## Friday, July 11, 2014

### The Trisection Garden

The Trisection Garden

There is a field-like structure; the "color garden" #R. Its elements are of the form (r,a) with r real and a, the ‘index’, equaling one of the three ‘colors’ r or g or b. (Red, green, blue.) These colors operate by the "pivot" #:
a#b = c       if and only if      a, b and c are all equal, or all different.

Pivot equals "the same or the third". If a and b are the same, then so is a#b; if a and b differ, then a#b is the third. Pivot is therefore only definable on triples.
r#r = r ;       r#g = b ;      r#b = g
g#r = b ;      g#g = g ;     g#b = r
b#r = g ;      b#g = r ;      b#b = b

We can define a#b  =  - (a+b)  mod 3, and this definition works, no matter how we map the three colors to the residues mod 3. This is due to pivot’s high symmetry:
P(x#y)   =   P(x) # P(y)
for P = any permutation of the colors.
Conversely,  pivot defines all color permutations:
three reflections:
x#r,   x#g,   x#b
and three rotations:
x    =    (x#r)#r    =   (x#g)#g    =   (x#b)#b
(x#r)#g    =    (x#g)#b    =    (x#b)#r
(x#g)#r    =    (x#r)#b    =    (x#b)#g
This makes pivot unique within 3-valued operators, and unique to 3-valued operators.

If c#x is a reflection of the colors (through color c), then
(c#x)  #     x       =      c

If R(x) is a rotation of the colors, then
R(x)   #    R2(x)    =   x
R2(x)  #     x         =   R(x)
x         #    R(x)    =  R2(x)

Pivot obeys these pivot laws:
a#a    =   a                                  recall
a#b    =   b#a                              commutativity
(a#b)#(c#d)    =    (a#c)#(b#d)   4-associativity
(a#b)#b   =   a                            cancellation
a#(b#c)   =   (a#b)#(a#c)            self-distribution

Define the garden operations by using pivot on the indices:
(r,a) + (s,b)   =   (r+s, a#b)
(r,a) * (s,b)   =   (r*s, a#b)
(r,a) - (s,b)   =   (r-s, a#b)
(r,a) / (s,b)   =   (r/s, a#b)
- (s,b)      =    ( - s, b )
1 / (s,b)    =    (1/s, b)

#R has three zeros and three units:    0r , 0g , 0b , 1r , 1g , 1b .

Define d(x)   =   x-x  ;   this ‘differential’ sends (r,i) to 0i .
Note that d(0i)  =  0i .
Define q(x)   =   x/x   ;   this ‘quotiental’ sends (r,i) to 1i .
Note that q(1i)  =  1i .

#R has three copies of R; the red reals, the green reals, and the blue reals. Operations on reals of the same color yield the same color; so each color line is closed. When reals of two different colors combine, the result is the third color.
Adding a zero changes only the number's color; ditto with multiplying by a one. Thus #R contains S3, the symmetry group of three objects:
three reflections:
x + 0r    =     x * 1r   ;
x + 0g     =     x * 1g   ;
x + 0b   =     x * 1b   ;
and three rotations:
x    =   (x+0r)+0r    =   (x+0g)+0g    =   (x+0b)+0b
=    (x*1r)*1r    =   (x*1g)*1g    =   (x*1b)*1b
(x+0r)+0g    =   (x+0g)+0b    =   (x+0b)+0r
=  (x*1r)*1g    =   (x*1g)*1b    =   (x*1b)*1r
(x+0g)+0r    =   (x+0b)+0g    =   (x+0r)+0b
=  (x*1g)*1r    =   (x*1b)*1g    =   (x*1r)*1b

#R obeys these ‘garden laws’:

x+y     =    y+x
x*y     =    y*x
(u+x)+(y+z)   =  (u+y)+(x+z)  =  u+x+y+z  4-associativity
(u*x)*(y*z)   =   (u*y)*(x*z)  =   u*x*y*z
(x+y) - y    =   x                                             cancellation
(x*y) / y    =   x
x*( y + z )   =    x*y + x*z                           distribution
(w+x)(y+z)   =   wy+wy+xy+xz                  FOIL
x+dx     =   x    =    x*qx                      relative identities
- x + x     =    dx                                   relative inverses
(1/x) * x   =    qx
x+(y+z)   =    (x+y)+(z+dx)                 differential associativity
dx    =   - dx    =    dx+dx     =    dx*dx   differential recall
d(xy) = d(x+y) = dx+dy  =  dx* dy   differential distribution
0i * x    =    0i + dx                            differential indices

We retain distribution, but must weaken associativity and relativize identity and inverse.

