Can rectangles really be halved by angling them?

[julian]

There was a recent post about "parroting Einstein".

There are these people I know who have got themselves a new idea about rectangles. They think that you can take a 2 by 1 rectangle and make it half the size, not by putting it into the distance but by angling it!

This seems like utter nonsense...if you try taking a 2 by 1 rectangle and try to halve the length and width by angling it you end up with a severely distorted diamond shape where one diagonal is much much bigger than the other. They want to claim otherwise and that it can be angled so that it is half the size in every regard (almost) with one diagonal being about 49% and the other 51%.

I've told them to take a pair of identical 2 by 1 rectangles, keeping one as the reference rectangle and angle the other, and try to do this idea of theirs...Months later and they still couldn't do it, and when I pointed it this out to them (and get this) to compensate for not being able to do it they told me that they are like Einstein discovering relativity! That it is a revolution.

They couldn't do it so they called themselves Einstein! Aside from this being bonkers, I had issues with this like rectangles having been around for millennia and are kind of of low-tech and anybody can get hold of them and that somebody might have noticed by now. Comparing it to relativity which is far more experimentally high-tech, inaccessible to most people and profound and even that was discovered over a hundred years ago. Since then we've had quite a few revolutions in science - that rectangles are going to `post-date' them? Just sounds like delusions of grandeur.

Here is their latest argument...O.K. so they couldn't do it with a pair of identical rectangles, so they have taken a situation of a pair of slightly non-identical rectangles with one being very very slightly bigger than the other. The bigger rectangle is at an angle - it looks no where near half the size of the `reference' rectangle and the diamond shape is already far too distorted...BUT they are THEN saying if only we had used identical rectangles this time then it would have magically worked! They mention the word "sensitive" as if it were an actual explanation but it is jut a word. They don't even seem to care what angle this rectangle is at.

I try telling them that you can ALWAYS get hold of a pair of slightly non-identical rectangles and as such you can't deduce anything from this let alone some bizarre `Einstein revolution' but they still think they are doing science. I tell them you could generalize this `argument' to anything...

 

[Mark44]

There was a recent post about "parroting Einstein".

There are these people I know who have got themselves a new idea about rectangles. They think that you can take a 2 by 1 rectangle and make it half the size, not by putting it into the distance but by angling it!

What do you mean by "angling" it?

This seems like utter nonsense...if you try taking a 2 by 1 rectangle and try to halve the length and width by angling it you end up with a severely distorted diamond shape where one diagonal is much much bigger than the other. They want to claim otherwise and that it can be angled so that it is half the size in every regard (almost) with one diagonal being about 49% and the other 51%.

I've told them to take a pair of identical 2 by 1 rectangles, keeping one as the reference rectangle and angle the other, and try to do this idea of theirs...Months later and they still couldn't do it, and when I pointed it this out to them (and get this) to compensate for not being able to do it they told me that they are like Einstein discovering relativity! That it is a revolution.

They couldn't do it so they called themselves Einstein! Aside from this being bonkers, I had issues with this like rectangles having been around for millennia and are kind of of low-tech and anybody can get hold of them and that somebody might have noticed by now. Comparing it to relativity which is far more experimentally high-tech, inaccessible to most people and profound and even that was discovered over a hundred years ago. Since then we've had quite a few revolutions in science - that rectangles are going to `post-date' them? Just sounds like delusions of grandeur.

Here is their latest argument...O.K. so they couldn't do it with a pair of identical rectangles, so they have taken a situation of a pair of slightly non-identical rectangles with one being very very slightly bigger than the other. The bigger rectangle is at an angle - it looks no where near half the size of the `reference' rectangle and the diamond shape is already far too distorted...BUT they are THEN saying if only we had used identical rectangles this time then it would have magically worked! They mention the word "sensitive" as if it were an actual explanation but it is jut a word. They don't even seem to care what angle this rectangle is at.

I try telling them that you can ALWAYS get hold of a pair of slightly non-identical rectangles and as such you can't deduce anything from this let alone some bizarre `Einstein revolution' but they still think they are doing science. I tell them you could generalize this `argument' to anything...

I am completely befuddled at what you're trying to say.

 

[phinds]

I do get what you are saying that they think they are doing. My advice is that you cannot change the minds of morons and trying to do so is an exercise in what the military call "pissing up a rope".

 

[julian]

by "angling" it I mean putting it at an angle.

They think they can put it at an angle to make it half the size in ever regard - almost - they think they can do it but with only a very very slight distortion.

