Where the first inversion is at 4-5.
The reason I do this is because of the relation to primes and to make the pattern seem more consistent. Then of course the next step would be exponent notation:
…
8 == ((1+1)^(1+1+1)
…
So the idea of choosing the correct method of patterning can change dramatically with small changes in rules, such that we can have:
8 == (1+1+1+1+1+1+1+1)
8 == ((1+1)*(1+1+1+1))
8 == ((1+1)^(1+1+1)
And the places of inversion will change according to method.
A question in an evolutionary context: Say the operators are symbols of evolutionary complexity, increasing from addition to multiplication to exponent, wouldn’t use be situational? Given specific circumstances could one “8” be more valid than another “8″, even though the understanding is that they amount to the same thing? Is the pattern one method will make more valid than the pattern of another?
I can’t help but think that the method is dependent on the amount of space in which we have to create the pattern, that the addition method is ok for a large posting width, and small numbers, but as the numbers increase and/or the posting width decreases, method becomes more important, maybe even requiring the creation (evolution) of more operators for more methods …
A little off-topic, but you can greatly shorten your descriptions above. In the case where you’re just adding 1′s, you could just as well leave off the pluses or the 1′s. So 8 == 11111111
And you don’t need the parens since the operations you’re actually performing always give higher precedence to addition than to multiplication. So then 8 == 11*11*11 == 11^111
And since you’re never multiplying by 1 or mutiplying/adding zero, you can use pluses instead of 1′s. So:
I think removing the parentheses can affect the overall pattern formation.
12 == (((1+1)^(1+1))*(1+1+1))
12 != +^+*++
Because we are changing the order of operations:
1+1^1+1*1+1+1
2^2*3
2^6
64
Also, I don’t think there is a way to use exponents and make 12 without using parentheses.
So if we simplify the method of notation, then some operations cannot be used for certain numbers. and this changes the pattern, and/or the availability of possible patterns.
This topic is discussed in D. Hofstadter’s book Godel, Escher, Bach: an Eternal Golden Braid.
Indeed, I hadn’t thought far enough ahead to worry about cases where order of operations would matter between exponentiation and multiplication. So parens would have to be acceptable where it’s ambiguous. What the default order of operations should be, can be left as an exercise for the reader.
as for your specific concern, if we left the order of operations as “+ then ^ then *”, +^+*++ would work fine.
For that, I think we’d need a stack with a line-termination character. In effect, we’d be removing two characters for one, which I suppose would be an improvement.
An observation about the number sequence:
...
3 == (1+1+1)
4 == (1+1+1+1)
5 == (1+1+1+1+1)
6 == (1+1+1+1+1+1)
7 == (1+1+1+1+1+1+1)
...
I would rewrite this as:
...
3 == (1+1+1)
4 == ((1+1)*(1+1))
5 == (1+1+1+1+1)
6 == ((1+1)*(1+1+1))
7 == (1+1+1+1+1+1+1)
...
Where the first inversion is at 4-5. The reason I do this is because of the relation to primes and to make the pattern seem more consistent. Then of course the next step would be exponent notation:
…
8 == ((1+1)^(1+1+1)
…
So the idea of choosing the correct method of patterning can change dramatically with small changes in rules, such that we can have:
8 == (1+1+1+1+1+1+1+1)
8 == ((1+1)*(1+1+1+1))
8 == ((1+1)^(1+1+1)
And the places of inversion will change according to method.
A question in an evolutionary context: Say the operators are symbols of evolutionary complexity, increasing from addition to multiplication to exponent, wouldn’t use be situational? Given specific circumstances could one “8” be more valid than another “8″, even though the understanding is that they amount to the same thing? Is the pattern one method will make more valid than the pattern of another?
I can’t help but think that the method is dependent on the amount of space in which we have to create the pattern, that the addition method is ok for a large posting width, and small numbers, but as the numbers increase and/or the posting width decreases, method becomes more important, maybe even requiring the creation (evolution) of more operators for more methods …
A little off-topic, but you can greatly shorten your descriptions above. In the case where you’re just adding 1′s, you could just as well leave off the pluses or the 1′s. So 8 == 11111111 And you don’t need the parens since the operations you’re actually performing always give higher precedence to addition than to multiplication. So then 8 == 11*11*11 == 11^111 And since you’re never multiplying by 1 or mutiplying/adding zero, you can use pluses instead of 1′s. So:
And so on
I think removing the parentheses can affect the overall pattern formation.
12 == (((1+1)^(1+1))*(1+1+1))
12 != +^+*++
Because we are changing the order of operations:
1+1^1+1*1+1+1
2^2*3
2^6
64
Also, I don’t think there is a way to use exponents and make 12 without using parentheses.
So if we simplify the method of notation, then some operations cannot be used for certain numbers. and this changes the pattern, and/or the availability of possible patterns.
This topic is discussed in D. Hofstadter’s book Godel, Escher, Bach: an Eternal Golden Braid.
Indeed, I hadn’t thought far enough ahead to worry about cases where order of operations would matter between exponentiation and multiplication. So parens would have to be acceptable where it’s ambiguous. What the default order of operations should be, can be left as an exercise for the reader.
as for your specific concern, if we left the order of operations as “+ then ^ then *”, +^+*++ would work fine.
We could keep the abbreviated syntax but avoid the necessity of parentheses by using post-op notation, couldn’t we?
For that, I think we’d need a stack with a line-termination character. In effect, we’d be removing two characters for one, which I suppose would be an improvement.
So:
The order of operations can be variable dependent on the number
The notation method effects pattern
The pattern can change if primes are always used
Etc.
How many ways can we change the rules? How many rules are there?
It is amazing just how much variation in pattern can be achieved just by changing and/or adding/subtracting rules.
There is the general rule of rules:
The fewer the rules, the less variable the pattern, and the inverse, the more rules the more variable the pattern.
Sorry for the late edit, but, I would also add that the fewer the rules the less adaptable as well...