# The generalization of Sierpinski carpet and Sierpinski triangle in $n$-dimensional space

## TL;DR

This paper extends the classical Sierpinski carpet and triangle fractals to n-dimensional space using affine transformations, enabling clear visualization of higher-dimensional fractals like the Menger sponge and Sierpinski simplex.

## Contribution

It introduces a general method for constructing and characterizing affine Sierpinski fractals in n-dimensional space, including explicit examples in four dimensions.

## Key findings

- Explicit construction of 4D Menger sponge and Sierpinski simplex
- Method applicable to broader class of fractals
- Enhanced understanding of higher-dimensional fractal geometry

## Abstract

We obtain a nature generalization for an affine Sierpinski carpet and Sierpinski triangle to $n$-dimensional space, by using the generations and characterizations of affinely-equivalent Sierpinski carpet. Exactly, in this paper, a Menger sponge and Sierpinski simplex in $4$-dimensional space could be drawn out clearly under an affine transformation. Furthermore, the method could be used to a much broader class in fractals.