April 29, 2023
“Is there some easy way of telling, by looking at a three-dimensional atlas, whether the manifold it represents is the three-dimensional surface of a four-dimensional sphere?”
The fourth spatial dimension is movement.
A static three-dimensional model of the solar system is a three-dimensional “surface” of its fourth-dimensional history of movement and change.
The point is where we start.
The line is the measurement between two points, or we may say it is the point “stretched” into the next dimension.
The square is the line “stretched” into the next dimension. It doesn’t have to be a square to be two-dimensional, but just needs to have a secondary property (width and length) that is measured spatially.
Again, the cube is the square “stretched” into the next dimension.
How, then, is the cube “stretched” into the next dimension? In all other cases we see the point, the line and the square retaining their properties, yet being “pulled” or “stretched” into a direction that its current form has no existence of.
How do we do that with a cube?
Imagine, if like the point, the line and the square, we “stretch” the cube in some way, retaining its current properties but copying its form over and over like we did with the point, line and square. We would see something like a trail of cubes all following each other, leaving infinite copies wherever the original cube goes.
What do we commonly associate with that sort of image? Movement.
We live with three-dimensional objects all around us, but we don’t see them leaving trails of themselves wherever they go, do we? No, but in a way they do. Dimensions are not physical, but measurable qualities, and a spatially measurable quality we can add to any three-dimensional object is its change over time — its position in space, or history. Where has it been, and where is it now? That is any object’s fourth spatial dimension. Space is needed for movement to be possible.
Movement is a new way to utilise space, thus a new spatial dimension which retains the properties of an object and adds a new measurable spatial quality — its motion in relation to other objects or itself or both. Measuring and observing it through recordings and memory is how we discover and define the ever-changing universe we live in. Everything moves in some way, and that is the principle in which our universe is built upon. Movement is energy.
The question poised was how to tell, by looking at a three-dimensional atlas, whether the manifold it represents is the three-dimensional surface of a four-dimensional sphere.
A sphere is three-dimensional, so I will simply replace the word with ‘model’.
A three-dimensional atlas I will refer to is a three-dimensional model of our solar system. A static model can be made, and, to be completely accurate, it must show the planets aligned in such a way that they once were, displaying every three-dimensional quality — sizes of planets, distances from each other, etc — but what does it need to be even more accurate? It needs its history of movement, and not just of its cyclical patterns, but of its complete formation. The static three-dimensional model, as accurate as it can be, is just a “surface” representation of one “point” in its constantly changing spatial form, thus a “shadow” of its four-dimensional history of existence.
Without a recording or memory of an object’s history, it can be impossible to accurately know its history of movement and change, but every bit of matter has a history, even if not imprinted in some measurable way, and much has been done to surmise the histories of the world we live in.
Comparing the infinite “shadows” of an object’s, or model’s, movement through space and time is how we measure its fourth-dimensional spatial quality — its history of movement and change.
Originally posted at https://firetonguezine.com/math/2023-04-29-response-to-poincare-conjecture.php
Leave a Reply