my big takeaway studying quaternions a little closer as of late, is that interpreting them as a 4D object is a red herring
they make a lot more sense when you think of them as a special encoding of an angle-axis model with special multiplication rules, or, as rotors in the framework of geometric algebra
treating the xyzw components as the same type of element in the same type of space just isn't helpful (to me), because they operate with completely different rules
w is just a regular ol number on the number line
x, y and z are very much not
w is just a regular ol number on the number line
x, y and z are very much not
x, y and z operate as a coefficient of a complex unit, usually called i, j and j separately, where each of those units square to -1
in the language of geometric algebra, they are the components of a 3D bivector (ie not a vector, and not just numbers)
in the language of geometric algebra, they are the components of a 3D bivector (ie not a vector, and not just numbers)
it's helpful to think of the basis for any system we have, and this is fundamental to geometric algebra:
{1} is a real number
{e₀, e₁, e₂} is a vector
{e₁₂, e₂₀, e₀₁} is a bivector
{1, e₀₁} is a complex number
{1, e₁₂, e₂₀, e₀₁} is a quaternion (rotor)
{1} is a real number
{e₀, e₁, e₂} is a vector
{e₁₂, e₂₀, e₀₁} is a bivector
{1, e₀₁} is a complex number
{1, e₁₂, e₂₀, e₀₁} is a quaternion (rotor)
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