Hausdorff dimension of some fractals

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Numbers

$F$

$\text{dim}_H(F)$

Apollonian gasket:

1.305688

comment: Value is only a heuristic estimate [1], https://en.wikipedia.org/wiki/Apollonian_gasket

Dragon curve's boundary:

1.523627086202492106277683935954216627284936383401193478138690909457921662895884106892664227464713942

comment: Exact value: $2\log_2\lambda$, where $\lambda$ is the real solution of $\lambda^3-\lambda^2-2 = 0$. https://en.wikipedia.org/wiki/Dragon_curve

Feigenbaum attractor:

3.5699456718709449018420051513864989367638369115148323781079755299213628875001367775263210342163

comment: https://oeis.org/A098587, https://en.wikipedia.org/wiki/Logistic_map

Koch curve:

1.261859507142914874199054228685521708599171280263760855741309887677370402761829610122345377098903491

comment: Exact value: $\log_3(4)$, https://en.wikipedia.org/wiki/Koch_curve

Sierpiński triangle:

1.584962500721156181453738943947816508759814407692481060455752654541098227794358562522280474918088242

comment: Exact value: $\log_2(3)$, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle

Definition

This table lists the Hausdorff dimension $\text{dim}_H(F)$ [2] of certain fractals $F$.

Parameters

$F$

— fractal

Comments

(1)

Readers are welcome to add more interesting examples.

References

[1]

Curtis T. McMullen, "Hausdorff dimension and conformal dynamics III: Computation of dimension", 1997. http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf},

Links

[2]

Wikipedia: Hausdorff dimension

[3]

Wikipedia: List of fractals by Hausdorff dimension

Data properties

Entries are of type: real number

How well the digits are known: heuristic

How they were obtained:

The table said so before this label existed: its own comment calls the first value a heuristic estimate. These are dimensions of fractals obtained numerically, and no error bound is claimed for them.