# Pascal pyramid in the space $\mathbf{H}^2\!\times\!\mathbf{R}$

## TL;DR

This paper introduces a new Pascal pyramid structure in the hyperbolic space ^2 , based on regular mosaics, extending properties of Euclidean and hyperbolic Pascal triangles to three dimensions.

## Contribution

It defines a novel Pascal pyramid in ^2  space using regular mosaics, and explores its properties and growth method.

## Key findings

- Defines Pascal pyramid in ^2  space
- Shows inheritance of properties from Pascal triangles
- Provides illustrative figures and growth method

## Abstract

In this article we introduce a new type of Pascal pyramids. A regular squared mosaic in the hyperbolic plane yields a $(h^2r)$-cube mosaic in space $\mathbf{H}^2\!\times\!\mathbf{R}$ and the definition of the pyramid is based on this regular mosaic. The levels of the pyramid inherit some properties from the Euclidean and hyperbolic Pascal triangles. We give the growing method from level to level and show some illustrating figures.