Hi all,

here's a little conundrum that has been bugging me for a while now.

If you give a little thought to it, it is apparent that an object that has zero length in the direction that it travels can only move in that direction in discrete jumps. That is to say, it cannot smoothly pass through every possible point between two positions but must do so in discrete steps.

Invoking the idea of making infinitely small movements in infinitely small periods of time doesn't solve this because no matter how small the step is it will still be greater than 0. I guess this can be shown mathematically by the statement: 1/infinity > 0

This is all well and good, and provides a nice model for lightspeed objects such as photons etc. where the object does not accelerate up to lightspeed but immediately starts moving at 'c' and, once moving, moves at the same constant velocity (for any given medium through which it travels). It would also fit with the phenomenon where the measured speed of such objects, regardless of the relative motion of the detectors, is always the same.

Time, of course, is also relevant to this. Not only must there be a difference in position for an object to have moved but there must also be a difference in time – if there's no difference in time it means that the object is just simultaneously in two places at the same time as opposed to having moved between the two positions.

If an object has moved between two positions and has done so in a discrete step it's then difficult not to wonder if time also progresses in discrete jumps as well and in fact, I think it could be argued that it does.

For example, if one considers an event occurring in time, we would generally think of it occurring over the period between the time that it started occurring and the time that it had finished occurring. We can look at this as saying that the proportion of the event that has occurred = 0 when the event starts, is 0.5 half way through the duration of the event and is 1 by the time the event has finished.

However, that starting value of 0 brings us back to a similar situation to the zero length object, where any change in value from 0 must be greater than 0 and therefore results in a discrete step. To be sure, in this case we're dealing with a zero thickness boundary rather than a zero length object but in either case there needs to be a discrete gap between them to prevent them from being in exactly the same place.

Actually, thinking about it even further, numbers themselves are also subject to this zero thickness boundary issue. While we can say that the number of values between say 0 and 1 is infinite, each value has zero 'thickness' and therefore every possible value requires a discrete separation from any other number.

Getting back to the conundrum part, once you've started looking into this zero thickness boundary stuff and it's implications, it's difficult to come up with a model for the normal sort of motion we see every day, where objects accelerate and appear to smoothly pass through every point between two positions. For example, for any object that exists, but does not exist in multiple places at the same time, we can say that at any one instant the proportion of it that exists at one location is 1 while at another location it is 0. However, if the object moves from the first position to the second that proportion must change and we're back to needing a step or gap to do so. A further interesting aspect of this is that it doesn't seem to just apply to the objects themselves, where the physical surface could be considered to be imprecise, but also if it is spinning, to it's axis and plane of rotation.

Just as a further bit of background interest, it's interesting to look at the different classes of n-dimensional objects and how they can move in zero length directions.

A zero dimensional object i.e. a point, can move in any direction and it's radiation pattern from a point of origin would be spherical, defining a volume. A one dimensional object i.e. a line, can move in any radial direction out from it's axis, in the shape of a circle and would define an area. A two dimensional object can only move in the direction of it's 'faces' and it's radiation pattern would be a line. A three dimensional object, in three dimensional space, doesn't appear to be able to move at all, nicely describing a point.

A three dimensional object in four dimensional space would have no problem because it would have zero thickness in the fourth dimension and if we apply it to four dimensional space/time it implies that three dimensional objects are moving in discrete steps, at 'c' but in the time dimension.

Heh:) – there's lots more to this – wave-function and indeterminate numbers type stuff seems to be a requirement , but I've gone on enough already and I'd appreciate hearing some other people's thoughts:)