# What does "points spanned by powers" mean in the Goffinet dragon definition?

The definition of the Goffinet dragon fractal given by Wolfram Mathworld refers to

plotting all points spanned by powers of the complex number p=0.65-0.3i

What does it mean for points to be "spanned by powers"?

I initially guessed that this simply meant that all the successive integer powers of p should be plotted. However, plotting $p$, $p^2$, $p^3$ simply gives a spiral heading towards zero, and not the fractal image displayed on the Wolfram page.

If you look at this book, which is referenced in the definition you linked to, you'll see that the points in the Goffinet dragon are all the points of the form $$\sum_{k = 1}^{\infty} a_k p^k$$ with $a_k$ either $0$ or $1$. Of course, for practical purposes the points actually plotted in a picture need to be cut off at some fixed power.
• @trichoplax Put another way - it's the set of all linear combinations of the numbers $p^k$ where the coefficients can be either zero or one. If you consider that these coefficients form a finite field, then this is exactly the linear the span of the $p^k$s. Jun 7, 2015 at 9:33