The Sedenion Loop

[Steve]

The inner loop for crunching sedenions has been computed. Recall that sedenions are generalizations of octonions which are generalizations of quaternions.  The space is 16 dimensions, but because of symmetry there is a redundancy of fractal images in many subspaces.  Like quaternions and octonions, there is total rotational symmetry about the real axis. The 2-D cross section of any plane through the real axis is the M-set. There are 112 possible pairs of dimension for 2-D cross sections, so there is a lot to explore.

The sedenions break the associative law: a(bc) is not always equal to (ab)c. Therefore sedenions are actually a "loop", not a group.  Also sedenions have zero divisors, which means that the product of two non zero elements can be zero. It will be interesting to look at planes that intersect zero divisors to see what happens.
 
I have looked at only a very few images so far.  The images below are from the s₃-s₇ plane at a point where most of the other 14 coordinate values are not zero. Each successive image is a zoom into the box shown. Each zoom is a factor of 10. There seems to be no geometric order to the fractal, but it is self similar to the limit of resolution - 10^-14 .
 

 

 

 
The following two images are 300 iterations rather than 200

 

 
 
 

[Krumel]

Omg, you did it.
You're crazy man

[Steve]

You're crazy man

Thank you. I consider that a highly regarded complement.

A word about how this was done:

The web site http://octonions-sedenions.tripod.com/srepresentation.html
gives a "somewhat more compact form by means of the following 35 sedenion cycles:"

        ( 1,2,3 ), ( 1,4,5 ), ( 1,7,6 ), ( 1,8,9 ), ( 1,11,10 ), ( 1,13,12 ), ( 1,14,15 )
        ( 2,4,6 ), ( 2,5,7 ), ( 2,8,10 ), ( 2,9,11 ), ( 2,14,12 ), ( 2,15,13 ),
        ( 3,4,7 ), ( 3,6,5 ), ( 3,8,11 ), ( 3,10,9 ), ( 3,13,14 ), ( 3,15,12 ),
        ( 4,8,12 ), ( 4,9,13 ), ( 4,10,14 ), ( 4,11,15 ),
        ( 5,8,13 ), ( 5,10,15 ), ( 5,12,9 ), ( 5,14,11 ),
        ( 6,8,14 ), ( 6,11,13 ), ( 6,12,10 ), ( 6,15,9 ),
        ( 7,8,15 ), ( 7,9,14 ), ( 7,12,11 ), ( 7,13,10 ).


I used  text editing formulae in a spread sheet to translate those permutation relations into these to these 226 group generators that my program can understand:

AB=C, BC=A, CA=B, AD=E, DE=A, EA=D, AG=F, GF=A, FA=G, AH=I, HI=A, IA=H, AK=J, KJ=A, JA=K, AM=L, ML=A, LA=M, AN=O, NO=A, OA=N,
BD=F, DF=B, FB=D, BE=G, EG=B, GB=E, BH=J, HJ=B, JB=H, BI=K, IK=B, KB=I, BN=L, NL=B, LB=N, BO=M, OM=B, MB=O, 
CD=G, DG=C, GC=D, CF=E, FE=C, EC=F, CH=K, HK=C, KC=H, CJ=I, JI=C, IC=J, CM=N, MN=C, NC=M, CO=L, OL=C, LC=O,
DH=L, HL=D, LD=H, DI=M, IM=D, MD=I, DJ=N, JN=D, ND=J, DK=O, KO=D, OD=K,   
EH=M, HM=E, ME=H, EJ=O, JO=E, OE=J, EL=I, LI=E, IE=L, EN=K, NK=E, KE=N,   
FH=N, HN=F, NF=H, FK=M, KM=F, MF=K, FL=J, LJ=F, JF=L, FO=I, OI=F, IF=O,   
GH=O, HO=G, OG=H, GI=N, IN=G, NG=I, GL=K, LK=G, KG=L, GM=J, MJ=G, JG=M,   

BA=~C, CB=~A, AC=~B, DA=~E, ED=~A, AE=~D, GA=~F, FG=~A, AF=~G, HA=~I, IH=~A, AI=~H, KA=~J, JK=~A, AJ=~K, MA=~L, LM=~A, AL=~M, NA=~O, ON=~A, AO=~N,
DB=~F, FD=~B, BF=~D, EB=~G, GE=~B, BG=~E, HB=~J, JH=~B, BJ=~H, IB=~K, KI=~B, BK=~I, NB=~L, LN=~B, BL=~N, OB=~M, MO=~B, BM=~O,
DC=~G, GD=~C, CG=~D, FC=~E, EF=~C, CE=~F, HC=~K, KH=~C, CK=~H, JC=~I, IJ=~C, CI=~J, MC=~N, NM=~C, CN=~M, OC=~L, LO=~C, CL=~O,
HD=~L, LH=~D, DL=~H, ID=~M, MI=~D, DM=~I, JD=~N, NJ=~D, DN=~J, KD=~O, OK=~D, DO=~K,   
HE=~M, MH=~E, EM=~H, JE=~O, OJ=~E, EO=~J, LE=~I, IL=~E, EI=~L, NE=~K, KN=~E, EK=~N,   
HF=~N, NH=~F, FN=~H, KF=~M, MK=~F, FM=~K, LF=~J, JL=~F, FJ=~L, OF=~I, IO=~F, FI=~O,   
HG=~O, OH=~G, GO=~H, IG=~N, NI=~G, GN=~I, LG=~K, KL=~G, GK=~L, MG=~J, JM=~G, GJ=~M,

