Huge update to README

README.md

@@ -9,13 +9,95 @@ A simple, fully static multivector implementation for Julia. While not the most
 allows for fast prototyping and implementation of geometric algebras and multivectors of arbitrary
 dimension or metric signature.
 
-Currently, the package exports the `CliffordNumber{Cl,T}` type, which represents an element of the
-Clifford algebra `Cl` with elements of type `T`, which have `L` elements. Future releases will
-include more storage-efficient data structures for common, sparser elements such as k-vectors and
-blades.
+# Clifford numbers
 
-Currently supported operations are addition (`+`), subtraction and negation (`-`), the geometric
-product (`*`), and the Hodge star (`⋆`).
+## Types
+
+This package exports `AbstractCliffordNumber{Q,T}` and its subtypes, which describe the behavior of
+multivectors with quadratic form `Q` and scalar type `T<:Union{Real,Complex}`. This is a subtype of 
+`Number`, and therefore acts as a scalar.
+
+`AbstractCliffordNumber{Q,T}` includes the following concrete subtypes:
+  * `CliffordNumber{Q,T,L}`, which represents the coefficients associated with all basis blades.
+  * `EvenCliffordNumber{Q,T,L}` and `OddCliffordNumber{Q,T,L}`, which represents multivectors with
+only basis blades of even or odd grade being nonzero. These are especially important when dealing
+with physically realizable Euclidean transformations (rotations and translations).
+  * `KVector{K,Q,T,L}`, which represents multivectors with only basis blades of grade `K` being
+zero. This is especially useful for representing common vectors and bivectors.
+
+## Promotion
+
+The type promotion system is heavily leveraged to minimize the memory footprint needed to represent
+the results of various operations. Promotion can convert the numeric types associated with two 
+`AbstractCliffordNumber{Q}` instances (for instance, the sum of `KVector{1,APS,Int}` and
+`KVector{1,APS,Float64}` is `KVector{1,APS,Float64}`), but it can also leverage grade information to
+promote to smaller types: the sum of `KVector{1,APS,Int}` and `KVector{3,APS,Int}` is an
+`OddCliffordNumber{APS,Int}`, but the sum of `KVector{1,APS,Int}` and `KVector{2,APS,Int}` are
+`CliffordNumber{APS,Int}`.
+
+## Indexing
+
+Although `AbstractCliffordNumber` instances are scalars, the `BitIndex{Q}` type can be used to
+retrieve coefficients associated with specific basis blades. The full set of `BitIndex{Q}` types for
+some `x::AbstractCliffordNumber` can be generated with `BitIndices(x)`, and this is a binary ordered
+vector of `BitIndex{Q}` objects.
+
+Mathematical operations are defined generically by working with the `BitIndex{Q}` objects associated
+with an `AbstractCliffordNumber{Q,T}`. Elementwise operations on each element of a `BitIndices`
+instance returns a `TransformedBitIndices`, a wrapper which lazily associates a function 
+
+## Operations
+
+The following mathematical operations are supported by this package:
+  * Addition (`+`), subtraction and negation (`-`)
+  * The geometric product (`*`)
+  * Left (`/`) and right (`\`) division, including rational division (`//`)
+  * The reverse (`~`), grade involution, and Clifford conjugation
+  * The modulus and absolute value (with `abs2` and `abs`)
+  * The wedge product (`∧`)
+  * The left (`⨼`) and right (`⨽`) contractions
+  * The dot product and the Hestenes dot product
+  * The commutator product (`×`)
+  * Exponentiation
+
+Some of the names or implementations of various operations may change in the near future.
+
+Note that `AbstractCliffordNumber{Q,T}` is a scalar type, so dotted operators do not apply any
+operations to each coefficient individually. However, they can be used to perform elementwise
+operations on collections of Clifford numbers.
+
+# Features to be added or improved
+
+## Type system
+
+  * A `StaticMultivector{Q,T,L}` type, allowing for non-static implementations.
+  * `numeric_type` will likely be renamed to `scalartype`.
+
+## Relationships between Clifford algebras
+
+  * Determining the even subalgebra associated with some Clifford algebra.
+  * Converting even multivectors of an algebra to multivectors in the even subalgebra.
+
+## Metric signatures
+  * A better API for metric signatures. The `QuadraticForm{P,Q,R}` type is limited in that the
+positive-squaring, negative-squaring, and zero-squaring basis blades are listed in that specific
+order.
+
+## Mathematical operations
+
+  * The implementation of the reverse operation: should we overload `Base.reverse`, `Base.conj`, 
+or do something else? What about other grade automorphisms?
+  * Better numerical accuracy and stability for some operations. In particular, properly leveraging
+`sinpi`, `cospi`, `hypot`, and other methods with better numerical accuracy and stability 
+internally.
+
+## Interoperability
+
+  * How do we handle the dot product of this package and the dot product of the `LinearAlgebra`
+standard library? This package has no dependencies, but the semantics should be compatible as much
+as possible.
+  * Secondarily, `StaticArrays.similar_type` and `CliffordNumbers.similar_type` have essentially
+identical behavior.
 
 [docs-stable-img]:  https://img.shields.io/badge/docs-stable-blue.svg
 [docs-stable-url]:  https://brainandforce.github.io/CliffordNumbers.jl/stable