By slight modification of the data of the Sierpiński gasket, keeping the open set condition fulfilled, we obtain self-similar sets with very dense parts, similar to fractals in nature and in random models. This is caused by a complicated structure of the open set and is revealed only under magnification. Thus, the family of self-similar sets with separation condition is much richer and has higher modelling potential than usually expected. An interactive computer search for such examples and new properties for their classification are discussed.
References
1.
C.
Bandt
, “Self-similar sets 5. Integer matrices and fractal tilings of
,” Proc. Am. Math. Soc.
112
, 549
–562
(1991
).2.
C.
Bandt
, “Local geometry of fractals given by tangent measure distributions
,” Monatsh. Math.
133
, 265
–280
(2001
).3.
C.
Bandt
and S.
Graf
, “Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff measure
,” Proc. Am. Math. Soc.
114
(4
), 995
–1001
(1992
).4.
C.
Bandt
and M.
Mesing
, Self-affine fractals of finite type. Convex and fractal geometry, 131-148, Banach Center Publ., 84, Polish Acad. Sci. Inst. Math., Warsaw, 2009
.5.
C.
Bandt
, N. V.
Hung
, and H.
Rao
, “On the open set condition for self-similar fractals
,” Proc. Am. Math. Soc.
134
, 1369
–1374
(2005
).6.
C.
Bandt
, D.
Mekhontsev
, and A.
Tetenov
, “A single fractal pinwheel tile
,” Proc. Am. Math. Soc.
146
, 1271
–1285
(2018
).7.
8.
P.
Duvall
, J.
Keesling
, and A.
Vince
, “The Hausdorff dimension of the boundary of a self-similar tile
,” J. London Math. Soc.
61
, 748
–760
(2000
).9.
K. J.
Falconer
, Fractal Geometry. Mathematical Foundations and Applications
(Wiley
, 1990
).10.
K. J.
Falconer
and J. J.
O'Connor
, “Symmetry and enumeration of self-similar fractals
,” Bull. London Math. Soc.
39
, 272
–282
(2007
).11.
12.
B.
Loridant
, “Crystallographic number systems
,” Monatsh. Math.
167
, 511
–529
(2012
).13.
K.-S.
Lau
, J. J.
Luo
, and H.
Rao
, “On the topology of fractal squares
,” Math. Proc. Cambridge Philos. Soc.
155
, 73
–86
(2013
).14.
D.
Mekhontsev
, IFSTile.com Version 1.7.0.2 (January 2018
) was used for this paper.15.
P. A. P.
Moran
, “Additive functions of intervals and Hausdorff measure
,” Math. Proc. Cambridge Philos. Soc.
42
, 15
–23
(1946
).16.
M.
Morán
, “Dynamical boundary of a self-similar set
,” Fund. Math.
160
(1
), 1
–14
(1999
).17.
H.-O.
Peitgen
, H.
Jürgens
, and D.
Saupe
, Chaos and Fractals, New Frontiers of Science
(Springer
, 1992
).18.
K.
Scheicher
and J. M.
Thuswaldner
, “Neighbors of self-affine tiles in lattice tilings
,” in Fractals in Graz 2001
, edited by P.
Grabner
and W.
Woess
(Birkhäuser
, 2003
), pp. 241
–262
.19.
A.
Schief
, “Separation properties for self-similar sets
,” Proc. Am. Math. Soc.
122
, 111
–115
(1994
).20.
R. S.
Strichartz
and Y.
Wang
, “Geometry of self-affine tiles I
,” Indiana Univ. Math. J.
48
(1
), 1
–24
(1999
).© 2018 Author(s).
2018
Author(s)
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