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I am currently wondering whether it is realistically possible to choose the topic "Fractals and fractal dimensions" for a seminar aimed at undergraduate students in the 2nd semester, with prerequisites only the basic first semester courses in Analysis and Linear Algebra.

These students do not yet know what a metric space, let alone a topological space is, and I feel that the seminar is not the right place to introduce these concepts. They do know the concepts of dimension of a (finite-dimensional) vector space as well as asymptotic behaviors (limits, sup/inf, boundedness etc.) of sequences of numbers in $\mathbb R$. Therefore I believe that as long as one restricts to subsets of the euclidean $\mathbb R^n$ the topic of fractional dimension and nice examples of fractals are in principle accessible to those students.

However, it seems hard to find accessible literature. Almost all books start with general topological or metric spaces and then use the corresponding notation and terminology throughout all later chapters. It would already be very helpful to find a book with a long, detailed chapter devoted to the special case of subsets of $\mathbb R^n$ (or even $\mathbb R^2$) which is at least partially independent of the rest of the book.

Comments are highly appreciated! Of course it might turn out that the topic is simply too complicated/ambitious for 2nd semester students. In that case I'd just choose a simpler, more standard topic.

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The book Fractal Geometry - Mathematical Foundations and Applications by Kenneth Falconer may be what you are looking for. As far as I recall, everything is done in $\mathbb{R}^n$ and he tries to keep the abstractions and technicalities to a minimum while retaining rigor.

Falconer has also written Fractals: A very short introduction, which may be motivating as well.

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  • $\begingroup$ The third edition of this book is exactly what I was looking for. Thanks for your answer! I think I'll try it, then. $\endgroup$
    – B K
    Commented Jul 22 at 18:58
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Erin Pearse's Introduction to dimension theory and fractal geometry may well be suited for this purpose. It introduces the various ways to define and measure a fractional dimension (box counting, Minkowski, Hausdorff), with worked out examples. A few technical terms do show up (Lipschitz continuity, Borel set), but these do not play a central role and the text should be understandable with only a basic knowledge of analysis.

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    $\begingroup$ That's indeed a very nice text I didn't know yet! It makes me more confident that it's doable. $\endgroup$
    – B K
    Commented Jul 21 at 20:34

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