I am very drunk ama

@wallhackio Tell me cool maths facts right now

@vaporeon_ give me a second I'll make a somewhat long write up in a sec

@vaporeon_ @vaporeon_ in physics you learn about vectors. we are told that vectors are the language of physics. in your very first semester of college (and likely high school physics) you learn about vectors. and yet vectors were only formalized in the early 1900s! what did physics even look like before vectors?

I haven't fully answered this question for myself but an important part of the history is the concept of the "quaternion". The famous Irish mathematician William Rowan Hamilton wanted to extend numbers to three dimensions. Of course, complex numbers can be understood as a "two-dimensional" analogue to numbers since you can plot the real and complex part of a number on a two dimensional plane. We write a complex number like x + iy. You might think that to extend numbers to three dimensions you would make a "super complex" number, something like x + iy + jz

However, Hamilton wanted numbers that exhibited all of the properties exhibited by real and complex numbers. Things like inverses, in which I can multiply a number by another number, and then divide by that same number to get the original number. He couldn't find a way to do this with numbers of the form x + iy + jz

The trick Hamilton came up with was to create numbers in four parts to represent three dimensions. That is, numbers were of the form a + bi + cj + dk. He called these numbers quarternions. The numbers associated with the i, j, and k were parts of three dimensions, and the extra number not associated with an i, j, and k was an auxiliary piece of the number that helped give the structure desirable mathematical properties. With this structure he was able to define multiplication is a way that created inverses! (Fascinatingly, a mathematician later proved that real numbers, complex numbers, and quaternions were the only numbers that could possibly satisfy all of the algebraic properties Hamilton was looking for!! No other mathematical structure was capable of having all of these properties! Hamilton was unknowingly looking for extremely privileged information!!!!)

When we multiply quaternions, we multiply them using the distributive rules of ordinary algebra:

(a + i*b + *j*c + *k*d) * (e + *i*f + *j*g + *k*h)
= ae + *i*af + *j*ag + *k*ah +
*i*be + *ii*bf + *ij*bg + *ik*bh +
*j*ce + *ji*bf + *jj*cg + *jk*ch +
*k*de + *ki*df + *kj*dg + *k*k*dh

To simplify further Hamilton introduced the following rules for the multiplication of i, j, and k with each other:

ii* = *jj* = kk = -1
ij = k
jk = i
ki = j

ji = -k
kj = -i
ik = -j

With these rules the multiplication of two quaternions simplifies to

(a + *i*b + *j*c + *k*d) * (e + *i*f + *j*g + *k*h)

= ae + i*af + *j*ag + *k*ah +
*i*be + *ii*bf + *ij*bg + *ik*bh +
*j*ce + *ji*bf + *jj*cg + *jk*ch +
*k*de + *ki*df + *kj*dg + *k*k*dh

= ae + *i*af + *j*ag + *k*ah +
*i*be - bf + *k*bg - *j*bh +
*j*ce - *k*bf - cg + *i*ch +
*k*de + *j*df - *i*dg - dh

= (ae - dh - cg - bf) + i(af + be + ch - dg) + j(ag - bh + ce + df) + k(ah + bg - bf + de)

Okay, so what? Well, let's consider the specific case where we multiply two numbers which only have nonzero parts in the i, j, and k parts. That is, a=e=0. The result is

(i*b + *j*c + *k*d) * (*i*f + *j*g + *k*h)
= -(dh - cg - bf) + *i(ch - dg) + j(-bh + df) + k(bg - cf)

Well hold on. The part without i, j and k looks like the negative of the dot product:

(*i*b + *j*c + *k*d) * (*i*f + *j*g + *k*h) = (dh - cg - bf)

And the part with i, j, and k looks like the cross product!!!

(i*b + *j*c + *k*d) ❌ (*i*f + *j*g + *k*h)
= *i(ch - dg) + j(-bh + df) + k(bg - cf)

It ends up that these two parts of quaternions were often useful in practical calculations, and the creation of vectors, the dot product, and the cross product resulted in the recognition of the usefulness of these two parts of quarternions!!!!!!

@wallhackio @vaporeon_ i ain't reading all that, happy for you or sorry that happened

@cam @wallhackio But quaternions are cool...
They do something something computer graphics rotating stuff with it, I do not know the details, perhaps the Clodsire can tell me more

I do have a problem with reading the formulae, though, I don't think you intended them to look like this

@vaporeon_ @cam yeah I did leave this comment haha. I didn't feel like deleting and remaining the post a bunch of times so I just let it be

regarding rotations I am aware quaternions are used for this purpose but it's something else details I haven't gone over yet