The concept of dimensions and their progression from a point to a hypersphere offers a compelling framework for understanding the structure of our universe. 

•   A point, devoid of dimensions, represents a pure location. 
  
• When allowed to move in one direction, it forms a line, embodying the first degree of freedom. 
  
• Extending this line orthogonally (at 90 degrees) creates a plane, introducing a second degree of freedom. 
• Further orthogonal movement transforms the plane into a volume, adding a third degree of freedom - our 3 spatial dimensions
• Now add another dimension at 90 degrees and there are 4 degrees of freedom - 4 spatial dimensions
  
  This progression illustrates how each new dimension represents an orthogonal push, introducing a new direction of movement. This idea is effectively captured in teaching narratives such as Edwin A. Abbott's "Flatland" and Dionys Burger's "Sphereland," which explore the challenges faced by beings with limited dimensions in comprehending higher-dimensional spaces. 
  
  In this context, dimensions are not mystical layers but rather the number of independent ways an entity can move. When considering three spatial dimensions, the notion of pushing equally in all directions from a central point naturally forms a sphere - a closed topology where every surface point is equidistant from the center. An expanding sphere is not merely a round object but a continuous succession of closed surfaces as its radius changes. This concept can be extended to a hypersphere, which is the result of extending a three-dimensional volume orthogonally into a fourth dimension.

 

A Hornby Train on a Higher‑Dimensional Track

How a simple childhood toy reveals the logic of higher dimensions

When I was a child, my grandfather — a train driver — gave me a Hornby clockwork wind‑up train set. It came with a circular track, a perfect loop. At the time, I simply enjoyed watching the train go round and round. Only later did I realise that this simple toy contains a profound lesson about dimensions.

Imagine a one‑dimensional being living on that track.
Their entire world is a line. They can only look “forward” and “back”. They have no concept of sideways, no ability to step off the track, no awareness that their line is actually curved.

To them:

• a straight line must be infinite
• a circular line feels infinite, because it never ends
• they cannot perceive the curvature that closes the loop

Now imagine their little train begins its journey.
Eventually, it returns to the same point it started from.
To us, this is obvious — it’s a circle.
To them, it is astonishing. They believe they have travelled in a straight line, yet somehow they are back where they began.

Frustrated, they wind the train faster, hoping that speed will break the cycle.
But something strange happens: the faster they go, the more they feel a mysterious force pushing them outward against the side of the carriage.

We know this as centrifugal force.
But to a one‑dimensional physicist, it is an inexplicable new “law of nature”:
a force that increases with speed, with no apparent cause.

Perhaps, in time, their scientists will build a “gravity simulator” — a hamster‑wheel‑like track — to reproduce this strange force.
Perhaps they will even derail the train in their attempts to escape the loop.

From our higher‑dimensional perspective, the explanation is simple:
their world is curved in a dimension they cannot perceive.

The lesson for us

We are in exactly the same position.

Our universe behaves as though the real “action” occurs in four spatial dimensions, while we are confined to perceiving only three. We experience the consequences — the “forces” — but not the underlying geometry.

Just as the one‑dimensional being interprets curvature as a mysterious force, we interpret higher‑dimensional geometry as:

• gravity
• inertia
• acceleration
• curvature of spacetime
• quantum phase
• spin
• charge

These may not be fundamental at all.
They may be projections of deeper structures in a higher‑dimensional space — just as the outward push on the Hornby train is a projection of circular motion into a one‑dimensional world.

In this view:

• our three‑dimensional physics is a shadow
• the true dynamics occur in four spatial dimensions
• what we call “forces” are geometric consequences
• and many of the mysteries of quantum mechanics arise from trying to interpret higher‑dimensional behaviour using lower‑dimensional intuition

The Hornby train is a child’s toy.
But it contains the seed of a profound idea:
a lower‑dimensional world can only ever experience the shadows of higher‑dimensional reality.I

Our universe can be conceptualized as the expanding boundary of a hypersphere, a three-dimensional universe embedded in a four-dimensional structure: A hypersphere. This perspective redefines our understanding of time. Rather than a static dimension, time is seen as emerging from the movement of a wavefront—the active boundary of the hypersphere. As the hypersphere's radius changes, the three-dimensional boundary of our universe expands, representing the ongoing transition from potential to reality. The three dimensional circle, as the time process continues, in a four dimensional space becomes a spiral.

This reframing of time as a process rather than a dimension offers a novel geometric viewpoint. Dimensions become orthogonal permissions, spheres are successions of closed topologies (digital), and our universe is the dynamic boundary of a hypersphere. Time, in this model, is the wavefront that continuously updates the universe's boundary, linking successive closed topologies into a continuum and shaping the unfolding of reality.