Big Ben In N Dimensions

[Hornbein]

The iconic Big Ben clock in London can be seen from any angle in the horizontal plane. It has four faces. How many faces do you need in N dimensions to get that same visibility?

Four!

A clock face can be seen as long as you are not in the same plane as that face, that is, the angle between the observer and the plane is at least 30 degrees. In mundane 3D we have xy and zy faces and an observer's location is [x,y,z]. We assume that vertical y is always more or less zero, as we only care about viewers who on the surface of the Earth. The minimal maximum angle is then when x=z, which means the angle of the observer to both faces is 45 degrees. x<>z increases the maximal angle.

Now we are in N dimensions with N>3. Any component in additional dimensions decreases the dot product, which increases the maximal angle with the clock faces and hence increases visibility. Taking to extremes, if the observer's location is entirely within these new dimensions then the dot product with both face planes xy and zy is zero, the angle between the observer and both clock faces is 90 degrees, and visibility is maximal to two faces.

Note that the clock's numerals and hands are ND or (N-1)D objects. They are arranged in a 2D pattern.

 

[.Scott]

In 3D, the clocks are arranged on the surfaces of a square vertical cylinder.

I would put my 4D Big Ben on a 4D globe with a 3D surface.
Let use w,x,y,z as our coordinate labels - with "z" being the vertical.
So working within the w,x,y terrain, 4 3D surfaces could best be arranged as the faces of a tetrahedral-hypercylinder - facing 0,0,1,0; 0,.816,-0.577,0; 0.707,-0.406,-0.577,0; -0.707,-0.406,-0.577,0.
The worse case would be a view looking at the tetrahedron cross-section point - for example from direction 0,0,-1,0. In that case, the dot product to any of the three best faces would be 0.577. Arcsin(0.577) is 35.3 degrees - so we are good.

But for 5D I am in trouble because I will be forming a 5D cylinder with a 4D cross-section and I cannot enclose a 4D cross section with only four 3D surfaces. The minimum number of 3D surfaces would be 5 tetrahedrons.

 

[Dr Wu]

It's a well attested fact that blind people having gained (or even regained) their sight through medical intervention often experience extreme difficulties in making sense of the visual data streaming into their eyes. This is explained by the atrophying of that part of the brain (the occipital lobe) that processes visual information.

Using the above as an analogy, it would be interesting to know what a human subject would experience upon being "immersed" into a world governed by N dimensions. A state of profound bewilderment would seem to be the immediate response, but what kind of bewilderment? What would this person actually see? Would he or she be so visually disorientated as to make it hard for them to even maintain their balance? Possibly this issue has already been addressed by Dr Who or Star Trek. Then there's The Politics of Ecstasy by Prof Timothy Leary. Ah, those were the days.

 

[Hornbein]

It's a well attested fact that blind people having gained (or even regained) their sight through medical intervention often experience extreme difficulties in making sense of the visual data streaming into their eyes. This is explained by the atrophying of that part of the brain (the occipital lobe) that processes visual information.

Using the above as an analogy, it would be interesting to know what a human subject would experience upon being "immersed" into a world governed by N dimensions. A state of profound bewilderment would seem to be the immediate response, but what kind of bewilderment? What would this person actually see? Would he or she be so visually disorientated as to make it hard for them to even maintain their balance? Possibly this issue has already been addressed by Dr Who or Star Trek. Then there's The Politics of Ecstasy by Prof Timothy Leary. Ah, those were the days.

My usual assumption is that ND is inhabited by ND people.

A 3D person in a 4D world would be overwhelmed by the big increase in information for which they have neither the hardware nor software to process. Maybe they could see only a 3D slice, which wouldn't be all that useful.

 

[Hornbein]

My usual assumption is that ND is inhabited by ND people.

A 3D person in a 4D world would be overwhelmed by the big increase in information for which they have neither the hardware nor software to process. Maybe they could see only a 3D slice, which wouldn't be all that useful.

It's a well attested fact that blind people having gained (or even regained) their sight through medical intervention often experience extreme difficulties in making sense of the visual data streaming into their eyes. This is explained by the atrophying of that part of the brain (the occipital lobe) that processes visual information.

