Arithmetic based fractals associated with Pascal's triangle.

T.W. Gamelin; Mamiron A. Mnatsakanian

Publicacions Matemàtiques (2005)

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

Abstract

top
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

top

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 -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.