[chriswarbo]

> In symplectic geometry an area is the fundamental quantity, whereas Euclidean geometry measures lengths and angles.

> yield the results in a coordinate, matrix and trigonometry-free manner

Some related ideas, for simplifying and generalising geometry:

Euclidean geometry is characterised by inner-product/symmetric-bilinear-form, shown in Section 2.1:

    𝐚𝐛 = a₁×b₁ + a₂×b₂

Where 𝐚 = a₁𝐱 + a₂𝐲 and 𝐛 = b₁𝐱 + b₂𝐲. This is just the first components multiplied together, plus the second components multiplied together; and is easily generalised to N dimensions:

    𝐚𝐛 = Σaₙbₙ

So far, so familiar. We tend to measure vectors using their length, which is the square-root of the vector's inner-product with itself, e.g.

    |𝐚| = √(𝐚𝐚)

However, this is quite restrictive: the inner-product only requires + and ×, which are well-defined for all sorts of fields (real numbers, complex numbers, finite fields, rational numbers, etc.); square-roots aren't so easy to define, which restricts Euclidean distance to only a few fields (e.g. real numbers and complex numbers).

Remarkably, we can do a lot of geometry without using length at all, hence not requiring square roots, and therefore generalising our results to many more fields. Instead, we just work with quantities like 𝐚𝐚 directly, which can be interpreted as the area of a square with side-length |𝐚| (AKA a "quadrance"). An obvious example is Pythagoras' theorem, which relates the quadrances of a right-triangle's sides.

This use of area is probably connected to symplectic geometry, but I haven't looked into that yet.

The approach described above is called Rational Trigonometry; which also avoids transcendental functions like cos/sin, by replacing angles with "spreads" (equivalent to the sin^2 of an angle), which range from 0 = parallel to 1 = perpendicular.

Looking again at the inner-product 𝐚𝐛, there's another degree of freedom lurking in there if we interpret it as matrix multiplication 𝐚𝐛ᵀ (the rules of matrix multiplication require us to transpose the 1×n row-vector 𝐛 into the n×1 column-vector 𝐛ᵀ).

By default, this matrix formulation doesn't alter the inner product: it's still Σaₙbₙ. However, it gives us the flexibility to introduce an n×n matrix 𝐌 in-between the vectors: 𝐚𝐌𝐛ᵀ

If 𝐌 is the identity matrix [[1, 0], [0, 1]] (denoted 𝐈 in the article), then we again keep the original behaviour. In this sense, Euclidean geometry is characterised by 𝐈 (encoding its symmetric bilinear form).

If we use other n×n matrices we get different geometries. In particular, the matrix [[1, 0], [0, -1]] gives us the "red" inner-product a₁×b₁ - a₂×b₂; and [[0, 1], [1, 0]] gives us the "green" inner-product a₁×b₂ + a₂×b₁. These are closely related to each other (one is a rotation of the other; both are 2D analogues of special-relativity), and to the "blue" Euclidean geometry. This colour-coding come from Chromogeometry, which studies their relations.

These are explained more in An Introduction to Rational Trigonometry and Chromogeometry (which I just submitted at https://news.ycombinator.com/item?id=30418194 )