 

[zoobyshoe]

It sounds to me like it might be a sort of running joke, like the Flat Earth society, or "planking". The point being to have some absurd premise but to pretend you're completely serious about it. You have to have a weird, Andy Kaufman kind of sense of humor to want to do it, but there are people like that.

 

[Mmm_Pasta]

Can't you "angle" it along the diagonal? That'll cut the rectangle in half. That's how I'm understanding the post anyway. I agree with Mark.

 

[collinsmark]

by "angling" it I mean putting it at an angle.

They think they can put it at an angle to make it half the size in ever regard - almost - they think they can do it but with only a very very slight distortion.

I'm still not understanding what you mean by "angling." Do you mean viewing the projection of a 2-dimensinoal rectangle when the rectangle is tilted at some angle in 3-dimensional space? That's the closes thing I can imagine given the description.

Have your acquaintances tried measuring the rectangle's shadow when the rectangle is held at various orientations with respect to the light source and shadow plane? (You can use the sun as a quick-and-dirty light source if you want one far from the rectangle.)

By the way, the mathematics involved in predicting the shape of the shadow have been well established for a long, long, time. There's nothing revolutionary in that respect. (Geometry and trigonometry are all that are necessary if you are working with flat surfaces, and point-light sources, and neglecting diffraction effects.)

But perhaps I am misinterpreting what your acquaintances are trying to do. Would it be possible to show a diagram of what you mean by this "angling"?

 

[collinsmark]

Here is something that doesn't [necessarily] involve shadows:

Start with a 2-unit by 1-unit rectangle. Orient it such that it is 1-unit across(horizontal) and 2-units up (vertical). In other words orient it such that a 1-unit side is sitting on the ground or on the table, and the 2-unit dimension is straight up, oriented 90^o from the horizontal plane. Now step back and view the thing face-on, perpendicular to the plane of the rectangle. Preferably view the rectangle from a distance through a telescope or binoculars, for reasons which should become clear in a moment [it's to reduce the effects of perspective]. You'll see a 2x1 rectangle, with the long side up, i.e., 1 unit in the horizontal axis and 2 units in the vertical axis.

Now tilt the rectangle toward the viewer or away from the viewer (it doesn't matter which) by rotating the rectangle along the side that is touching the ground or the table. Angle it such that the rectangle now makes an angle with the horizontal of 14.4775^o (approximately).

Now look at the rectangle again, from the same location as before with binoculars or a telescope, and you will see what seems to be 1x2 rectangle with 1 unit along the horizontal axis and 0.5 units along the vertical axis (approximately).

[Edit: Or if you don't have a telescope or binoculars, you can do the same thing by looking at the rectangle's shadow. Orient the shadow plane such that it is perpendicular to the line created from the light source to the rectangle. A far away light source (like the sun) might work best, in order to reduce perspective distortion.]

[Another edit: Or if you want the physical rectangle to always remain perpendicular to the horizontal (suppose you already have a clip & stand that holds it 90^o from the table), then do the same thing except start with the long (2-unit) axis being along the horizontal and the short axis (1-unit) being along the vertical. After viewing it for the first time (perpendicular to the plane of the rectangle), twist the rectangle by rotating it along the vertical axis by 75.5225^o (approximately). Now you'll see what seems to be a rectangle that's 1 unit up, and 0.5 units across (approximately). You'll still need to view it from far away though (binoculars/telescope), or use a far away light source if you are using the shadow method.]

[Even yet another edit: by the way, if you are wondering where I got those angles from,
$$14.4775^o \approx \sin^{-1} \left( \frac{1}{4} \right)$$
$$75.5225^o \approx \cos^{-1} \left( \frac{1}{4} \right)$$
where $\sin^{-1}$ and $\cos^{-1}$ are the inverse sine and inverse cosine functions respectively.]

 

[collinsmark]

With respect to how Einstein might fit into this whole thing, there might be a loose and subtle relationship.

When analyzing Einstein's special theory of relativity, Lorenz transformations are invariably involved somehow. Lorentz transformations are very similar to the exercise outlined in my previous post except (a) it involves the three dimensional projections of objects that exist within four dimensional spacetime [objects such as you and me, rectangles cut out of cardboard, and pretty much everything else], and (b) the math is a little different in that it involves hyperbolic functions instead of trigonometric functions (hyperbolic functions such as sinh and cosh instead of sin and cos; and instead of the normal Pythagorean theorem of $c^2 = a^2 + b^2$, things are of the form $\Delta s^2 = - \Delta t^2 + \Delta x^2$ (and that last equation assumes the speed of light is normalized, such as being in units of 1 light year per year, or one light second per second).

