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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.

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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.

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  • $\begingroup$ This makes sense now - thank you. I had tried to view the relevant pages before asking the question, but they were not included in the preview I was using. The one you link to gives the full explanation and a recursive approach to plotting. $\endgroup$ Jun 6, 2015 at 16:01
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    $\begingroup$ @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. $\endgroup$ Jun 7, 2015 at 9:33

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