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[](https://docs.rs/geonum)
[](https://crates.io/crates/geonum)
[](https://coveralls.io/github/mxfactorial/geonum?branch=main)
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# geonum
scaling numerical simulation with the [geometric number](https://gist.github.com/mxfactorial/c151619d22ef6603a557dbf370864085) spec
### features
- dot product, wedge product, geometric product
- inverse, division, normalization
- million-dimension geometric algebra with O(1) complexity
- multivector support and trivector operations
- rotations, reflections, projections, rejections
- exponential, interior product, dual operations
- meet and join, commutator product, sandwich product
- left-contraction, right-contraction
- anti-commutator product
- grade involution and clifford conjugate
- grade extraction
- section for pseudoscalar (extracting components for which a given pseudoscalar is the pseudoscalar)
- square root operation for multivectors
- undual operation (complement to the dual operation)
- regressive product (alternative method for computing the meet of subspaces)
- automatic differentiation through angle rotation (v' = [r, θ + π/2]) (differential geometric calculus)
- replaces set theory foundations with direct geometric representations
- transforms category theory abstractions into simple angle transformations
- unifies discrete and continuous math through a common geometric framework
- provides physical geometric interpretations for abstract mathematical concepts
- automates away unnecessary mathematical formalism using length-angle representation
- enables scaling precision in statistical modeling through direct angle quantization
- supports time evolution via simple angle rotation (angle += energy * time)
- provides statistical methods for angle distributions (arithmetic/circular means, variance, expectation values)
- enables O(1) machine learning operations that would otherwise require O(n²) or O(2^n) complexity
- implements perceptron learning, regression modeling, neural networks, and activation functions
- replaces tensor-based neural network operations with direct angle transformations
- enables scaling to millions of dimensions with constant-time ML computations
- eliminates the "orthogonality search" bottleneck in traditional tensor based machine learning implementations
### use
```
cargo add geonum
```
see `tests` to learn how geometric numbers unify and simplify mathematical foundations including set theory, category theory, and algebraic structures
### benches
#### rank-3 tensor comparison
| tensor (O(n³)) | 2 | 1.06 µs |
| tensor (O(n³)) | 3 | 2.25 µs |
| tensor (O(n³)) | 4 | 3.90 µs |
| tensor (O(n³)) | 8 | 7.92 µs |
| tensor (O(n³)) | 16 | 69.54 µs |
| geonum (O(1)) | any | 9.83 ns |
geonum achieves constant O(1) time complexity regardless of problem size, 400× faster than tensor operations at size 4 and 7000× faster at size 16, eliminating cubic scaling of traditional tensor implementations
#### extreme dimension comparison
| traditional ga | 10 | 543.40 ns (partial) | O(2^n) = 1024 components |
| traditional ga | 30 | theoretical only | O(2^n) = 1 billion+ components |
| traditional ga | 1000 | impossible | O(2^1000) ≈ 10^301 components |
| traditional ga | 1,000,000 | impossible | O(2^1000000) components |
| geonum (O(1)) | 10 | 134.12 ns | O(1) = 2 components |
| geonum (O(1)) | 30 | 153.64 ns | O(1) = 2 components |
| geonum (O(1)) | 1000 | 2.08 µs | O(1) = 2 components |
| geonum (O(1)) | 1,000,000 | 2.94 ms | O(1) = 2 components |
geonum enables geometric algebra in million-dimensional spaces with constant time operations, achieving whats mathematically impossible with traditional implementations (requires more storage than atoms in the universe)
#### multivector ops
| grade extraction | 1,000,000 | 130.69 ns | O(2^n) |
| grade involution | 1,000,000 | 157.18 ns | O(2^n) |
| clifford conjugate | 1,000,000 | 112.90 ns | O(2^n) |
| contractions | 1,000,000 | 266.31 ns | O(2^n) |
| anti-commutator | 1,000,000 | 246.24 ns | O(2^n) |
| all ops combined | 1,000 | 826.57 ns | impossible at high dimensions |
geonum performs all major multivector operations with exceptional efficiency in million-dimensional spaces, maintaining sub-microsecond performance for grade-specific operations that would require exponential time and memory in traditional geometric algebra implementations
### tests
```
cargo fmt --check # format
cargo clippy # lint
cargo test --lib # unit
cargo test --test "*" # feature
cargo bench # bench
cargo llvm-cov # coverage
```
### docs
```
cargo doc --open
```
### todo
- blade classification (identifying geometric types like points, lines, planes)
- rotor estimation algorithms for transforming between sets of geometric objects
- specialized projective geometric algebra (PGA) support
- broadcasting support for operating on multiple objects (point clouds)