03/04/2017, 10:31 AM

Hi, everyone!

The oldest dimensional base units like i[0]=1, i[1]=i (imaginary unit), i[2] and i[3] (quaternions), i[4] ... i[7] (octonions), i[8] ... i[15] (sedenions) ... etc. is called ancient-dimensional base units (by me). The less old dimensional base units where the indeces can be any real is simply called old-dimensional base units. Its PARI/gp code looks like this one:

So i[x] i[y] = twist(x;y) i[x xor y]

And for instance the graph of i[y] = i[1] i[x] looks like this:

https://www.dropbox.com/s/81rxif99w7j50u...D.bmp?dl=0

As you can see the sign of the function is alternating.

Okey, so these were the old-dimensional numbers. The so-called hyperdimensional base units are base units whose indeces can be complex, quaternion ... so simply old-dimensional numbers. For example these are hyperdimensional numbers:

i[1+i+i[2]]

i[i[0.5]-3]

i[0.3+i[100]-i[7]+i[pi]]

My question is that how to multiplicate two hyperdimensional base units according to the T matrix above in the code?

I think we should extrapolate the T matrix to 3D matrix, but how?

The oldest dimensional base units like i[0]=1, i[1]=i (imaginary unit), i[2] and i[3] (quaternions), i[4] ... i[7] (octonions), i[8] ... i[15] (sedenions) ... etc. is called ancient-dimensional base units (by me). The less old dimensional base units where the indeces can be any real is simply called old-dimensional base units. Its PARI/gp code looks like this one:

Code:

`twist(p,q)={`

L=ceil(1.4426950408889634*log((max(q+1,p+1))));

res=1;sta=1;

if(round(p)!=p||round(q)!=q,p*=2^60;q*=2^60;L+=60);

p=round(p);q=round(q);

T=[1,2,3,4;2,2,5,6;3,6,3,5;4,5,6,4;5,6,6,6;6,5,5,5];

for(i=0,L,j=L-i-1;k=2*bittest(p,j)+bittest(q,j)+1;sta=T[sta,k]);

if(sta==4||sta==6,return(-1),return(1));

}

xor(x,y)={return(bitxor(round(x*2^60),round(y*2^60))/2^60)}

And for instance the graph of i[y] = i[1] i[x] looks like this:

https://www.dropbox.com/s/81rxif99w7j50u...D.bmp?dl=0

As you can see the sign of the function is alternating.

Okey, so these were the old-dimensional numbers. The so-called hyperdimensional base units are base units whose indeces can be complex, quaternion ... so simply old-dimensional numbers. For example these are hyperdimensional numbers:

i[1+i+i[2]]

i[i[0.5]-3]

i[0.3+i[100]-i[7]+i[pi]]

My question is that how to multiplicate two hyperdimensional base units according to the T matrix above in the code?

I think we should extrapolate the T matrix to 3D matrix, but how?