A^2=~1, B^2=~1, C^2=~1, D^2=~1, E^2=~1, F^2=~1, G^2=~1, H^2=~1, I^2=~1, J^2=~1, K^2=~1, L^2=~1, M^2=~1, N^2=~1, O^2=~1


The program inputs these group generators and computes the Java sedenion recursion formula:

for (kk = 0; kk < loops; kk++) {
    e0=E[0]; e1=E[1]; e2=E[2]; e3=E[3]; e4=E[4]; e5=E[5]; e6=E[6]; e7=E[7]; e8=E[8];
    e9=E[9]; e10=E[10]; e11=E[11]; e12=E[12]; e13=E[13]; e14=E[14]; e15=E[15];

   E[0] = point[0] + e0*e0 - e1*e1 - e2*e2 - e3*e3 - e4*e4 - e5*e5 - e6*e6 - e7*e7 - e8*e8 - e9*e9
                - e10*e10 - e11*e11 - e12*e12 - e13*e13 - e14*e14 - e15*e15;
   E[1] = point[1] + e2*e3 + e4*e5 - e6*e7 + e8*e9 - e10*e11 - e12*e13 + e14*e15 + 2.0f*( + e0*e1 );
   E[2] = point[2] - e1*e3 + e4*e6 + e5*e7 + e8*e10 + e9*e11 - e12*e14 - e13*e15 + 2.0f*( + e0*e2 );
   E[3] = point[3] + e1*e2 + e4*e7 - e5*e6 + e8*e11 - e9*e10 - e12*e15 + e13*e14 + 2.0f*( + e0*e3 );
   E[4] = point[4] - e1*e5 - e2*e6 - e3*e7 + e8*e12 + e9*e13 + e10*e14 + e11*e15 + 2.0f*( + e0*e4 );
   E[5] = point[5] + e1*e4 - e2*e7 + e3*e6 + e8*e13 - e9*e12 + e10*e15 - e11*e14 + 2.0f*( + e0*e5 );
   E[6] = point[6] + e1*e7 + e2*e4 - e3*e5 + e8*e14 - e9*e15 - e10*e12 + e11*e13 + 2.0f*( + e0*e6 );
   E[7] = point[7] - e1*e6 + e2*e5 + e3*e4 + e8*e15 + e9*e14 - e10*e13 - e11*e12 + 2.0f*( + e0*e7 );
   E[8] = point[8] - e1*e9 - e2*e10 - e3*e11 - e4*e12 - e5*e13 - e6*e14 - e7*e15 + 2.0f*( + e0*e8 );
   E[9] = point[9] + e1*e8 - e2*e11 + e3*e10 - e4*e13 + e5*e12 + e6*e15 - e7*e14 + 2.0f*( + e0*e9 );
   E[10] = point[10] + e1*e11 + e2*e8 - e3*e9 - e4*e14 - e5*e15 + e6*e12 + e7*e13 + 2.0f*( + e0*e10 );
   E[11] = point[11] - e1*e10 + e2*e9 + e3*e8 - e4*e15 + e5*e14 - e6*e13 + e7*e12 + 2.0f*( + e0*e11 );
   E[12] = point[12] + e1*e13 + e2*e14 + e3*e15 + e4*e8 - e5*e9 - e6*e10 - e7*e11 + 2.0f*( + e0*e12 );
   E[13] = point[13] - e1*e12 + e2*e15 - e3*e14 + e4*e9 + e5*e8 + e6*e11 - e7*e10 + 2.0f*( + e0*e13 );
   E[14] = point[14] - e1*e15 - e2*e12 + e3*e13 + e4*e10 - e5*e11 + e6*e8 + e7*e9 + 2.0f*( + e0*e14 );
   E[15] = point[15] + e1*e14 - e2*e13 - e3*e12 + e4*e11 + e5*e10 - e6*e9 + e7*e8 + 2.0f*( + e0*e15 );

   magn = E[0]*E[0]+E[1]*E[1]+E[2]*E[2]+E[3]*E[3]+E[4]*E[4]+E[5]*E[5]+E[6]*E[6]+E[7]*E[7]+E[8]*E[8]
                +E[9]*E[9]+E[10]*E[10]+E[11]*E[11]+E[12]*E[12]+E[13]*E[13]+E[14]*E[14]+E[15]*E[15];
   fpCounter++;
   if ( magn > escape)  break;
}