Pascal's Repertorium of Higher Mathe-

Geometry of Binomial Coefficients

This note describes the geometrical pattern of zeroes and ones obtained by reducing modulo two each element of Pascal's triangle formed from binomial coefficients. When an infinite number of rows of Pascal's triangle are included, the limiting pattern is found to be "self-similar," and is characterized by a "fractal dimension" log2 3. Analysis of the pattern provides a simple derivation of the ...

Sierpinski’s Triangle and the Prouhet-Thue-Morse Word

Sierpinski's triangle is a fractal and the Prouhet-Thue-Morse word is suÆciently chaotic to avoid cubes. Here we observe that there is at least a tenuous connection between them: the Sierpinski triangle is evident in Pascal's triangle mod 2 whose inverse, as an in nite lower-triangular matrix, involves the Prouhet-Thue-Morse word. Pascal's triangle mod 2 (Fig. 1b) is a discrete version of the f...

Construction of Zero Cross Correlation Code using a type of Pascal's Triangle Matrix for Spectral Amplitude Coding Optical Code Division Multiple Access networks

In this paper a new method to construct zero cross correlation code with the help of Pascal's triangle pattern called Pascal's Triangle Matrix Code (PTMC) for Spectral Amplitude Coding Optical Code Division Multiple Access (SAC-OCDMA) system is successfully developed. The advantages of this code are simplicity of code construction, flexibility of choosing code weight and number of users. The nu...

Title: Model-based Design of Clinical Information Systems

Pascal’s ring, cardinal points, and refractive compensation

Pascal's ring is a hexagon each of whose corners represents one of the six cardinal points of an optical system and whose sides represent relationships of relative axial position of the cardinal points. Changes to the ring represent the axial displacements of the cardinal points of the visual optical system of an eye that are caused when a spectacle lens compensates for the ametropia. Pascal's ...