Balls bouncing in a box – in two time dimensions

Let’s kick off this blog with a demonstration of the effects of multiple time dimensions.

We simulate 12 conventional objects bouncing around in a box. By “conventional” I mean that each object is at a well defined position in space and time.

We will start with one dimension of time and then extend it to two and plot the paths of one of the objects. We see that even though the each particle has a definite position at each time coordinate (t1,t2), there is an uncertainty about the direction in which it is currently moving. Each particle also has multiple histories as to how it got to this point.

Let’s first look at the one dimensional time case.


Here we have the 12 objects in a box, 6 blue, 6 red. Similar colours repel, different colours attract. There is no collision detection nor friction. We show the state of the box for the first 10 times, and we are tracing the path of one of the objects.

We are starting from a very highly ordered state, at later times the state in the box is less well ordered. We can see that time flows from left to right. This simulation with only one dimensional time shows the familiar common-sense, Newtonian picture of the objects. At each time each object is in a well defined place. The black line traces one unique history for the object that we are monitoring.

Now let us have a look at the same system with 2 time dimensions.


If we start from the highest ordered state in the top left corner and we want to move along to successively less ordered states, we could move from left to right in the top row, or we could move down the left column, or we could follow a path in between, for example diagonally across or at an angle.

The initial conditions for the vertical and horizontal time directions are very slightly different. (If we made the initial vertical and horizontal components the same, we would get a perfectly symmetric pattern.)

As you can see on the following diagram, which shows one particular space at some time (t1,t2), there is a very definite position for the black ball, but there are several ways it could have got there, and in fact, it did. There is an inherent uncertainty about its momentum. This is a natural consequence of having more than one time dimension and not a designed-in ‘feature’ of our simulation. There are multiple histories associated with each ball.


If we don’t know which (t1,t2) time we are at, but only on which generation we are (meaning distance from the origin, i.e. t1+t2) then it gets even more interesting. At generation 12 we could be on any of the 12 times on the diagonal of the 2-dimensional diagram. If we plot the location of the black particle in each of these boxes on top of each other, using an opacity of 1/12 we get the following picture.


The location of the object is smeared into a cloud. If we test for the position of the object we get the situation of the previous diagram, where we know where the object is, but can’t say by which path it got there. The probability of finding it at a particular location is determined by the blackness level of the cloud picture.

These are natural consequences of having more than one time dimension.

The code that produced these pictures is available on the downloads page.

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