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))
= 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?
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