Dual quaternion

[ker2x]

http://altdevblogaday.com/2011/11/16/dual-quaternion/

If anyone understand.... please ?

[Syntopia]

There was some discussion on dual and hyper-dual (4D) numbers some months ago, which might be of interest: http://www.fractalforums.com/general-discussion-b77/hyper-dual-numbers/msg35859/#msg35859

Dual Quaternions seems to be 8D quantities, but I'm not sure they are useful for fractals - there are quite a lot of zeroes in their multiplication table.

[s31415]

These "dual" number are called Grassmann numbers in theoretical physics and are typically used to describe "fermionic" particles and fields (while usual numbers are used to describe "bosonic" particle and fields). Their main property is that their multiplication is anti-commutative instead of commutative. So in particular, if you have a one-dimensional space of Grassmann numbers, generated by a variable e, then you have e^2 = -e^2 = 0. This makes Grassmann numbers rather boring, because the most general function of a Grassmann variable you can write is linear: f(e) = a + be, where a and b are real or complex. This should be contrasted with functions of a commuting variable x, for which you can write a whole Taylor series f(x) = a_0 + a_1 x + a_2 x^2 +... Apart from this you can do with them more or less all you can do with commuting variables: addition, multiplication, even integration. It gets slightly more interesting when you consider several anticommuting variables, but I still don't think there is potential for creating fractals with them.

See this wikipedia article for a bit more information:
http://en.wikipedia.org/wiki/Grassmann_number

Sam

[Syntopia]

Interesting, but I'm not sure I understand the link? Dual numbers (as defined in http://simonstechblog.blogspot.com/2011/11/dual-number.html) do commute. This is also true for Hyper-Dual numbers. Dual Quaternions neither commute nor anti-commute in general (they contain ordinary Quaternions as a subset). The "dual unit" of a dual number does satisfy e^2=0, but that is only one part of a dual number.

One interesting application of dual numbers is as a computational device to find the derivative of functions (http://en.wikipedia.org/wiki/Automatic_differentiation). And I actually think this could be interesting in relation to finding the numerical gradient of the DE (instead of the Buddhi/Makin approach, where you introduce an arbitrary epsilon step length to find the gradient).

I have tried replacing complex and quaternion Mandelbrot/Julia system with dual and hypercomplex numbers, but the results were not very interesting.

[s31415]

The coefficients a and b in
http://simonstechblog.blogspot.com/2011/11/dual-number.html
are ordinary real numbers which do commute. Epsilon is a Grassmann variable which commute with the real numbers but anti-commute with itself (hence epsilon^2 = 0). As the second equation of
http://altdevblogaday.com/2011/11/16/dual-quaternion/
shows, dual quaternions are obtained by replacing the real numbers by quaternions. The Grassmann variable epsilon now commutes with quaternions, but still anti-commutes with itself. I couldn't find a handy reference about hyperdual numbers, so I don't know about them.

You can generalize this by adding other Grassmann variables commuting with the coefficients but anti-commuting among themselves.

The application to compute derivatives might be interesting indeed.

Sam

[Syntopia]

Yes, I guess you could call the epsilon unit a grassman number since it anti-commutes with itself - but remember that it also commutes, since A*B=B*A=0 for all dual numbers with zero real component. And all dual numbers do commute with other dual numbers (except the Dual Quaternions, naturally), and are actually very useful in Taylor series (this is the way the method for doing derivatives is established).

[s31415]

Yes, they also commute because there is only one Grassmann variable (or "number"). Dual numbers are the same as the Grassmann algebra generated by a single Grassmann variable.

Functions of a single Grassmann variable are at most linear, of the form a + b epsilon. On the other hand, if you consider a function from the dual numbers to themselves, or equivalently from the Grassmann algebra to itself, then you get what is described in the article about differentiation.