This four dimensional spacetime is usually attributed to Herman Minkowski rather than Einstein though. Albert Einstein was a former student of Minkowski. But it was Minkowski that took Einstein's relativity and started treating Lorenz contractions as projections in 3-dimensions from a 4-dimensional spacetime.

So if your acquaintances are suggesting that angling this rectangle is somehow related to Einstein's work in relativity, then yes, I suppose there is a small relationship, sort of. Understanding the experiment with angling a rectangle is first step toward understanding Einstein's special theory of relativity, in terms of four dimensional spacetime.

 

[julian]

Can't you "angle" it along the diagonal? That'll cut the rectangle in half. That's how I'm understanding the post anyway. I agree with Mark.

They want to make the whole thing half the size including BOTH diagonals (well nearly half the size - make one diagonal look 51% the size and the other say 49% the size). Angling it along a diagonal wouldn't change that diagonal obviously, so that wouldn't do what they want it to do.

 

[julian]

I'm still not understanding what you mean by "angling." Do you mean viewing the projection of a 2-dimensinoal rectangle when the rectangle is tilted at some angle in 3-dimensional space? That's the closes thing I can imagine given the description.

Yes.

 

[julian]

Here is something that doesn't [necessarily] involve shadows:

Start with a 2-unit by 1-unit rectangle. Orient it such that it is 1-unit across(horizontal) and 2-units up (vertical). In other words orient it such that a 1-unit side is sitting on the ground or on the table, and the 2-unit dimension is straight up, oriented 90^o from the horizontal plane. Now step back and view the thing face-on, perpendicular to the plane of the rectangle. Preferably view the rectangle from a distance through a telescope or binoculars, for reasons which should become clear in a moment [it's to reduce the effects of perspective]. You'll see a 2x1 rectangle, with the long side up, i.e., 1 unit in the horizontal axis and 2 units in the vertical axis.

Now tilt the rectangle toward the viewer or away from the viewer (it doesn't matter which) by rotating the rectangle along the side that is touching the ground or the table. Angle it such that the rectangle now makes an angle with the horizontal of 14.4775^o (approximately).

Now look at the rectangle again, from the same location as before with binoculars or a telescope, and you will see what seems to be 1x2 rectangle with 1 unit along the horizontal axis and 0.5 units along the vertical axis (approximately).

[Edit: Or if you don't have a telescope or binoculars, you can do the same thing by looking at the rectangle's shadow. Orient the shadow plane such that it is perpendicular to the line created from the light source to the rectangle. A far away light source (like the sun) might work best, in order to reduce perspective distortion.]

[Another edit: Or if you want the physical rectangle to always remain perpendicular to the horizontal (suppose you already have a clip & stand that holds it 90^o from the table), then do the same thing except start with the long (2-unit) axis being along the horizontal and the short axis (1-unit) being along the vertical. After viewing it for the first time (perpendicular to the plane of the rectangle), twist the rectangle by rotating it along the vertical axis by 75.5225^o (approximately). Now you'll see what seems to be a rectangle that's 1 unit up, and 0.5 units across (approximately). You'll still need to view it from far away though (binoculars/telescope), or use a far away light source if you are using the shadow method.]

[Even yet another edit: by the way, if you are wondering where I got those angles from,
$$14.4775^o \approx \sin^{-1} \left( \frac{1}{4} \right)$$
$$75.5225^o \approx \cos^{-1} \left( \frac{1}{4} \right)$$
where $\sin^{-1}$ and $\cos^{-1}$ are the inverse sine and inverse cosine functions respectively.]

It is not allowed to be that simple...there are markings on the rectangle, say an arrow, so that you always know what is the longer edge of the rectangle and what is the shorter edge.

 

[collinsmark]

It is not allowed to be that simple...there are markings on the rectangle, say an arrow, so that you always know what is the longer edge of the rectangle and what is the shorter edge.

Is it allowed for the rectangle to be moving very fast?

For example, if when stationary, assume that you measure the dimensions of a rectangle (some physical rectangle cut out of cardboard for example, measured in the same reference frame as the rectangle) to be 2 units horizontal by 1 unit vertical.

Then allow it move very fast along the horizontal, you could then measure the same exact rectangle to be 0.5 units along the horizontal and still 1 unit along the vertical. Is this what they meant?

Special relativity allows this, but the rectangle must be moving pretty fast.

