The Update Rule tab looks like,
The update rule is set, by default, to the Pascal's Triangle update rule. So if you never use an update rule other than Pascal's Triangle you have no need to alter this information. On the other hand, if you wish to experiment with other update rules here is how the updating system is set up.
The rule is any valid group theoretic expression that uses variable names for the group elements and integers for the exponents. Juxtaposition for the group operation is not recognized, you must use a * to denote a group operation. Powers are denoted by ^ and any negative powers must be put in parentheses. Variable names can be any string of letters. Numbers, underscores and special characters are not permitted. For example, the following are legitimate update rule expressions, a*b, a^2*b, b*a*b^(-1), a*b*a^(-1)*b^(-1), and a^3*b^(-4). The following are not legitimate expressions, a^2b and a*b*a^-1*b^-1.
The update grid is how you tell the program where to get the elements that
are to be used in the update formula. There must be one and only one X in the
entire grid and it must be on the last row of the grid, but it may be in any column.
The position of the X tells the program that this is the cell to be updated. The
other table entries are names of variables used in the update rule. When the program
generates its data it will start at the X and substitute the respective cell entries
in for the corresponding variables in the formula to determine the value of
the new cell. For example, say we are working with Z5, infinite
automaton, using a column alignment, a seed of just a single 1 and the following
update rule.
From the grid, when we are updating a cell we will take the element from the last row and one cell to the left and substitute that element in for a in the update formula and we will take the element from the last row and one cell to the right and substitute that element in for b in the update formula. We then evaluate the formula and place the result in the X cell.
Graphing the first several rows of this automaton gives the following image.
Note that each cell is the sum (mod 5) of the elements to its NW and NE.
The update rule also allows for the inclusion of constants. Constants must be
group elements and are designated in the constants list at the bottom of the
tab, not in the update rule expression.
To create, delete and edit a constant you use the toolbar buttons, discussed
below. For now, let's assume that we have Z5, infinite
automaton, using a column alignment, a seed of just a single 1 and the following
update rule.
Just as in the last example, each cell is the sum (mod 5) of the elements to
its NW and NE but this time we add 3 (mod 5) to that result.
As a last example, say we are working with the Quaternions, default
element is the identity, seed is "i j" and has the following update rule.
When graphed, we have the following image.
There are eight tools for the update rule tab.
Simply type in a constant name and group element value. The name follows the same restrictions as the variable names, all letters. The value must be a legitimate group element value for whatever group you are currently working with. The syntax of the constant name and the value are not checked when you input or edit a constant. To check the syntax select the syntax checking tool.