# Arithmetic based fractals associated with Pascal's triangle.

• Volume: 49, Issue: 2, page 329-349
• ISSN: 0214-1493

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## Abstract

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Our goal is to study Pascal-Sierpinski gaskets, which are certain fractal sets defined in terms of divisibility of entries in Pascal's triangle. The principal tool is a carry rule for the addition of the base-q representation of coordinates of points in the unit square. In the case that q = p is prime, we connect the carry rule to the power of p appearing in the prime factorization of binomial coefficients. We use the carry rule to define a family of fractal subsets Bqr of the unit square, and we show that when q = p is prime, Bqr coincides with the Pascal-Sierpinski gasket corresponding to N = pr. We go on to describe Bqr as the limit of an iterated function system of partial similarities , and we determine its Hausdorff dimension. We consider also the corresponding fractal sets in higher-dimensional Euclidean space.

## How to cite

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Gamelin, T.W., and Mnatsakanian, Mamiron A.. "Arithmetic based fractals associated with Pascal's triangle.." Publicacions Matemàtiques 49.2 (2005): 329-349. <http://eudml.org/doc/41572>.

@article{Gamelin2005,
abstract = {Our goal is to study Pascal-Sierpinski gaskets, which are certain fractal sets defined in terms of divisibility of entries in Pascal's triangle. The principal tool is a carry rule for the addition of the base-q representation of coordinates of points in the unit square. In the case that q = p is prime, we connect the carry rule to the power of p appearing in the prime factorization of binomial coefficients. We use the carry rule to define a family of fractal subsets Bqr of the unit square, and we show that when q = p is prime, Bqr coincides with the Pascal-Sierpinski gasket corresponding to N = pr. We go on to describe Bqr as the limit of an iterated function system of partial similarities , and we determine its Hausdorff dimension. We consider also the corresponding fractal sets in higher-dimensional Euclidean space.},
author = {Gamelin, T.W., Mnatsakanian, Mamiron A.},
journal = {Publicacions Matemàtiques},
keywords = {Ecuaciones funcionales; Sistemas de funciones iteradas; Fractales; Dimensión de Hausdorff; Sierpinski gasket; Pascal triangle; carry rule; iterated function system; Hausdorff dimension},
language = {eng},
number = {2},
pages = {329-349},
title = {Arithmetic based fractals associated with Pascal's triangle.},
url = {http://eudml.org/doc/41572},
volume = {49},
year = {2005},
}

TY - JOUR
AU - Gamelin, T.W.
AU - Mnatsakanian, Mamiron A.
TI - Arithmetic based fractals associated with Pascal's triangle.
JO - Publicacions Matemàtiques
PY - 2005
VL - 49
IS - 2
SP - 329
EP - 349
AB - Our goal is to study Pascal-Sierpinski gaskets, which are certain fractal sets defined in terms of divisibility of entries in Pascal's triangle. The principal tool is a carry rule for the addition of the base-q representation of coordinates of points in the unit square. In the case that q = p is prime, we connect the carry rule to the power of p appearing in the prime factorization of binomial coefficients. We use the carry rule to define a family of fractal subsets Bqr of the unit square, and we show that when q = p is prime, Bqr coincides with the Pascal-Sierpinski gasket corresponding to N = pr. We go on to describe Bqr as the limit of an iterated function system of partial similarities , and we determine its Hausdorff dimension. We consider also the corresponding fractal sets in higher-dimensional Euclidean space.
LA - eng
KW - Ecuaciones funcionales; Sistemas de funciones iteradas; Fractales; Dimensión de Hausdorff; Sierpinski gasket; Pascal triangle; carry rule; iterated function system; Hausdorff dimension
UR - http://eudml.org/doc/41572
ER -

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