PascGalois JE Help: 2-D Cellular Automata Counts/Period Tab


The Counts/Period Tab

If you wish to count the number of elements in a sequence of levels, the number of elements in a sub-pyramid or frustum, or find the period or death of a finite automaton this tab may be of use to you. The calculations here are independent of those in the image panel so you can count elements in sequences of levels that extend beyond the number of levels you input in the options tab. Furthermore, since the data is regenerated each time a count is done the system does not need to store more levels than what is needed and hence one can do counts on far more levels than would be possible to graph. This tab also has more advanced options for finding the period or death of a finite automaton.

At the top you will see a tab system that allows the user to select a counting mode. The modes are level counting, sub-pyramid counting or frustum counting.

The Counting Modes

In the level counting mode at the top of the tab is the range of levels to be used. This range is inclusive. In this mode the program will count all of the elements on each level in the range.

In the sub-pyramid counting mode, pictured below, the user selects the range of levels, an anchor position and the way that the counting region is increased as you move to the next level.


The first entry in the level range is the anchor level, this is where the sub-pyramid will start. As with the counting of the one-dimensional automata if the beginning level is larger than the ending level the sub-pyramid will be pointing downward. The user then selects the position of the "top" of the sub-pyramid. This can be a little tricky because if you change the level range the position numbering of a point may also change. For example, use the default 4-rule with a single seed element and graph 20 levels. If you place the mouse over the element on the first level in the image tab you will see that the position is row 20 and column 20. Now if you change the number of levels to 30 and regraph the automaton the position of this "top" point is row 30 and column 30. So care must be taken when selecting this position. The position numbering system that is used in the counting is the same as the numbering system used in the graphing routines. Furthermore, the numbering that is used is the same as with the image of the largest level in the range of levels that are counted. That is, if you are counting a sub-pyramid that is between levels 23 and 47 the numbering that is used is exactly the same as if you were to graph the first 47 levels of the automaton. The user should use the graphs to get a feel for the numbering system before trying to count a large pyramid without the aid of a graph.

The region increments allow the user to specify how the counting region is increased as you progress to the next level. The region can also be expanded as either a rectangle or a diamond. For example, say that the green "*" is at row 10 and column 10. If we set all increments to 1 and use the rectangular region the counting regions will be in blue.


Under the same setup but using the diamond regions our counting regions would be.


As with the one-dimensional automaton counts the increments can be 0 and negative as well. For example, if you set the down to -1, keep the rest of the increments as 1 and use a rectangular region you will count the elements on one face of the pyramid.

In the frustum counting mode, pictured below, the user selects the range of levels, an anchor rectangle and the way that the counting region is increased as you move to the next level.


This mode works the same way that the sub-pyramid counting mode works except that the increase in region is always rectangular and the anchor is a rectangular region instead of a single cell. The rectangular region is selected by specifing the upper left position and lower right position of the rectangle.

The Counts/Period Toolbar

The Counts/Period Toolbar consists of seven options (five for infinite automata).

The Element List Toolbar

The element list toolbar has four simple options.


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