Math Writing by Stuart Hood

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2023-04-29 - A Response to the Poincare Conjecture

2023-05-03 - Motion - The Fourth Spatial Dimension

2023-05-05 - Laws of Motion

2023-05-05 - Time - The Measure of Duration

2023-06-08 - Motion - The Fourth Spatial Dimension: Part II

2023-06-08 - Letter to Clay Mathematics Institute 		Motion - The Fourth Spatial Dimension
May 3, 2023

What are the properties of motion? It is movement in space. An object can change its position in relation to another object by a change in distance or rotation on its axis. An object can move in relation to itself through the transformation of its shape by distortion.

Motion is a property of physical reality that is measured by the three lower spatial dimensions plus a property of time.

How do we spatially measure motion? An object’s pathway, or history of position in space, is measurable by three dimensional qualities, but for complete accuracy it also needs time. What we see at any moment is the universe’s three-dimensional qualities, but its fourth dimensional spatial qualities are known through our memory and methods of recording, or possibly by deducing from what instances of change have already occurred.

In a three-dimensional model we can document a point’s movement through a succession of x, y, z, t measurements. In relation to another object, we can say that it is a position of x, y, z to it, and if we want to track any motion between the two we take the x, y, z of each interval of time during when the motion occurred. In relation to itself, we take the changing three-dimensional measurements and pair them with a point in time.

“Squaring” Motion


Here we square dimensions to commonly represent them, but an object doesn’t need to be exactly squared, per se, in order to have the dimensional properties.

A line takes on one dimensional quality. We can measure it to 2m. Pulling that 2m line out another 2m in a different spatial dimension creates 4 “square” metres, which are four 1 by 1 squares. Pulling the 4m2 squares out another 2m in a different spatial dimension creates a cube of eight 1 by 1 by 1 cubes. How do we do the next representation? Like before, let’s pull the 8m3 cube 2m in another spatial dimension by moving it. Now we have the 8m3 cube of the first position and the 8m3 cube of the second position which total to 24 or 16m4.

This does not work if we change the original size to 3m, though. We would get 3m, 9m2, 27m3 and then the two copies of 27m3 to make 54m4 instead of 34 which is 81. So what is the “squaring” equivalent of motion? Stretching it in the same way as we did the others didn’t add up mathematically. If we try the same thing with a size of 1 we also get different results: 1m, 1m2, 1m3, then the two copies of the 1m3 are 2m4, not the mathematically anticipated 1m4. Perhaps the “squaring” of motion should include how many copies we record of the three-dimensional object’s path. With the measurements we are using, as the numbers rise we are simply getting more detailed about the same shape, cutting it up into more and more segments, increasing the divisions within it. We can do the same with its path of motion. Counting the original position as zero we take a snapshot every time it moves 1m. So a 1m3 cube moves 1m and we have one instance of change to measure — 1m4. An 8m3 cube moves 2m and we take a snapshot every 1m giving us two instances of change to measure — 8 times 2 which is 16m4. A 27m3 cube moves 3m and we take a snapshot every 1m giving us three 27m3 cubes, or 81m4. It takes three steps of the base measurement to move out of itself.

Nevertheless, these are just exercises in symmetry. As a line is squared evenly, the square is cubed evenly, how is the cube “moved” evenly?
Going back to the pairing of three-dimensional measurements with points in time, we can do it in different ways. For the sake of fluidity, the time measurement should be a repeating, constant interval.

The changes of the x, y, z dimensions indicate that motion has occurred, and the regular ‘t’ interval are the snapshots of states between motion. In each instance we are looking at a three-dimensional snapshot, or “surface,” of the motion that transpired in between them. The accuracy of information of the motion depends on how many snapshots or surfaces we can see within a period of time. Like watching a slow motion replay of a sporting video broadcast, we can more accurately determine at what time an object was at a certain position (like a puck crossing a goal line before the time of the period expires) in better ways with higher and higher frame rates. Now, the frames aren’t the motion dimension itself, but the lower-dimensional representation of it, and by playing them in succession like a video we can partially recreate the fourth-dimensional motions they measure (capture).

What is the “frame rate” of reality? Is it connected to the speed of light?
Previously we expanded each dimension by the same size as the original line. In the fourth-dimension of motion, we moved it in a direction at a distance of the same size. What would it look like to instead expand and contract by the same size?

A cube contracting its own size would progressively get smaller toward zero. Expanding, it would grow to double its size.

This is what the “shadow of a hypercube” looks like — the tesseract. Only it is a collection of “shadows” of its path of movement.

What are the extremities we know of motion? Light. Energy. Motion is the building block of the universe. It is known that everything vibrates. Without it, the three lower spatial dimensions would be of no use. It is intrisically tied to everything in existence.