OMIC's N-nion's site

..."observed" and written by Async, async@centrum.cz;

*** NEWS *** NEWS **** NEWS *** - 4th dimesion "broken"

see END OF PAGE.

*** NEWS *** NEWS **** NEWS *** 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 eix = (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 eix+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, ii = -j, ij = i, ji = -i, jj = 1 N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. 3D spiral  ID, ii = -1, ij = i, ji = i, jj = 1 N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. singularity spiral  ID,ii = -1, ij = -j, ji = 0, jj = 0 N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. loop with tail  ID,ii = -j, ij = 0, ji = i, jj = 1 N/A; mixture of sin, cos, exp, sinh, cosh, lin, const. sickle  Quaternion eix+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,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,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, 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, 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, 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, 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, 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, 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  nscope.zip

or contact Async at async@centrum.cz, or jense@centrum.cz

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.
see these, such beautiful...   