..."observed" and written by Async, async@centrum.cz;
N-nions are derived from complex numbers. N-nion is N-dimensional number which consist of a+ib+jc+kd+... parts.
Nomenclature:
N=1: real
N=2: complex
N=3: trinion
N=4: quaternion
N=5: pentonion
N=6: hexnion
N=7: heptnion
N=8: octonion
...
...if you invent better names, write me!NOTE: a+ib+ic+.. a is chanel no. 0, b is chanel no. 1, c is chanel no. 2...
Addition, substraction and multiplication is equal to complex's rules.
e.g. Triternion:
(a+ib+jc) + (x+iy+jz) = (a+x) + i(b+y) + j(c+z)
(a+ib+jc) - (x+iy+jz) = (a-x) + i(b-y) + j(c-z)
(a+ib+jc) * (x+iy+jz) = (ax) + i(ay+bx) + j(az+cx) + [ii](by) + [ij](bz) + [ji](by) + [jj](cz)[ii],[ij],[ji],[jj] must be defined. Allowed assignments for Trinion [XY]: {0,1,-1,-i,i,-j,j} It's allowed non-unit assignment e.g. i*i= 0.5i... this assignment affect only x-axis scale (when using e^{(ix+jy+...)}).
Division hasn't been resloved.
Exponential e^{(ix+jy+...)} was defined as complex:
e^{(ix+jy+...)}=(1+ix+jy+...)(1+ix+jy+...)(1+ix+jy+...)... x,y limit to zero.
Computed results of "infinite" series (1+ix+jy+...)(1+ix+jy+...)(1+ix+jy+...)...:Complex e^{ix} | = (1+ix)(1+ix)...(i+ix) | |||
i,j,k,.. rules | "internal functions: each "chanel" " | object(space): | plot of functions: | plot of space: |
i*i = 0 | real: linear y=1 i: linear y=x | line | ||
i*i = 1 | real: y=cosh(x) i: y=sinh(x) | hyperbola | ||
i*i = -1 | real: y=cos(x) i: y=sin(x) | circle | ||
i*i = -i | real: y=1 i: y=1-exp(-x) | line | ||
i*i = i | real: y=1 i: y=-1+exp(x) | line |
Trinion e^{ix+iy} | = (1+ix+jy)(1+ix+jy)...(i+ix+jy) | |||
i,j,k,.. rules | "internal functions: each "chanel" " | object(space): | plot of functions: | plot of space(3D plot): |
ID[523], ii = -j, ij = i, ji = -i, jj = 1 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | 3D spiral | ||
ID[569], ii = -1, ij = i, ji = i, jj = 1 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | singularity spiral | ||
ID[37], ii = -1, ij = -j, ji = 0, jj = 0 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | loop with tail | ||
ID[544], ii = -j, ij = 0, ji = i, jj = 1 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | sickle |
Quaternion e^{ix+iy+iz} | = (1+ix+jy+iz)(1+ix+jy+iz)...(i+ix+jy+iz) | |||
i,j,k,.. rules | "internal functions: each "chanel" " | object(space): | plot of functions: | plot of space(3D plot, "chanels" defined by postfix (file name) XYZ): |
ID[3141592661], ii = -1 ij = -j ik = k ji = -1 jj = 1 jk = i ki = -k kj = k kk = 0 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | stairs to the heaven | ||
ID[3141592674], ii = ii = j ij = j ik = k ji = -1 jj = 1 jk = i ki = -k kj = k kk = 0 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | spiral spoon | ||
ID[3141592683], ii = j ij = -k ik = k ji = -1 jj = 1 jk = i ki = -k kj = k kk = 0 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | flower | ||
ID[7443894661], ii = -k ij = 0 ik = -k ji = 0 jj = 0 jk = 0 ki = -i kj = k kk = 1 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | enlacing spiral | ||
ID[28718281849], ii = i ij = i ik = -j ji = -j jj = -j jk = i ki = -1 kj = 1 kk = 1 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | sipiral with exit | ||
ID[3141592657], ii = -k ij = i ik = k ji = -1 jj = 1 jk = i ki = -k kj = k kk = 0 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | tornado | ||
ID[2718281828], ii = -1 ij = k ik = -j ji = 0 jj = -i jk = k ki = -1 kj = 1 kk = 0 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | tornado | ||
ID[103885895], ii = -1 ij = k ik = -j ji = -k jj = -1 jk = i ki = j kj = -i kk = -1 |
N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. | true quaternion |
Download Nnion-scope to observe new N-nions!:
nscope.zip
Note: pictures encoding: DDN#####...###SCXYZ
DD = dimension, e.g. complex =02
N = leading char. of ID of i*i,j*i,k*j... assignments.
S = specials: C= {I,S}, I= internal functions, S=plot
plot: (XY)Z = X,Y,Z axes "chanel" (real=0,i=1,j=2,k=3..) assignment.
e.g. 02N1S01.gif: complex, i*i=-1, 2D plot - circle, real=X,imag=Y axis.
*** NEWS *** NEWS **** NEWS ***
N-nion scope comes with 4D space - 4th dimesion = color.