calculus, Geometry, Triginometry, and logic.
The drawing is flawed...the line was drawn erroniously by human hand.

Solve the problem as if the triangles were having the same ratio. This problem really does exist.

Eventually You will have to take the limits of the perimeter as area approached the 32.5. and some other integration calculations to realise the correlation is as i stated long ago. "having to do with overlapping perimeters and the sum of the areas"

 

[kylebisme]

na, it doesnt work if you use a real triangle.

 

[TuxDave]

Yeah, you can easily see that they are not triangles at all. Look at the slopes of the of each of the two smaller triangles. The red triangle has a slope of 3/8 and the other has a slope of 2/5. So they will form a bulge depending on which position you place them at.

 

[Omegachi]

different slope buddy.

 

[sao123]

Forget the drawing...Change the slope, make the triangles use the same ratio. Then solve the problem.

Its a real problem out of a real calculus book...Its difficult, but it works.