A SHORTCUT TO
WRITING THE
COMPLEMENTARY / CYCLIC SEQUENCE Please be
aware that,
despite the attention
given to proofing and proofing again ,there may persist some
mistakes (
"may" that is like when a bookseller state " books may have
reminder
marks" : you can be
dead certain the book has them
! )
Get the
complementary
( ← )
means from RIGHT
to LEFT
( → ) means from LEFT
to RIGHT 0 2
4
1 3 #
#
#
##
Write
the top line relating to the bottom
rim-------------------------------------------- 0
2
4
1
3
0 2
4
1
3 0
2
4
1 3
BOTTOM rim sequence written ( ← )
TO BE READ
( → ) 0
2
4
1 3
0 2
x @
@ @
@
@ @
@
@
Reverse the top
line to write the bottom line ( relating to top rim-------------------
BOTTOM rim sequence written ( ← ) TO BE READ
( → ) 0
2
4
1 3
0 2
x @
@ @
@
@ @
@
@ x2
0
3
1 4
2
0
TOP rim sequence written ( → ) TO BE READ
READ (←)
Add the
coding-----------------------------------------------------------------------------------
BOTTOM rim sequence written ( ← ) TO BE
READ
( → ) 0
2
4
1 3
0 2 x
this is relating to the
ODD
↖
B.R.T.L @
/
\
/ \
/ \
@ x2
0
3
1 4
2
0 this is relating to the
EVEN
↙
T.L.B.R
TOP rim sequence written ( → ) TO BE READ
READ (←)
------------------------------------
BOTTOM rim sequence written ( ← ) TO BE READ
( → )
U
O
U
O U
O
U 0
2
4
1 3
0 2 x
this is relating to the
ODD
↖
B.R.T.L @
/
\
/ \
/ \
@ x2
0
3
1 4
2
0 this is relating to the
EVEN
↙
T.L.B.R
U
O
U O
U
O U
TOP rim sequence written ( → ) TO BE READ
READ (←)
------------------------------------
This will be reduced to
------------------------------------
BOTTOM rim sequence written ( ← ) TO BE READ
( → ) 0
1
2
3 4
5 6
7
number of CROSSING ROW on left
none U
O U
O
U
O none 0
2
4
1 3
0 2 x
this is relating to the
ODD
↖
B.R.T.L @
\
/ \
/ \
/
@ x
2
0
3
1 4
2
0 this is relating to the
EVEN
↙
T.L.B.R
none
O U
O
U O
U
none 7
6
5
4 3
2 1
0 number
of CROSSING ROW on right
TOP rim sequence written ( → ) TO BE READ
READ (←)
------------------------------------
the 2 extreme 0-@
pair
being just there as place-holding items ( just as digit 0 is just a
place
holding digit giving meaning to the other digits 1 to 9 in
numbers ! )
A SHORTCUT TO
WRITING THE COMPLEMENTARY / CYCLIC SEQUENCE
this will add a short cut to the first phase of above
We know how to compute the place away from 0
of a given number.
But that give them in disarray.
It would be nice to get the first after 0 then the second, then the
third...
The value of the increment span ( modular) between 2
successive numbers must be an
integer.
This integer must be when multiplied by (-L) mod B [or of
(L)mod B in the other case] an
integer itself.
IS for increment span between 2 consecutive ( spatially ) numbers
IS = f(B) / (-L)mod B - for the complementary (
for the periodic = f(B) / (+L)mod B )
or rather
IS = f(B) / ((B-L ) modulus B)) for the complementary
( = f(B) / (B - ((B-L ) modulus B))
for the periodic
)
we must use (-L)mod B or ((B-L ) modulus
B)) as "great modular step"
even / even leads to integer
odd / odd may lead to integer
even / odd may lead to integer
odd
/ even does not lead to integer so to cover that in case B is ODD by
precaution we
add 1 and we will eventually "tweak" the formula for
correction
so it become IS = f(B) + 1 / (-L) mod B
so trying IS = (B +1) / (-L) mod B is not
satisfying
if (B +1) < (-L) mod B
so to ensure that (B+1) will always be ( in the most economical fashion
) > (-L) mod B
we need a multiplying factor for B that must be kept at it barest
minimum (Occam' razor )
IS = ((mf * B ) + 1 ) / (-L)mod B
for the complementary
and for the periodic
IS = ((mf * B ) + 1 ) / (+L)mod B
so to disambiguate
IScomp = ((mf * B )
+ 1 ) / (-L)mod B ISper
= ((mf * B ) + 1 ) / (+L)mod B
OR my preferred
IScomp = ((mf * B ) + 1 ) /
((B-L ) modulus B)) ISper
= ((mf
* B ) + 1 ) / (B - ((B-L ) modulus B)) (mf) must be an
integer, the smallest making IS an integer step leading to a closed
path , in other word a circuit.
this works
You get the complementary at a fast pace and the just reverse it ! to
get the periodic
0
10 7
4 1 11
8
5 2 12
9 6
3
the other way is
DW ( Digit Written) then its place after 0 is DW *
(L) modulus B (too late to change
but instead of Digit Written it should have been Number Written)
(1
* (-17)) mod 13 = -1 * 9
==
-17 mod 13 ===
9th place
(2 *
(-17)) mod 13 = 2 * 9
==
-34 mod 13 ===
5th place
(3
* (-17)) mod 13 = 3 * 9
==
-51 mod 13 ===
1rst place
(4
* (-17)) mod 13 = 4 * 9
==
-68 mod 13 ===
10th place
...
(12 * (-17)) mod 13 = 12 * 9
== -204 mod 13 ===
4th place