i finally sort of understand what a tensor might be, well i know en...
Axioms 0.1. Let us have coordinates G in Euclidean space, which are a defor-
mation of the coordinates R with the deformation gradient
G =R
1 0
1
√
3
1
(1)
then points
a
∗
γ
n
M
∈G, for a, n ∈Z
+
0
, M ∶={1, 2}, which you will denote as follows
Γ
m
a
∗
, where m ∈ Z
+
0
, then G
a
∗
1
∶=←→
a
∗
γ
n
1
a
∗
γ
k
1
, where n ≠ k, these relationships
apply
(i)
a
∗
γ
n
1
∈G
a
∗
1
,
a
∗
γ
k
2
∈G
a
∗
2
, n ≠k, where ∢
a
∗
γ
n
1
a
∗
γ
k
2
=
π
3
(ii)
a
∗
γ
n
M
∀
a
∗
γ
k
M
(S(
a
∗
γ
n
M
)=S(
a
∗
γ
k
M
))⇒
a
∗
γ
n
M
=
a
∗
γ
k
M
(iii) ∀
a
∗
γ
n
M
(
a
∗
γ
n
M
+
a
∗
γ
0
M
=
a
∗
γ
n
M
,
a
∗
γ
0
1
+=
a
∗
γ
0
2
,, but ∀n, k ≠0,
a
∗
γ
n
1
≠
a
∗
γ
k
2
(iv) ∀
a
∗
γ
n
M
∀
k
M
(
a
∗
γ
n
M
+S(
a
∗
γ
k
M
)=S(
a
∗
γ
n
M
+
a
∗
γ
k
M
)=S(
a
∗
γ
n+k
M
)
∀G
n∗
M
G
M
, G
1
∪G
2
≅G
n∗
1
,
n
∗
γ
k
M
∈G
n∗
M
, while Γ
ψ(n)
=Γ
0
k∗
, then Γ
n
∗a
=
a
∗
γ(x, y)=
(
a
∗
γ
x
1
,
a
∗
γ
y
2
)
Definition 0.1 (ψ function). Let function ψ(x) is a surjective-noninjective
function at x that satisfies the following relations
(i) For x ∈Z
+
,
x =1 +
∑
i<x
ψ(i) (2)
(ii) For x ∈Z
+
,
ψ(x)=
1 , x =0
n , n ∈Z
+
, 6(n −1)+1 ≤x ≤6(n −1)+4
n +1 , n ∈Z
+
, x =6(n −1)+5
n +2 , n ∈Z
+
, x =6(n −1)+6
(3)
Subsequently, the model needs to introduce the notion of oriented vectors.
e
0
=
Γ
1
Γ
2
,
e
x
=
Γ
ψ(x)
Γ
ψ(x+1)
(4)
then you define a rotation for which n ∈Z
+
0
; k ∈Z
+
; Γ
ψ(n)
=Γ
0
k
∗
, where h is fixed
point
Γ
ψ(n)
Γ
ψ(n+1)
×
(n mod 6)
3
π ×h =
Γ
0
k
∗
Γ
3h
2
+[n mod 6−3]h+1
k
∗
(5)
1
so
e
1
=
e
0
×
2
3
π ×ψ(1)=
Γ
2
ψ(1)
Γ
3
ψ(1)
×
−
1
2
−
√
3
2
√
3
2
−
1
2
,n =1 (6)
e
n
=
e
n−1
×
1
3
π ×ψ(n)=
Γ
ψ(n−1)
ψ(n −1)
Γ
ψ(n)
ψ(n −1)
×
1
2
−
√
3
2
√
3
2
1
2
,n >1 (7)
then the relation holds
Γ
ψ(n)
Γ
ψ(n+1)
=
Γ
0
k
∗
Γ
3ψ(n)
2
+2ψ(n)+1
k
∗
(8)
2
i finally sort of understand what a tensor might be, well i know enough now that i know the superscripts are not exponents