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## #1 2008-04-27 13:05:47

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### 3D weirdness

Here's something I just realised when idly staring at my ceiling trying to comprehend group theory.

Stand roughly in the middle of the room (which should be cuboid-shaped. Switch rooms if it isn't.), and look at a corner above you.

It should have 3 angles meeting up with each other - two on walls and one on the ceiling. The angle on the ceiling should appear to be a bit more than 90 degrees.

However, this is true for all four of the ceiling-corners, and thus your ceiling is a quadrilateral whose angles sum >360 degrees. Crazy ceilings.

Why did the vector cross the road?
It wanted to be normal.

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## #2 2008-04-27 13:16:20

MathsIsFun
Registered: 2005-01-21
Posts: 7,664

### Re: 3D weirdness

Ahhh... the missing element is the turning of the head. You should see all corners at once I think

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #3 2008-04-27 16:01:20

mikau
Member
Registered: 2005-08-22
Posts: 1,504

### Re: 3D weirdness

Yeah, if you could get all 4 angles in view at once, you should have a trapezoid that does not defy the rule.

You do present an interesting point though.. how do we know what the angles do when we're not looking? They might be stealing candy!

A logarithm is just a misspelled algorithm.

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## #4 2008-04-27 16:07:53

MathsIsFun
Registered: 2005-01-21
Posts: 7,664

### Re: 3D weirdness

mikau wrote:

...They might be stealing candy!

Of course! That explains the mystery!

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #5 2008-04-27 16:50:05

Member
Registered: 2008-01-16
Posts: 147

### Re: 3D weirdness

its true if you consider it to be a relativistic space. XD.

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## #6 2008-04-27 22:39:01

integer
Member
Registered: 2008-02-21
Posts: 79

### Re: 3D weirdness

spherical trigonometry!

You are viewing/measuring the angles in a plane normal to your line of sight.

The sum of the angles of a triangle on a sphere is greater than 180 degrees.
Thus, the sum of the angles of a quadrilateral on a sphere will be greater than 360 degrees.

You can "prove" to some that the sum of the angle of a square is greater than 360 degrees.
Of course, those are then same persons who believe in the "proofs" that 1 = 2 and 1 = 0.

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## #7 2008-04-27 22:49:23

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: 3D weirdness

integer wrote:

The sum of the angles of a triangle on a sphere is greater than 180 degrees.

I knew that bit, from the puzzle about the man who walks S, E, N and ends up in the same place, before seeing a polar bear. (The angles of his turns would be 90, 90 and >0)

What confuses me is that the ceiling is a plane, and so surely spheres shouldn't enter into it?

Why did the vector cross the road?
It wanted to be normal.

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## #8 2008-04-28 03:51:44

Member
Registered: 2008-01-16
Posts: 147

### Re: 3D weirdness

ceiling is not a plane, its a fragment on a sphere, since it's parellel to the ground.

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## #9 2008-04-28 06:38:40

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: 3D weirdness

Well yes technically, but the sphere is so huge (and so has curvature so tiny) that the ceiling could reasonably be approximated as a plane.

I wouldn't think that you'd get such a massive discrepancy from the expected 360 degree sum just because the ceiling might be very very very slightly curved.

Why did the vector cross the road?
It wanted to be normal.

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## #10 2008-04-28 15:27:45

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

### Re: 3D weirdness

You think that's weird?
Try taking a nice pair of binoculars
into the car with you when you are
not the driver.  Take some hilly and
will be unforgettable.  I won't even
try to explain it because it blows
the mind.

igloo myrtilles fourmis

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## #11 2008-04-28 18:00:06

Member
Registered: 2008-01-16
Posts: 147

### Re: 3D weirdness

mathsyperson wrote:

Well yes technically, but the sphere is so huge (and so has curvature so tiny) that the ceiling could reasonably be approximated as a plane.

I wouldn't think that you'd get such a massive discrepancy from the expected 360 degree sum just because the ceiling might be very very very slightly curved.

Hmm,you got a point... isn't that some kind of visual fallacy, then?

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## #12 2008-04-29 03:41:51

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

### Re: 3D weirdness

Okay, well the binoculars thing reminds me of Lorenz transforms
when things get shorter at the speed of light.

But the angle in the ceiling is to be expected because an angle in a plane will
be viewed at its smallest at the true angle, but from other angles it can be
as large at 179.9999 if you are almost in the plane, like a fly on the ceiling.

igloo myrtilles fourmis

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## #13 2008-04-29 03:45:36

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

### Re: 3D weirdness

Woops, correction, there are places where the angle actually gets smaller than the true angle too I just noticed while walking around looking at my thumb and index finger turning it all around.

igloo myrtilles fourmis

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## #14 2008-04-29 03:47:20

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

### Re: 3D weirdness

If you are "inside" the triangular perpendicular prism formed by the angle, then
the angle can get larger, but if you outside this triangular space, so "outside the angle", then the angle can get smaller.

igloo myrtilles fourmis

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## #15 2008-04-29 04:09:18

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

### Re: 3D weirdness

Woops, there are really 4 triangular prisms;
make an X from the original V angle by extending
the lines to cross.  Now I think now if you
are inside the 2 opposite  prisms with the same
size angle, where one is the original one, then
the viewed angle can get larger.  But if you
in 1 of the other 2 opposing angles, call them
the "left" and "right" extensions from the original
prism, then the angle can get smaller.
This is just my observation, no mathematical
conclusions yet, sorry.  And it may be wrong.

Last edited by John E. Franklin (2008-04-29 04:09:48)

igloo myrtilles fourmis

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