Hyperbolic Pascal pyramid

TL;DR

This paper introduces the hyperbolic Pascal pyramid, a new mathematical structure based on hyperbolic geometry, extending previous Pascal-like figures into three dimensions within hyperbolic space.

Contribution

It defines and explores the properties of the hyperbolic Pascal pyramid, generalizing hyperbolic Pascal triangle and pyramid concepts to three dimensions.

Findings

Describes the construction of the hyperbolic Pascal pyramid.

Analyzes the growth and values of elements in the pyramid.

Provides illustrative figures of the pyramid's levels.

Abstract

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 and Pascal's arithmetic pyramid. We describe the growing of hyperbolic Pascal pyramid considering the numbers and the values of the elements. Further figures illustrate the stepping from a level to the next one.