 

[julian]

There is no moving at all. There could be finite size effects from not using a telescope - generally though the `angled' rectangle must be above the reference rectangle.

 

[dipole]

If you're so convinced they're wrong, why don't you just prove it? Take a little bit of trigonometry and you should be able to easily show it is or isn't possible. Until then you're just making wild claims as much as they are. Plus I still have no idea what the hell you're actually talking about.

 

[collinsmark]

It is not allowed to be that simple...there are markings on the rectangle, say an arrow, so that you always know what is the longer edge of the rectangle and what is the shorter edge.

There is no moving at all. There could be finite size effects from not using a telescope - generally though the `angled' rectangle must be above the reference rectangle.

If that's the case, and if motion is not allowed, then I don't think it can be done.

If you did allow motion, you could put the rectangle tall side up for example*, and angle it toward or away from the viewer to be 30^o with respect to the horizontal plane. Now it is projected as a square (sin30^o = 1/2), being 1 unit tall and 1 unit wide. Then have it move along the horizontal (with the velocity vector perpendicular to the angle of the viewer) at $\frac{\sqrt{3}}{2}$ times the speed of light, and Lorenz contractions would cause the horizontal width to be compressed by 1/2. That way, it is measured as 1 unit tall, and 1/2 unit wide, and the markers on the rectangle's edges still conform to the original.

*(Actually it wouldn't matter which side is "up," in the way I describe it. It could be rotated to any arbitrary angle (but still perpendicular to the viewer), so long as the rest is adhered to, such as tilting the thing 30^o from the horizontal toward or away from the viewer, and the velocity remaining perpendicular to the viewer and along the horizontal.)

But without motion, none of this would be possible.

 

[zoobyshoe]

If you're so convinced they're wrong, why don't you just prove it? Take a little bit of trigonometry and you should be able to easily show it is or isn't possible. Until then you're just making wild claims as much as they are. Plus I still have no idea what the hell you're actually talking about.

This would be a practical demonstration of what I think he's saying (he'll have to confirm or deny):

On poster board, draw a 1x2 rectangle. Inside it draw another 1x2 rectangle all of whose dimensions are 1/2 the original.

Cast the shadow of a separate 1x2 rectangle onto the larger of the two drawn rectangles. Adjust the distance until the shadow is congruent with the larger rectangle.

No, only by tilting and rotating the separate rectangle, not by adjusting its distance from the poster board, cause the shadow to become congruent with the smaller, 1/2 dimension 1x2 rectangle.

 

[julian]

I have tried taking a pair of 2 by 1 rectangles and making one half the length and width of the other by putting it at an angle but all that happens is that you get a very distorted diamond shape where one diagonal is much longer than the other.

 

[julian]

There is an obvious trig calculation I can do. Easier to explain with diagrams. Might LATEX it tomorrow.

 

[zoobyshoe]

This would be a practical demonstration of what I think he's saying (he'll have to confirm or deny):

On poster board, draw a 1x2 rectangle. Inside it draw another 1x2 rectangle all of whose dimensions are 1/2 the original.

Cast the shadow of a separate 1x2 rectangle onto the larger of the two drawn rectangles. Adjust the distance until the shadow is congruent with the larger rectangle.

No, only by tilting and rotating the separate rectangle, not by adjusting its distance from the poster board, cause the shadow to become congruent with the smaller, 1/2 dimension 1x2 rectangle.

Last sentence should read: "Now, only by tilting and rotating the separate triangle..."

 

[zoki85]

There are these people I know who have got themselves a new idea about rectangles. They think that you can take a 2 by 1 rectangle and make it half the size, not by putting it into the distance but by angling it!

 

[EnSlavingBlair]

Tell them it's perfectly possible to do, all you have to do is distort the rectangle (ie. fold it!) ;)

 

[julian]

Nearly written up pdf file...you get a very very distorted shape. No bending allowed either - rather plastic the rectangle. Already know it is nonsense because I have already tried it with actual rectangles. Nobody has ever heard of this nonsense - you don't get people going oh from these seats the tennis court is half the size in every regard...Not just in tennis - not heard of in architecture, maths, painting ect

 

[julian]

If you take a 2 by 1 rectangle and rotate about the z-axis by 60 degrees and then the y-axis by 60 degrees to try and half the width and length - the projected diagonals turn out to be 81% and 25%...so very distorted shape.

Trivial typo: the second page it should say "The y component will be -1/2 cos 60 = -1/2 1/2". It is O.K. because I write down the correct thing in the next equation.