All of these laws preserve the ‘stratified variable count, mod 3’; a.k.a. the weight. The weight counts each variable appearance +1 times, mod 3, if it occurs at even number of +’s and *’s deep; and it counts each variable appearance -1 times, mod 3, if it occurs at an odd number of +’s and *’s deep.
Take, for instance, the cancellation law:
(x + y) - y   =   x
On the left side y occurs once on level one, and once on level two; this gives a weight of 1-1 = 0. On the right side y does not occur at all; again, weight = 0. And the letter x has weight 1 on both sides.
Consider the distribution law:
x*(y+z) = (x*y)+(x*z)
On the left side x occurs once on level one; this gives a weight of -1. On the right side x occurs twice on level two; this gives a weight of 1+1 = -1 mod 3.

Now consider the differential association law:
x+(y+z)   =    (x+y)+(z+dx)
On the left, x occurs once at level one; weight = -1. On the right, x occurs once at level two , and twice at level three (in dx = x-x) ; so weight = 1-1-1 = -1.

In general, weight of x   =    ( number of even-level x’s)  -  ( number of odd-level x’s), modulo three.
We can also define weight inductively. Given expressions A and B on which we’ve defined the weight w, then define the weight on these combinations of A and B:
w(-A) =  w(1/A) =  w(A)   ;
w(A+B)  =  w(A*B)  =  w(A-B)  =  w(A/B)  =   – w(A) – w(B)
=  w(A) # w(B),   mod 3
Pivot re-emerges!

Defined either way, weight is preserved by the garden laws; so any identity derived from them also preserves each variable’s weight. A syntactic notion has semantic effects!

We can construct an isomorphic copy of #R on the standard reals; the “trisection garden”. It starts with these functions:

f  -1(x)    =   tan(3*arctan(x))    =   (x3-3x)/(3x2-1) .
s(x)     =   tan( (pi/3) + arctan(x) )  =  (x + root(3))/(1 - x root(3))

f -1(x)   =    tan( (-pi/3)  +  (arctan(x))/3  )
f0(x)    =    tan(                (arctan(x))/3  )
f1(x)    =    tan( (pi/3)  +  (arctan(x))/3  )

f  -1(x)   is the ‘angle tripling’ function; it is the slope of a line with triple the angle from horizontal of a line with slope x.  The function s is the slope of a line turned 60 degrees counterclockwise from a line with slope x. The f’s are angle trisection functions; they are the slopes of lines with one-third the angle from horizontal of a line with slope x. But because adding 180 degrees to an angle yields the same slope, the f’s are separated by 60 degree turns; the function s.
f  -1(x) has three branches, defined over domains  ( -1/0, -1/ root(3)  ,  (- 1/ root(3) , 1/ root(3) ) and ( 1/ root(3) , 1/0) .  Therefore let us also define the index function i(x) thus:
i(x)      =      -1 mod 3    on  ( -1/0, -1/ root(3))
0 mod 3    on   (- 1/ root(3)  , 1/ root(3)  )
1 mod 3    on   ( 1/ root(3)   ,  1/0)

