# On the upper embedding of symmetric configurations with block size 3

## TL;DR

This paper investigates conditions for embedding symmetric configurations with block size 3 into orientable surfaces, ensuring triangular faces and a single large face, and explores embeddability limits with new findings and corrected data.

## Contribution

It provides a sufficient condition for such embeddings, verifies it for configurations up to 19 points, and identifies non-embeddable configurations on 21 points.

## Key findings

- All configurations on up to 19 points satisfy the embedding condition.
- Existence of a 21-point configuration that cannot be embedded in any orientation.
- Revised the table of configuration counts, correcting previous data.

## Abstract

We consider the problem of embedding a symmetric configuration with block size 3 in an orientable surface in such a way that the blocks of the configuration form triangular faces and there is only one extra large face. We develop a sufficient condition for such an embedding to exist given any orientation of the configuration, and show that this condition is satisfied for all configurations on up to 19 points. We also show that there exists a configuration on 21 points which is not embeddable in any orientation. As a by-product, we give a revised table of numbers of configurations, correcting the published figure for 19 points. We give a number of open questions about embeddability of configurations on larger numbers of points.