# Apollonian packings in seven and eight dimensions

## TL;DR

This paper extends the study of Apollonian packings to seven and eight dimensions, demonstrating they retain key properties like tangency, space-filling, and integer curvatures, similar to lower-dimensional cases.

## Contribution

It generalizes Apollonian packings to higher dimensions beyond six, confirming their key geometric and number-theoretic properties in seven and eight dimensions.

## Key findings

- Hyperspheres are tangent or non-intersecting in 7D and 8D.
- The packings fill the hyperspace in these dimensions.
- Hyperspheres can have integer curvatures in a suitable perspective.

## Abstract

In an earlier work, we proposed a generalization for the Apollonian packing in arbitrary dimensions and showed that the resulting object in four, five, and six dimensions have properties consistent with the Apollonian circle and sphere packings in two and three dimensions. In this work, we investigate the generalization in seven and eight dimensions and show that they too have many of the properties of those in lower dimensions. In particular, the hyperspheres are tangent or do not intersect; they fill the hyperspace; the object includes a maximal cluster of mutually tangent hyperspheres; and there exists a perspective where all hyperspheres in the object have integer curvatures.