Distance, Structure, and Permissible Trajectories in Four-Dimensional Space

More about Distance, Structure, and Permissible Trajectories in Four-Dimensional Space

Abstract

In four-dimensional geometry, metric distance describes how far two points lie apart, but it does not determine how motion may occur between them. This paper separates continuous distance from discrete permissibility by modeling motion inside a completed structure made of adjacent compartments—like a staircase built from drawers. We show that lawful trajectories follow adjacency rules rather than straight-line shortest paths. When projected into lower dimensions, these lawful paths appear stair-like or spiral-like, not as inefficiencies, but as consequences of structure.

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1. Metric distance in four dimensions

Let a point in four-dimensional space be:

p = (x, y, z, w)

The Euclidean distance between two points p1 and p2 is:

d = sqrt( (x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2 + (w2 - w1)^2 )

Differential calculus generalizes local distance with a metric tensor g_ij. In an unconstrained space, shortest paths (geodesics) are the ones that minimize total length.

But this assumes every direction is accessible. Compartmental structures break that assumption.

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2. Trajectories and path length

A trajectory in four-dimensional space can be written as:

gamma(t) = ( x(t), y(t), z(t), w(t) )

Its path length is:

L = integral over t of sqrt( (dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 + (dw/dt)^2 )

If there are no restrictions, geodesics minimize L.

In a drawer-staircase structure, however, motion is not continuous through the glowing volume. It happens by discrete transitions: drawer to drawer.

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3. Structural constraints as adjacency rules

Let the space be divided into compartments (drawers) S0, S1, S2, …

A move is permitted only if it goes between adjacent drawers. Define:

A(S_i, S_j) = 1 if S_j is adjacent to S_i
A(S_i, S_j) = 0 otherwise

We encode permissibility along the path as a constraint C:

C(gamma(t)) = 1 if gamma(t) stays inside drawers and only crosses via allowed adjacencies
C(gamma(t)) = 0 otherwise

So even if two drawers are close in Euclidean distance, you cannot “cut through” the structure. You must follow the rails: adjacency, order, stepwise access.

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4. Constrained motion

The motion problem becomes:

Minimize L subject to C(gamma(t)) = 1

Solutions are generally not straight lines. They are lawful paths that climb compartment by compartment.

Projected into 3D, they look like stair-steps.
Embedded in 4D, they can look like spirals or helices: paths that wrap through a constrained system rather than crossing it directly.

The path is longer in metric distance, but it is the only valid one.