The space of the zero-octonions
September 15, 2016
Zero-octonions
The zero-octonions are the expanded number system of the zero-quaternions by the Cayley-Dickson
construction of the imaginary number h (h
2
= 0). (Zero-octonions: http://blog.hani.co.kr/reslieu/58124)
(a + bh)(c + dh) = (ac + h 2d*b) + (bc* + da)h,
h
2
=0
(a + bh)(c + dh) = ac + (bc* + da)h (a, b, c and d are the quaternions)
☓
Zero-octonion multiplication table
i
i
–1
k
j
j
j
k
jℓ
kℓ
–ℓ
–iℓ
0
0
–j
iℓ
–ℓ
–kℓ
–i
–1
kℓ
–jℓ
iℓ
jℓ
0
0
–1
ℓ
–iℓ
–jℓ
–kℓ
jℓ
kℓ
ℓ
–iℓ
kℓ
iℓ
k
–k
iℓ
ℓ
ℓ
–kℓ
–jℓ
iℓ
i
jℓ
0
0
ℓ
0
kℓ
0
0
0
0
0
0
jℓ
–ℓ
0
0
0
The zero-octonions can include the triangular operations in the 3-dimensional Euclidean space as the zeroquaternions. (The geometry of the zero-quaternions: http://blog.hani.co.kr/reslieu/60448)
= −1,
= −1,
= −1
=
, =
, =
= −1
( ℓ) = 0, ( ℓ) = 0, ( ℓ)( ℓ) = 0
( ℓ) = − ℓ, ( ℓ) = ℓ
( ℓ) = − ℓ, u( ℓ) = ℓ
x = cos
y = cos
+ cos ( + a ℓ + b ℓ)
+ cos ( + c ℓ + d ℓ)
xy = (cos + sin
+ sin
ℓ + sin
ℓ) (cos + sin
+ sin
ℓ + sin
ℓ)
= cos( + ) + sin( + )
+{a sin cos β + c cos sin β + (d − b) sin sinβ} ℓ + {b sin cos β + d cos sin β + (a − c) sin sin β} ℓ
= cos( + ) + sin( + )( +
a sin cos β + c cos sin β + (d − b) sin sinβ
b sin cos β + d cos sin β + (a − c) sin sin β
ℓ+
ℓ)
sin( + )
sin( + )
The triangular operation of the zero-quaternions is counter-clockwise. But the triangular operation of the
zero-octonions in the 3-dimensional Euclidean space is clockwise.
Written by Sung-tae Lim
Blog: http://blog.hani.co.kr/reslieu/
E-mail: [email protected]
1
The space of the zero-octonions
September 15, 2016
kℓ
+
jℓ
y
x
iℓ
1
1
√2
+
+
2
2
2
1
1
√2
=−
+
+
2
2
2
√2
√2
=−
+
2
2
=
x = cos + sin ( + ℓ)
3
3
y = cos + sin ( − ℓ)
3
3
xy = {cos + sin ( + ℓ)}{cos + sin ( − ℓ)}
3
3
3
3
1 √3 √2
1
1
1
1
1 √3 √2
1
1
1
1
√3
√2
√3
√2
={ +
+
+
+
−
+
+
ℓ}{ +
+
+
−
−
+
+
2
2 2
2
2
2
2
2
2
2
2 2
2
2
2
2
2
2
1
√6
√3
√3
√6
√3
√3
√6
√3
√3
√6
√3
√3
= (1 +
+
+
−
ℓ+
ℓ+
ℓ)(1 +
+
+
+
ℓ−
ℓ−
ℓ)
4
2
2
2
2
2
2
2
2
2
2
2
2
1 √6
3√2
3√2
√3
√3
=− +
+
+
−
ℓ+
ℓ
2
4
4
4
8
8
1 √3 √2
1
1
√3
√2
√2
=− +
{
+
+
+
−
ℓ+
ℓ }
2
2 2
2
2
2
2
2
2
2
√3
= cos
+ sin
+
ℓ
3
3
2
Written by Sung-tae Lim
Blog: http://blog.hani.co.kr/reslieu/
E-mail: [email protected]
2
ℓ}
The space of the zero-octonions
September 15, 2016
The space of the zero-octonions
Therefore, the non-zero part of the zero-octonions is the axis of the angle on the Euclidean space.
Furthermore the zero-octonions have the rotation property as the octonions.
(x1x2)y = x1(x2y)
x(y1y2) = (xy1)y2
(x1y)x2 = x1(yx2)
It means that the zero-octonions are the 4-dimensional Euclidean space and two zero-octonions rotate on
the Euclidean plane as the zero-quaternions. The external angle of the multiplied zero-octonions can be the
spherical angle and the sum of the triangle angle on the zero-octonions can be smaller than 180°.
kℓ
ℓ
+
jℓ
y
x
iℓ
The 4-dimensional Euclidean space of the zero-octonions is different from the prediction of the 4dimensional space by the principle of the performance of equivalent forms.
(Question: http:// blog.hani.co .kr/reslieu/58150)
Written by Sung-tae Lim
Blog: http://blog.hani.co.kr/reslieu/
E-mail: [email protected]
3