Big Ben In N Dimensions.

[PatrickPowers]

Everybody knows about Big Ben, the London clock with four faces. How many faces need N dimensional Big Ben have?

It turns out with N>2 the answer is always four. Let's say there are two clock faces are in the e12 plane, facing in opposite directions, and a similar two in the e34 plane. Then if one is a reasonable distance away from the clocks (you can't see from inside or if very close to Big Ben) then one will have an angle of at least 45 degrees with at least one of the faces. Adding more dimensions doesn't change the situation. Any length in the remaining N-4 dimensions can only serve to increase the maximum angle with the two planes. The greater of the two angles will always be 45 degrees or more.

Example: Observer is at e1. They cannot see the e12 face but are perpendicular to the e34 face.

Observer is at e1+e3. They have an angle of 45 degrees with each of the two planes. Any decrease in either angle causes an increase of the other angle. So there is always an angle of at least 45 degrees. Adding more dimensions may increase and never decreases these angles. All the new dimensions are perpendicular to both clocks so anyone solely in some combination of these new dimensions sees them both full on.

[quickfur]

That's wrong. In n dimensions, Big Ben should have 2(n-1) faces. In 4D, it would have 6 faces, facing each of the 6 cardinal horizontal directions.

[PatrickPowers]

That's wrong. In n dimensions, Big Ben should have 2(n-1) faces. In 4D, it would have 6 faces, facing each of the 6 cardinal horizontal directions.

The math says otherwise. It IS weird.

[quickfur]

You're making the wrong assumption that the distance from the tower is at least a certain number in all directions. That's not necessarily true.

Consider for example the 4D clock tower, with clocks facing the ±X and ±Y directions. Let's say W is the vertical. Place the tower on the origin, and the observer at <0, 0, 0, 100> (in the order <w, x, y, z>). Then he will not be able to see any of the 4 clock faces, because the only cube facet of the tower he will see is the one facing +Z. All the others are obscured by this facet (as can be proven by projecting the tesseractic clock level of the tower along +Z).

[PatrickPowers]

Let's say you have a 2D playing card suspended in 4D space in the wx plane. Then anyone in the yz plane Is perpendicular to every point in it, as the dot product of their vectors is always zero. [w,x,0,0] dot [0,0,y,z] = 0. It would be possible to make a complete orbit of the card while remaining perpendicular to it the entire time. Would the card appear the same during this entire voyage? I'm not sure but I don't see why not.

Here in our 3D world a clock face is plainly visible as long as our line of sight has at least a 30 degree angle with the plane of the clock. Whether that works in 4D I'm not sure but it seems plausible. So let's say that you have a clock face in the wx plane and another in the yz plane, both centered at the origin. Assume you are some minimum distance from the origin. Then no matter where you are, the dot product with one of those faces is going to be at most sqrt(2)/2. You are going to be a minimum of 45 degrees away from at least one of those two faces, rendering it readable. The "worst" you can do is be at [a,a,a,a] whereupon you are 45 degrees from each face. Should you add more dimensions then that doesn't affect this minimum, indeed if a 6D observer were in their uv plane then the dot product with both faces would be zero so they would be 90 degrees away from both faces no matter where they were in that plane.

To us it seems quite strange to have a horizontal clock face but in 4D it appears to arise naturally. As long as you can see it, the orientation doesn't matter much. Of course the hands and numbers of the clock would be 4D, only their arrangement is 2D.

[PatrickPowers]

Some more detail.

a dot b = ||a|| ||b|| cos theta, so theta = arccos( a dot b / ||a|| ||b|| ). Let's say you have b=[w,x,0,0] as the plane of a clock face and the observer is at a=[a,a,a,a]. Take the plane of the clock face to be unit length while a is greater than unit length. theta = arccos( a dot b / sqrt(2)a ) = arccos( a / sqrt(2)a = arccos( 1/sqrt(2) ) = 45 degrees.

Let's say we are in 8D and a=[a,a,a,a,a,a,a,a]. Then we get theta = arccos( 1/sqrt(4) ) which is sixty degrees.

Let's say we are in 8D and a=[0,0,a,a,a,a,a,a]. Then we get theta = arccos(0) which is ninety degrees, the most favorable possible angle. Wild, eh? Completely the opposite of what I expected.

Now I'm headed off for a Tokyo art exhibition which I expect to be fantastic, followed by an equally exquisite concert by the young winners of a competition for avaunt guard classical composers.

[Klitzing]

quickfur is assuming n-1 dimensional clock hyperplanes,
patrick is assuming still 2 dimensional ones ...
--- rk

[quickfur]

Ahh, that's where the misunderstanding came from.

IMO, retaining purely 2-dimensional clock faces in higher dimensions doesn't make much sense. Because geometrically-speaking, a 2D surtope of a clock tower amounts to just an edge or corner of the tower; you'd hardly display time in a clock tower by writing it as dots along the edge of the tower since it'd almost never be noticed, and even if noticed would be so fine as to be practically illegible.

When we 3D beings look at, say, a flower, our attention centers primarily on the middle (the stamen, stigmata, etc.), surrounding which are the petals. A 2D being, OTOH, looking at the projection of a 3D flower, sees only the jagged edges traced out by the outline of the petals, and fails to notice the middle part at all. Similarly, in a 2D world native 2D creatures would design their clocks such that all information is represented on the edge of the clock -- that being the only part they can see -- whereas we 3D beings looking at such a construct would see primarily only the internal mechanisms inside the clock which fills the view from our 3D POV, and likely miss the information represented around its edges.

So, you need an (n-1)-dimensional hyperplane for the display to occupy sufficient volume in the observer's field-of-view in order for it to be visible and legible, even if the information itself is displayed in the equivalent of a multi-prism of a 2D clock face (i.e., the Cartesian product of a 2D clock face with an (n-3)-dimensional cube).

[PatrickPowers]

Ahh, that's where the misunderstanding came from.

IMO, retaining purely 2-dimensional clock faces in higher dimensions doesn't make much sense. Because geometrically-speaking, a 2D surtope of a clock tower amounts to just an edge or corner of the tower; you'd hardly display time in a clock tower by writing it as dots along the edge of the tower since it'd almost never be noticed, and even if noticed would be so fine as to be practically illegible.

When we 3D beings look at, say, a flower, our attention centers primarily on the middle (the stamen, stigmata, etc.), surrounding which are the petals. A 2D being, OTOH, looking at the projection of a 3D flower, sees only the jagged edges traced out by the outline of the petals, and fails to notice the middle part at all. Similarly, in a 2D world native 2D creatures would design their clocks such that all information is represented on the edge of the clock -- that being the only part they can see -- whereas we 3D beings looking at such a construct would see primarily only the internal mechanisms inside the clock which fills the view from our 3D POV, and likely miss the information represented around its edges.

So, you need an (n-1)-dimensional hyperplane for the display to occupy sufficient volume in the observer's field-of-view in order for it to be visible and legible, even if the information itself is displayed in the equivalent of a multi-prism of a 2D clock face (i.e., the Cartesian product of a 2D clock face with an (n-3)-dimensional cube).

As I said, the dots and hands are ND, not 2D. They are arranged in a 2D pattern. You could easily see ND objects arranged in even a 1D pattern like a straight line.