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Hyperdimensions
#1
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:
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)}
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?
Xorter Unizo
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#2
I really do not know what you are talking about ...

Regards

Tommy1729
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#3
(03/09/2017, 01:23 PM)tommy1729 Wrote: I really do not know what you are talking about ...

Regards

Tommy1729

I call the complexes, quaternions, octonions, sedenions ... etc. ancient dimensional numbers. (But it is not so important.)
Look:
https://en.wikipedia.org/wiki/Quaternion

https://en.wikipedia.org/wiki/Octonion
https://en.wikipedia.org/wiki/Sedenion
And here is a summation for N-dimension:
http://jwbales.us/
We have a 2^n dimensional space with base units. These are the dimensional base units whose multiplication is that I am interested in. I found some formulas to calculate it. But I must step toward into the so-called hyperdimensional spaces with its hyperdimensional base units.
I would like to know how I can multiplicate two of hyperdimensional base units.
Are we closer to understanding?
Xorter Unizo
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