Using the above as an analogy, it would be interesting to know what a human subject would experience upon being "immersed" into a world governed by N dimensions. A state of profound bewilderment would seem to be the immediate response, but what kind of bewilderment? What would this person actually see? Would he or she be so visually disorientated as to make it hard for them to even maintain their balance? Possibly this issue has already been addressed by Dr Who or Star Trek. Then there's The Politics of Ecstasy by Prof Timothy Leary. Ah, those were the days.

Maintaining balance would be impossible. Crawling would be the only possibility.

 

[sbrothy]

I vote for this thread to be admitted as a runner-up for the (as yet non-existing) award: “Most Gratuitously Geeky Thread of the Year”!

(I hope I used “gratuitous” correctly here. It is of course “tongue in cheek”. I’m just poking fun.)

 

[Hornbein]

I vote for this thread to be admitted as a runner-up for the (as yet non-existing) award: “Most Gratuitously Geeky Thread of the Year”!

(I hope I used “gratuitous” correctly here. It is of course “tongue in cheek”. I’m just poking fun.)

They laughed at my theories....

 

[Dr Wu]

Just a thought: would someone blessed with 4D vision* be able to witness first hand the gravitational distortions affecting the fabric of spacetime? This GR-tweaked ability would enable such a person to glance up at the sun and see it embedded in its own gravity well. An eye-splitting/mind-bending prospect, for sure, yet one that presents intriguing (not to say challenging) possibilities from a writing POV.

* Or should this properly be 5D?

 

[Hornbein]

Just a thought: would someone blessed with 4D vision* be able to witness first hand the gravitational distortions affecting the fabric of spacetime? This GR-tweaked ability would enable such a person to glance up at the sun and see it embedded in its own gravity well. An eye-splitting/mind-bending prospect, for sure, yet one that presents intriguing (not to say challenging) possibilities from a writing POV.

* Or should this properly be 5D?

That would be a different kind of 4D vision, one well worth exploring. Where might such a thing lead? Would looking at an object reveal its past? An immobile object like a rock, would one be able to see thousands of years worth of its past?

What I have in mind when I write 4D is actually (4+1)D, four Euclidean dimensions and one of time.

As long as I have you on the phone, what happens if one extends our 3+1 GR to 4+1 GR? assuming that this is possible. I heard Ed Witten say that the 1/r^2 Newtonian approximation is owing to the three Euclidean dimensions, so I have loosely assumed that 4+1 GR would be 1/r^3.

 

[DaveC426913]

it would be interesting to know what a human subject would experience upon being "immersed" into a world governed by N dimensions. A state of profound bewilderment would seem to be the immediate response, but what kind of bewilderment? What would this person actually see?

In Flatland the book, a Flatlander was lifted out of his plane and shown the wonders of the 3D world. I took issue with that.

The Flatlander would still only be able to see a slice of the 3D world. They would see things change inexplicably - growing, shrinking, sliding around, etc. It would make no sense and be completely overwhelming.

 

[Hornbein]

In Flatland the book, a Flatlander was lifted out of his plane and shown the wonders of the 3D world. I took issue with that.

The Flatlander would still only be able to see a slice of the 3D world. They would see things change inexplicably - growing, shrinking, sliding around, etc. It would make no sense and be completely overwhelming.

I felt the same way, reading it at age 14 or so. Not only that, all their internal fluids would instantly exit their bodies. I was dissatisfied with the Flatland approach. Fifty years later like Frank Sinatra I did it my way.

I suspect that higher dimensional geometry was an influence on Lewis Carroll's Alice In Wonderland with its 2D people. In 1864 that topic was just getting started.

 

[pines-demon]

The iconic Big Ben clock in London can be seen from any angle in the horizontal plane. It has four faces. How many faces do you need in N dimensions to get that same visibility?

I don't get it what is this property of the clock in the Big Ben that allows somebody to see it from every angle? Does it not have just 4 synchronized clocks?

And I guess we are assuming the base is always a square?

 

[Hornbein]

I don't get it what is this property of the clock in the Big Ben that allows somebody to see it from every angle? Does it not have just 4 synchronized clocks?

And I guess we are assuming the base is always a square?

I'm saying the Big Ben clock is a single clock with four faces. It has a square base.

 

[pines-demon]

I'm saying the Big Ben clock is a single clock with four faces. It has a square base.

Ok, but it still has four clock faces? This might be a dumb quesiton but I thought I was missing something. Also, just for the sake of understanding the question, if the base was an equilateral triangle, would that visibility property change?

