Hyperbolic Pascal simplex

TL;DR

This paper introduces the hyperbolic Pascal simplex, a new geometric structure in 4D hyperbolic space, generalizing hyperbolic Pascal triangles and pyramids, with detailed growth and visualization.

Contribution

It presents the first definition and analysis of the hyperbolic Pascal simplex, extending hyperbolic Pascal structures into four dimensions.

Findings

Defined the hyperbolic Pascal simplex in 4D hyperbolic space

Analyzed the growth of elements within the structure

Provided illustrative figures of the simplex's development

Abstract

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