Evaluating the exact infinitesimal values of area of Sierpinski's carpet and volume of Menger's sponge
Chaos, Solitons and Fractals 42: 3042–3046 (2009)
| Abstract | Very often traditional approaches studying dynamics of self-similarity processes are not able to give their quantitative characteristics at infinity and, as a consequence, use limits to overcome this difficulty. For example, it is well know that the limit area of Sierpinski’s carpet and volume of Menger’s sponge are equal to zero. It is shown in this paper that recently introduced infinite and infinitesimal numbers allow us to use exact expressions instead of limits and to calculate exact infinitesimal values of areas and volumes at various points at infinity even if the chosen moment of the observation is infinitely faraway on the time axis from the starting point. It is interesting that traditional results that can be obtained without the usage of infinite and infinitesimal numbers can be produced just as finite approximations of the new ones. | |||||||||
| Keywords | Sierpinski’s carpet | |||||||||
| Categories | ||||||||||
| Options |
|
|||||||||
Download options| PhilPapers Archive |
Upload a copy of this paper Check publisher's policy on self-archival Papers currently archived: 5,701 |
| External links |
  Try with proxy. |
| Through your library | Configure |
Similar books and articlesErik Palmgren (1998). Developments in Constructive Nonstandard Analysis. Bulletin of Symbolic Logic 4 (3):233-272.
Véronique Izard, Pierre Pica, Elizabeth S. Spelke & Stanislas Dehaene (2008). Exact Equality and Successor Function: Two Key Concepts on the Path Towards Understanding Exact Numbers. Philosophical Psychology 21 (4):491 – 505.
Scott R. Harris & Kerry O. Ferris (2009). How Does It Feel to Be a Star? Identifying Emotions on the Red Carpet. Human Studies 32 (2):133 - 152.
Boško Živaljević (1992). Lusin-Sierpiński Index for the Internal Sets. Journal of Symbolic Logic 57 (1):172-178.
Yaroslav Sergeyev (2010). Lagrange Lecture: Methodology of Numerical Computations with Infinities and Infinitesimals. Rendiconti Del Seminario Matematico dell'Università E Del Politecnico di Torino 68 (2):95–113.
Yaroslav Sergeyev (2009). Numerical Computations and Mathematical Modelling with Infinite and Infinitesimal Numbers. Journal of Applied Mathematics and Computing 29:177-195.
Yaroslav Sergeyev (2009). Numerical Point of View on Calculus for Functions Assuming Finite, Infinite, and Infinitesimal Values Over Finite, Infinite, and Infinitesimal Domains. Nonlinear Analysis Series A 71 (12):e1688-e1707.
Yaroslav D. Sergeyev (2008). A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities. Informatica 19 (4):567-596.
Yaroslav Sergeyev (2007). Blinking Fractals and Their Quantitative Analysis Using Infinite and Infinitesimal Numbers. Chaos, Solitons and Fractals 33 (1):50-75.
Analytics
Monthly downloads |
Added to index2009-12-03Total downloads5 ( #160,428 of 549,124 )Recent downloads (6 months)1 ( #63,361 of 549,124 )How can I increase my downloads? |
My notes
Discussion
