Abstract
In this chapter we present some other definitions of the dimension of a geometric object. The chapter concludes with a discussion of the uses and abuses of the fractal dimension of a geometric object.
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Notes
- 1.
A.M. Lyapunov (1857–1918) was a Russian mathematician.
- 2.
Let \(F: {\mathbb {R}}^m \rightarrow {\mathbb {R}}^k\) so \(F(x) = \bigl (F_1(x), F_2(x), \ldots , F_k(x) \bigr )\), where each \(F_i: {\mathbb {R}}^m \rightarrow {\mathbb {R}}\). The Jacobian matrix J of F is the k by m matrix of partial derivatives defined by J ij = ∂F i∕∂x j for i = 1, 2, …, k and j = 1, 2, …, m.
- 3.
Let J be a square matrix. Then the real number λ is an eigenvalue of J if for some nonzero vector u (called an eigenvector of J) we have Ju = λu.
- 4.
The basin of attraction [Farmer 83] of an attractor Ω is the closure of the set of initial conditions that approach Ω.
- 5.
The title of this section is a rephrasing of the title of [Halley 04].
- 6.
Morphology is the branch of biology concerned with the form and structure of organisms.
- 7.
Adapted from http://en.wikipedia.org/wiki/Axon.
- 8.
Microglia are cells occurring in the central nervous system of invertebrates and vertebrates. These cells mediate immune responses in the central nervous system by clearing cellular debris and dead neurons from nervous tissue through the process of phagocytosis (cell eating). http://www.britannica.com/EBchecked/topic/380412/microglia.
- 9.
In the song “Over the Rainbow”, immortalized by Judy Garland in the movie “The Wizard of Oz”, the interval between the first two syllables “some” and “where” is an octave.
- 10.
In the end of the first line of the “Star Spangled Banner”, for the notes “you” and “see”, the pitch of “see” is a fourth higher than “you”.
- 11.
In the song “Twinkle Twinkle Little Star”, the pitch of the third and fourth notes (the second “twinkle”) is a fifth higher than the pitch of the first two notes (the first “twinkle”). That is, the interval is a fifth.
- 12.
“Shazam” is a registered trademark of Apple Inc. and its affiliates.
- 13.
Timbre is the quality given to a sound by its overtones: such as the resonance by which the ear recognizes and identifies a voiced speech sound, or the quality of tone distinctive of a particular singing voice or musical instrument. https://www.merriam-webster.com/dictionary/timbre.
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Rosenberg, E. (2020). Other Dimensions. In: Fractal Dimensions of Networks. Springer, Cham. https://doi.org/10.1007/978-3-030-43169-3_20
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