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....