Philosophy Explorer wrote: ↑Sun Feb 22, 2015 5:19 am
For about six months I ve been thinking about numbers arranged in a cyclic pattern.
For example we have 1
...................................2 3
Another example is 1
................................4 2
..................................3
If we interpret these numbers as bases and exponents, we can say (using the diamond pattern) that one is functioning as an exponent for the number four and as a base for the number two when moving in a clockwise direction.
How important this is to math I don't know nor do I know if someone else ever investigated this. To me it looked interesting enough to list which may stimulate thought.
PhilX
PS There is another meaning, more common, for the term cyclic number.
I have observed similarities to what you are observing. If viewed as a cyclic function the number one would have to exists through a "self-reflection" process where it is spatially equivalent to a 1 dimensional point reflecting into itself:
One reflecting upon itself maintains itself while manifesting as approximates 2 and -1.
1 ≡ 1 → 1,2,-1
One reflecting upon itself maintains itself as an act of stability and unity.
a) 1 ≡ 1 → 1 ∵ (1 ≡ 1) = (◻ = 1)
Simultaneously it manifests "2" because 1 reflecting 1 is structurally congruent to 2.
b) 1 ≡ 1 → 2 ∵ 1 ≡ 1 ≅ 2
Simultaneously it manifests "-1" because 1 reflecting 1 is an approximate of 2 and this approximation is the deficiency of 1 from 2.
c) 1 ≡ 1 → -1 ∵ 1 ≡ 1 → 1 ≈ 2
As all number is composed upon a self-reflecting one, all 1n follows the same form and function
(1,2,-1) ≡ (1,2,-1) → (-2,-1,0,1,2,3,4)
*****With all positive numbers manifesting at double the rate of the negative.
a) 1 ≡ 2 → 3
b) 1 ≡ -1 → 0
c) 2 ≡ 2 → 4
d) 2 ≡ -1 → 1
e) -1 ≡ -1 → -2
The question I have is which came first the form (1,2,3,etc.) or the function (+,-, etc.)?
We can observe that:
Through self reflection as "cycling":
1) addition cycling (reflecting) upon itself results in multiplication as the addition of addition. This is considering that addition may be viewed strictly as a structural extension of 1 as a Positive.
2) multiplication cycling upon itself results in exponention.
3) subtraction cycling upon itself results in division as the subtraction of subtraction. This is considering that subtraction may be viewed strictly as a structural extension of 1 as a Negative.
4) division cycling upon itself results in roots.
Through dualities as "rotations":
⟨+|-⟩ → +1- → ⟨+|-⟩
****with "1" being the axis from which it begins as arithmetic and subtraction are the first degrees of numerical "curvature".
****Addition and subtraction exist if and only if there is "1".
****Addition and subtraction can be viewed as equivalent to the reflection of a Positively valued 1 or a Negatively valued 1.
⟨*|/⟩ → x2/ {or (+1-) ≡ (+1-)} → ⟨*|/⟩
**** with "2" being the axis from which it begins as multiplication and division are the second degrees of numerical "curvature".
**** Multiplication and Division exist if and only if there is "1 reflecting upon itself (cycling)" as 2. 2 dividing itself results in the 1 as a further foundation for standard multiplication and division.
Considering that all cycles seem to stem from Pi in theory their should be some cyclic connection as a result of 1 reflecting upon itself. I am still working on that aspect. The Pythagoreans had a theory where "3" was actually the "first" number and it came before 1 or at the same time as 1. If this holds true than in theory Pi may hold a key to the origins of mathematics and the nature of spatial properties.
If number is viewed as a spatial element conducive to a "point" and nothing more Pi results in an infinite particle wave function that in theory would hold all real world potential "realities" (as all physical realities are composed of particle waves).
example:
....................1
....................1
....................1
....................1.....1
................1...1.....1
.......1.......1...1.....1
1......1.......1...1.....1
1 ..1..1...1..1...1..1..1
1 ..1..1...1..1...1..1..1
3. 1 4 1 5 9 2 6
If you fold the above on itself, which I will not post because of time, it looks like a continuing mushroom effect.