Cell Complex

Aims

Libraries and tools for topological and geometric modeling.

Docs

- A Recursive Combinatorial Description of cell-complex
- A paper about the definition of cell-complex in this project

Modules

- npm install cell-complex

- API Docs

- Try examples

- Contents:
- int
- num
- euclid
- combinatorial-game
- cell-complex
- homology

$3

#### examples/int-module.js:

`` javascript
let assert = require ("assert") .strict

let ut = require ("cell-complex/lib/util")
let int = require ("cell-complex/lib/int")

{
/**
* generic row_canonical_form
* i.e. hermite_normal_form for integers
*/

let A = int.matrix ([
[2, 3, 6, 2],
[5, 6, 1, 6],
[8, 3, 1, 1],
])

let B = int.matrix ([
[1, 0, -11, 2],
[0, 3, 28, -2],
[0, 0, 61, -13],
])

assert (
A.row_canonical_form () .eq (B)
)
}

{
/**
* generic diag_canonical_form
* i.e. smith_normal_form for integers
*/

let A = int.matrix ([
[2, 4, 4],
[-6, 6, 12],
[10, -4, -16],
])

let S = int.matrix ([
[2, 0, 0],
[0, 6, 0],
[0, 0, -12],
])

assert (
A.diag_canonical_form () .eq (S)
)
}

{
/**
* solve linear diophantine equations
*/

let A = int.matrix ([
[1, 2, 3, 4, 5, 6, 7],
[1, 0, 1, 0, 1, 0, 1],
[2, 4, 5, 6, 1, 1, 1],
[1, 4, 2, 5, 2, 0, 0],
[0, 0, 1, 1, 2, 2, 3],
])

let b = int.vector ([
28,
4,
20,
14,
9,
])

let solution = A.solve (b)

if (solution !== null) {
solution.print ()

assert (
A.act (solution) .eq (b)
)
}
}
`

$3

- with config-able epsilon for numerical stability

#### examples/num-linear-algebra.js:

` javascript
let assert = require ("assert") .strict

let ut = require ("cell-complex/lib/util")
let num = require ("cell-complex/lib/num")

{
/**
* reduced_row_echelon_form reduces pivots to one
* while respecting epsilon for numerical stability
*/

let A = num.matrix ([
[1, 3, 1, 9],
[1, 1, -1, 1],
[3, 11, 5, 35],
])

let B = num.matrix ([
[1, 0, -2, -3],
[0, 1, 1, 4],
[0, 0, 0, 0],
])

A.reduced_row_echelon_form () .print ()

assert (
A.reduced_row_echelon_form () .eq (B)
)
}
`

$3

- module theory) over euclidean ring
- for generic matrix algorithms

$3

- a game engine for n-player perfect information games
- example games:
- tic-tac-toe
- hackenbush -- demo

$3

- cell-complex based low dimensional algebraic topology library

$3

- cellular homology of cell-complex

#### examples/four-ways-to-glue-a-square.js:

- The following pictures are made by Guy Inchbald, a.k.a. Steelpillow

!Flatsurfaces.svg

` javascript
let cx = require ("cell-complex/lib/cell-complex")
let hl = require ("cell-complex/lib/homology")
let ut = require ("cell-complex/lib/util")
`

!Spherecycles1.svg

` javascript
class sphere_t extends cx.cell_complex_t {
constructor () {
super ({ dim: 2 })
this.attach_vertexes (["south", "middle", "north"])
this.attach_edge ("south_long", ["south", "middle"])
this.attach_edge ("north_long", ["middle", "north"])
this.attach_face ("surf", [
"south_long",
"north_long",
["north_long", "rev"],
["south_long", "rev"],
])
}
}
`

!Toruscycles1.svg

` javascript
class torus_t extends cx.cell_complex_t {
constructor () {
super ({ dim: 2 })
this.attach_vertex ("origin")
this.attach_edge ("toro", ["origin", "origin"])
this.attach_edge ("polo", ["origin", "origin"])
this.attach_face ("surf", [
"toro",
"polo",
["toro", "rev"],
["polo", "rev"],
])
}
}
`

!Kleincycles1.svg

` javascript
class klein_bottle_t extends cx.cell_complex_t {
constructor () {
super ({ dim: 2 })
this.attach_vertex ("origin")
this.attach_edge ("toro", ["origin", "origin"])
this.attach_edge ("cross", ["origin", "origin"])
this.attach_face ("surf", [
"toro",
"cross",
["toro", "rev"],
"cross",
])
}
}
`

!Projectivecycles1.svg

` javascript
class projective_plane_t extends cx.cell_complex_t {
constructor () {
super ({ dim: 2 })
this.attach_vertexes (["start", "end"])
this.attach_edge ("left_rim", ["start", "end"])
this.attach_edge ("right_rim", ["end", "start"])
this.attach_face ("surf", [
"left_rim", "right_rim",
"left_rim", "right_rim",
])
}
}

`

- calculate homology groups):

` javascript
let report = {
"sphere": hl.report (new sphere_t ()),
"torus": hl.report (new torus_t ()),
"klein_bottle": hl.report (new klein_bottle_t ()),
"projective_plane": hl.report (new projective_plane_t ()),
}

ut.log (report)

let expected_report = {
sphere:
{ '0': { betti_number: 1, torsion_coefficients: [] },
'1': { betti_number: 0, torsion_coefficients: [] },
'2': { betti_number: 1, torsion_coefficients: [] },
euler_characteristic: 2 },
torus:
{ '0': { betti_number: 1, torsion_coefficients: [] },
'1': { betti_number: 2, torsion_coefficients: [] },
'2': { betti_number: 1, torsion_coefficients: [] },
euler_characteristic: 0 },
klein_bottle:
{ '0': { betti_number: 1, torsion_coefficients: [] },
'1': { betti_number: 1, torsion_coefficients: [ 2 ] },
'2': { betti_number: 0, torsion_coefficients: [] },
euler_characteristic: 0 },
projective_plane:
{ '0': { betti_number: 1, torsion_coefficients: [] },
'1': { betti_number: 0, torsion_coefficients: [ 2 ] },
'2': { betti_number: 0, torsion_coefficients: [] },
euler_characteristic: 1 }
}
`

Community

- We enforce C4 as collaboration protocol -- The C4 RFC
- Style Guide -- observe the style of existing code and respect it
- Code of Conduct
- Source code -- github
- CI -- travis-ci

Contributing

- Prepare: npm install
- Compile: npx tsc
- Compile and watch: npx tsc --watch
- Run all tests: npx ava
- Run specific test file: npx ava -sv `

License

- GPLv3