# A formula for the area of a triangle: Useless, but explicitly in Deep Sets form

arXiv: https://arxiv.org/abs/2503.22786 · HTML: https://mlsanity.com/paper/2503.22786/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 21 references that are in the map.
- First posted on arXiv 28 Mar 2025.

## Builds on (most cited first, 3 of 3)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)

## Abstract (arXiv)

Any permutation-invariant function of data points vecri can be written in the form ρ(sumiφ(vecri)) for suitable functions ρ and φ. This form - known in the machine-learning literature as Deep Sets - also generates a map-reduce algorithm. The area of a triangle is a permutation-invariant function of the locations vecri of the three corners 1≤ i≤ 3. We find the polynomial formula for the area of a triangle that is explicitly in Deep Sets form. This project was motivated by questions about the fundamental computational complexity of n-point statistics in cosmology; that said, no insights of any kind were gained from these results.