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Originally Posted by only_human
I have recently reread Wolframs "A New Kind of Science" and with all the Cellular Automata in it I wanted to develop a feel for binary patterns on paper and possibly to assign a more arithmetic meaning to them.

I've read it, too. (Well, part way through  then I browsed.) The trap Wolfram seems to have fallen into (based on my reading and the comments of others) was getting so wrapped up in his own interpretation of his patterns that he neglected to adequately check back with others' results, before publishing, to find out whether he had reinvented the wheelbarrow.
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By looking at bits on paper and assigning them this positional value to them I hope to also develop a feel for when they can not be put into the form of a rectangle of a composite number (therefore prime).
I also wanted to fiddle with patterns that are fully filled with a result. The fact that in absolute value hex FF x FF = FE01 leaves me feeling a bit vaguely dissatisfied.

Does 99 x 99 = 9801 or any other (b1) x (b1) = b
^{2}  2b + 1 leave you dissatisfied?
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I hope by having a representation method that can be fully filled as this one would be for that result might make patterns a bit more meaningful to me. In this 2d representation, FF x FF would occupy an 8 bit by 8 bit square full of ones. Of course that is devoting 64 bits for a value that could be held in 16 bits in an ordinary product result.

... because your representation shows all the intermediate steps before the final summation, whereas the product shows only what's left after the summation.