 

[Hornbein]

Ok, but it still has four clock faces? This might be a dumb quesiton but I thought I was missing something. Also, just for the sake of understanding the question, if the base was an equilateral triangle, would that visibility property change?

Four clock faces, one for each side of the square. For a triangular base the minimum angle between an observer and a face would be 30. So yes that just barely qualifies for my somewhat arbitrary definition of "visible."

 

[pines-demon]

Four clock faces, one for each side of the square. For a triangular base the minimum angle between an observer and a face would be 30. So yes that just barely qualifies for my somewhat arbitrary definition of "visible."

And you want to have symmetrical faces right? you can get angles larger than 30 degrees with an isosceles triangle for example.

 

[Hornbein]

And you want to have symmetrical faces right? you can get angles larger than 30 degrees with an isosceles triangles for example.

With an isosceles the minimum angle goes under 30 degrees, so poor visibility from some points.

 

[DaveC426913]

Oh. OK. 30 degrees is an arbitrary practical viewing angle.

But technically, anything greater than zero is viewable, right?

 

[Hornbein]

Oh. OK. 30 degrees is an arbitrary practical viewing angle.

But technically, anything greater than zero is viewable, right?

It's a clock so the purpose is to be able to read the time. With Big Ben it could be from miles away. With 30 degree angle that maximum distance is halved.

 

[DaveC426913]

It's a clock so the purpose is to be able to read the time. With Big Ben it could be from miles away. With 30 degree angle that maximum distance is halved.

Yes. I'd say we assumed this was an idealized question of geometry, rather than a practical one of user experience.

And we didn't know, when starting out, the nature of those constraints (hard or soft). Put another way: your chickens are not spherical and they do not live in a vacuum.

 

[pines-demon]

It's a clock so the purpose is to be able to read the time. With Big Ben it could be from miles away. With 30 degree angle that maximum distance is halved.

What do I do if I am looking at a vertex? Let's say we put the 6 clock faces in the faces of a cube, what happens when looking at three faces at the same time? What do I use as a valid angle? A valid solid angle?

I am wondering this because the Big Ben could be basically a cuboid with 6 faces but we remove faces that are perpendicular to 1D gravity, so we get 4 valid faces. If we do the same with a hypercube and 1D gravity we have 12 faces left. Why is this not the answer? Is it the vertices?

 

[Hornbein]

What do I do if I am looking at a vertex? Let's say we put the 6 clock faces in the faces of a cube, what happens when looking at three faces at the same time? What do I use as a valid angle? A valid solid angle?

I am wondering this because the Big Ben could be basically a cuboid with 6 faces but we remove faces that are perpendicular to 1D gravity, so we get 4 valid faces. If we do the same with a hypercube and 1D gravity we have 12 faces left. Why is this not the answer? Is it the vertices?

That's what you would think. But linear algebra shows otherwise. That's why the question is interesting : the result is so surprising. You would think the more dimensions the more faces you would need but instead the extra dimensions improve the visibility of the four faces. To see a clock face you need to be more or less perpendicular to it. The dimensions are perpendicular to one another so the more dimensions, the more perpendicularity.

 

[pines-demon]

That's what you would think. But linear algebra shows otherwise. That's why the question is interesting : the result is so surprising. You would think the more dimensions the more faces you would need but instead the extra dimensions improve the visibility of the four faces. To see a clock face you need to be more or less perpendicular to it. The dimensions are perpendicular to one another so the more dimensions, the more perpendicularity.

So we keep the Big Ben as it is no matter the dimension? Make sense. But then wouldn't the visibility of an equilateral-triangle base clock increase too?

 

[Hornbein]

So we keep the Big Ben as it is no matter the dimension? Make sense. But then wouldn't the visibility of an equilateral-triangle base clock increase too?

Yes it would.

Home clocks on a wall would be 2D no matter how many dimensions you've got.

 

[Hornbein]

So with a clock, the more dimensions you have the more perpendicular you would tend to be to that clock. Now instead consider standing on a planet looking at the sun. The more dimensions you have the less perpendicular the the sun tends to be to the surface, that is, the lower in the sky it is likely to be. This is because the surface of the planet is N-1 dimensional. There is only one perpendicular direction w.r.t. that and the Sun isn't that likely to be there, while with the clock has the many N-2 dimensions perpendicular to it. This seems weird to us because we are used to both the clock and the surface of Earth having the same dimensionality.