A large (apparently, infinite!) number of internal symmetries of
the lattice ZN is identified. Their presence stems from one interesting
observation. It concerns Cellular Automation. If we take a secondorder
reversible Automata  in its simplest version (the so-called
Fredkin’s Simplified Automata whose behaviour depends solely
on the shape of a certain neighbourhood in ZN), and run it under
two certain conditions (arbitrary and arbitrary -”transliterated”),
then they will continue going completely “symmetrical”, all the way,
from the Point of Origin to the Point of the Mirror. (What exactly
is this “symmetry” is described in detail in [2-7]). This property is
performed for some neighbourhoods (or as we write - “Masks”)
(we call such neighbourhoods Perfect), and for others – no (not
Perfect). (Figure 1). The paradox of the situation is that there
are a lot of Perfect neighbourhoods! For example, seven Perfect
neighbourhoods with their properties are reflected in Appendix 1.
The analysis of this feature leads to the fact that we can “get rid” of
the Cellular Automation and come to very interesting structures,
the so-called Transition Tables (TT) [4-6]. (The Cellular Automation
needs only to build the TT themselves, then they begin to “live their
lives”). We have counted a total of 8(!) properties of TT, some of
which are very paradoxical. We will list them with conventional
names along with the short commentary:
a) «Substitutions». (Transition tables are six tables made up of
six-digit numbers. The number of columns is equal to the number
of cells in the Mask. The number of rows (R) is some basic constant
for each Mask. All 5 other Transition Tables are obtained from the
first Table by corresponding substitutions).
b) «Graph». (For our proof, a graph composed of TT with nodes
at Mask cells must satisfy some simple property).
c) “Closed Lotto game”. (The main paradoxical property! Can
be formulated as an independent integer colouring problem.
Our TT miraculously solve it!)
d) “Constant accurate calculation”. (Another completely
new paradoxical property! Just like property 3, it can be easily
checked on the computer).
e) «Inverse transformation». (One additional TT property
which clearly has some binding nature).
f) “Division of TT rows into two parts; “right” and “left””.
(Reference to the third property. The corresponding diagram
clearly shows the division).
g) “The main law of conservation”. (Back to where we started
– to Cellular Automation! For any single CA, empirically find a
law of conservation: w0(τ) + w1(τ) = w1(τ+1) - w0(τ+1). Here w0
and w1 are the number of cells in the Automata obtained from
the “left” and “right” rows summed throughout the Automata;
τ is the time) .
h) There are related lectures on the «Alex Kornushkun»
YouTube channel. Today there are 5 of them. We are preparing
the sixth. Please pay attention to the first Lecture and the
program key_5m3.exe (the link is given in the description
of the Lectures). The program (along with the Lecture itself)
reproduces a step-by-step proof of the Perfection of this Mask.
Note how computer proof leads us to the desired result “in the
happiest way”! Clearly, this can’t just be a coincidence.