The fN’s range over just those intervals:
i(  fN(x) )   =    N

The function s(x) has period three:
s(s(s(x)))   =    x     ;        s(s(x))  =  s-1(x)  =  - s(-x)
It shifts the index of a number up by one modulo three:
i(s(x))    =    i(x) + 1   mod 3 .
It also shifts the index of the fN’s up by one, modulo three:
s(fN(x))    =    fN+1(x)
fN(x)        =    sN(f0(x))
The function f -1  is derivable from s:
f -1(x)       =      x + s(x) + s(s(x))     =    - x*s(x)*s(s(x)).
Therefore f  -1(x) “absorbs” s:
f  -1(s(x))  =  s(x)+s(s(x))+s(s(s(x)))  =  s(x)+s(s(x))+x = f -1(x)
The fN’s are inverses of the three branches of f -1:
fN(f -1(x))   =    x      if    i(x) = N
f -1(fN(x))   =    x      for all x.
The f N’s  solve this cubic equation:
y3  - 3xy2  - 3y  +  x    =   0
This implies, among other things:
f -1 + f0 + f1              =      3x
f -1 * f0 * f1               =     - x
f -1*f0 + f0*f1 + f1*f -1      =   - 3
By solving the cubic, we find that:
f N (x)
=   x + 2*cuberoot(x2+1)* Realpart( W(1-N)*cuberoot(x-i))
where W  =  (-1 + i root(3))/2   ;   a cube root of unity.

Now define these trisection operators:

x #+ y   =   f i(x)#i(y) ( f -1(x) + f -1(y) )   =   s i(x)#i(y) (f0(f -1(x) + f -1(y))
x #* y   =   f i(x)#i(y) ( f -1(x) * f -1(y) )    =   s i(x)#i(y) (f0(f -1(x) * f -1(y))
x #- y    =   f i(x) # i(y) ( f -1(x) - f -1(y) )   =  s i(x) # i(y)(f0(f -1(x) - f -1(y))
x #/ y    =   f i(x) # i(y) ( f -1(x) / f -1(y) )   =  s i(x) # i(y) (f0(f -1(x) / f -1(y))
#- y    =         f  i(y) ( - f -1(y) )          =    s  i(y) (f0 (- f -1(y))
#1/ y    =         f  i(y) ( 1 / f -1(y) )        =    s  i(y) (f0 (1 / f -1(y))

These are field operations conjugated by angle trisection, with indices pivoting. These equations imply that:

s(x) #+ s(y)   =   s( x #+ y )
s(x) #* s(y)    =  s( x #* y )
s(x) #- s(y)     =  s( x #- y )
s(x) #/ s(y)     =  s( x #/ y )

s2(x) #+ s2(y)    =    s2( x #+ y )
s2(x) #* s2(y)    =    s2( x #* y  )
s2(x) #- s2(y)     =    s2( x #- y  )
s2(x) #/ s2(y)      =    s2( x #/ y  )

s(x) #+ y    =    x  #+ s(y)    =   s2( x #+ y )
s(x) #* y    =    x  #* s(y)    =    s2( x #* y )
s(x) #- y     =    x  #- s(y)   =    s2( x #- y )
s(x) #/ y     =    x  #/ s(y)    =    s2( x #/ y )

s2(x) #+ y   =    x #+ s2(y)    =    s( x #+ y )
s2(x) #* y   =    x #* s2(y)    =    s( x #* y )
s2(x) #- y    =    x #- s2(y)     =    s( x #- y )
s2(x) #/ y    =    x #/ s2(y)      =    s( x #/ y )

#- s(y)       =     s( #- y )
#1/ s(y)      =     s( #1/ y )
#- s2(y)     =     s2( #- y )
#1/ s2(y)    =     s2( #1/ y )

That defines the trisection garden. It is an image of #R, via this isomorphism:

(r,a) —> sa( f0(r) )

- where { s^red, s^green, s^blue } is matched one-to-one with {identity,s,s2}. There are 3! = 6 ways to match colors to Z mod 3, and they all yield the same trisection garden.

The trisection garden is like a field, with cancellation, distribution and FOIL, but associativity is weakened to quadruple, and inverse and identity are relativized, to cover three parts of the line.
It involves angle trisection, trigonometry, the cubic,  and the symmetries of the triple. Its indices operate by the pivot, which is unique to three-element sets. This has 3 written all over it. Surely something this pretty has a use! Perhaps in geometry? Quark theory?  Political science?