• Corpus ID: 12655689

The Toothpick Sequence and Other Sequences from Cellular Automata

  title={The Toothpick Sequence and Other Sequences from Cellular Automata},
  author={David L. Applegate and O. Pol and N. J. A. Sloane},
  journal={arXiv: Combinatorics},
A two-dimensional arrangement of toothpicks is constructed by the following iterative procedure. At stage 1, place a single toothpick of length 1 on a square grid, aligned with the y-axis. At each subsequent stage, for every exposed toothpick end, place an orthogonal toothpick centered at that end. The resulting structure has a fractal-like appearance. We will analyze the toothpick sequence, which gives the total number of toothpicks after n steps. We also study several related sequences that… 

On the Number of ON Cells in Cellular Automata

If a cellular automaton (CA) is started with a single ON cell, how many cells will be ON after n generations? For certain \odd-rule" CAs, including Rule 150, Rule 614, and Fredkin’s Replicator, the

Nim Fractals

It is shown that the game of Nim can be viewed as a cellular automaton, where the total number of counters divided by 2 can be considered as a generation in which P-positions are born.

The Pioneering Role of the Sierpinski Gasket

T he famous Sierpinski gasket, shown in figure 1, appeared in Italian mosaics in the 13th century. But it was not named the Sierpinksi gasket then; in fact, Wacław Sierpiński was not yet born. He was

Odd-Rule Cellular Automata on the Square Grid

This work classify all the odd-rule CAs dened by specifying a neighborhood for each cell, with the rule that a cell turns ON if it is in the neighborhood of an odd number of ON cells at the previous generation, and otherwise turns OFF.

The Sierpinski Triangle and The Ulam-Warburton Automaton

The pioneering role that the Sierpinski triangle plays in the Ulam-Warburton automata is explained and a number of pictures are shown along the way.

Ulam-Warburton Automaton - Counting Cells with Quadratics

This paper is about a sequence of quadratic functions that enumerate the total number of ON cells up to and including generation $n$ of the Ulam-Warburton cellular automaton, where $n$ has the form

Exploring microwave absorption by non-periodic metasurfaces

In recent years there has been a large body of work investigating periodic metasurface microwave absorbers. However, surprisingly few investigations have focused on the absorption performance of

Cohomology of $\mathbb{N}$-graded Lie algebras of maximal class over $\mathbb{Z}_2$

We compute the cohomology with trivial coefficients of Lie algebras $\mathfrak{m}_0$ and $\mathfrak{m}_2$ of maximal class over the field $\mathbb{Z}_2$. In the infinite-dimensional case, we show



Statistical mechanics of cellular automata

Analysis is given of ''elementary'' cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to p definite rules involving the values of its nearest neighbors.

A new kind of science

A New Kind of Science, written and published by Stephen Wolfram, is the outcome of the studies he conducted systematically upon cellular automata, a class of computer model which may be visualized as a set of memory locations, each containing one bit.

Two-dimensional cellular automata

A largely phenomenological study of two-dimensional cellular automata is reported. Qualitative classes of behavior similar to those in one-dimensional cellular automata are found. Growth from simple

Computation in Cellular Automata: A Selected Review

Cellular automata (CAs) are decentralized spatially extended systems consisting of large numbers of simple identical components with local connectivity. Such systems have the potential to perform

The On-Line Encyclopedia of Integer Sequences

  • N. Sloane
  • Computer Science
    Electron. J. Comb.
  • 1994
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.

Fractal geometry

Editor's note: The following articles by Steven G. Krantz and Benoit B. Mandelbrot have an unusual history. In the fall of 1988, Krantz asked the Bulletin of the American Mathematical Society Book

Mathematical constants

  • S. Finch
  • Mathematics
    Encyclopedia of mathematics and its applications
  • 2005
UCBL-20418 This collection of mathematical data consists of two tables of decimal constants arranged according to size rather than function, a third table of integers from 1 to 1000, giving some of

Chaos and Fractals

My talk is a survey of the mathematical work done in our laboratory in the past 10 years. We are not mathematical biologists; rather, we work in mathematics, while maintaining a close relationship

Personal communication

solver), A tree in the integer lattice, Problem 10360

  • Amer. Math. Monthly,
  • 1994