A FORMULA FOR THE AREA OF A TRIANGLE:USELESS, BUT EXPLICITLY IN DEEP SETS FORM

Any permutation-invariant function of data points 𝑟 → 𝑖 can be written in the form 𝜌 ⁡ ( ∑ 𝑖 𝜙 ⁡ ( 𝑟 → 𝑖 ) ) 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 𝑟 → 𝑖 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. △ △ △ The area Δ of a triangle can be written in many ways. Here are a few: where the quantities used are defined in Figure 1, apart from the side lengths 𝑎 , 𝑏 , 𝑐 = | 𝑎 → | , | 𝑏 → | , | 𝑐 → | and the semi-perimeter 𝑠 = ( 𝑎 + 𝑏 + 𝑐 ) / 2 . Formula (1) is the usual introduction to the subject, traceable back to

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• Henry Zimmerman