Hyperbolic Pascal triangles

Fibonacci words in hyperbolic Pascal triangles

The hyperbolic Pascal triangle HPT 4,q (q ≥ 5) is a new mathematical construction, which is a geometrical generalization of Pascal’s arithmetical triangle. In the present study we show that a natural pattern of rows of HPT 4,5 is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the h...

The Pascal-de Moivre Triangles*

The coefficients of the Pascal triangle were generalized in 1756 by de Moivre [5]. Each row of a Pascal triangle contains a sequence of numbers that are the coefficients of the power series expansion for the binary expression (l + x)^. The de Moivre formula [2], [4], [5], [6] derives the coefficients of the power series for the generalized expansion of (1 + x + x + • • • + x^"^). Thus, for inte...

Tea Tasting and Pascal Triangles

Hyperbolic Pascal pyramid

In this paper we introduce a new type of Pascal’s pyramids. The new object is called hyperbolic Pascal pyramid since the mathematical background goes back to the regular cube mosaic (cubic honeycomb) in the hyperbolic space. The definition of the hyperbolic Pascal pyramid is a natural generalization of the definition of hyperbolic Pascal triangle ([2]) and Pascal’s arithmetic pyramid. We descri...

Hyperbolic Pascal simplex

In this article we introduce a new geometric object called hyperbolic Pascal simplex. This new object is presented by the regular hypercube mosaic in the 4-dimensional hyperbolic space. The definition of the hyperbolic Pascal simplex, whose hyperfaces are hyperbolic Pascal pyramids and faces are hyperbolic Pascals triangles, is a natural generalization of the definition of the hyperbolic Pascal...