Engine prototype (#13)

This PR adds code for a Julia-language prototype of a configuration solver, in the `engine-proto` folder. It uses Julia version 1.10.0.

### Approaches
Development of this PR tried two broad approaches to the constraint geometry problem. Each one suggested various solution techniques. The Gram matrix approach, with the low-rank factorization technique, seems the most promising.

- **Algebraic** *(In the `alg-test` subfolder).* Write the constraints as polynomials in the inversive coordinates of the elements, and use computational algebraic geometry techniques to solve the resulting system. We tried the following techniques.
  - **Gröbner bases** *(`Engine.Algebraic.jl`).* Symbolic. Find a Gröbner basis for the ideal generated by the constraint equations. Information about the solution variety, like its codimension, is then relatively easy to extract.
  - **Homotopy continuation** *(`Engine.Numerical.jl`).* Numerical. Cut the solution set along a random hyperplane to get a generic zero-dimensional slice, and then use a fancy homotopy technique to approximate the points in that slice.

  A few notes about our experiences can be found on the [engine prototype](wiki/Engine-prototype) wiki page.
- **Gram matrix** *(in the `gram-test` subfolder).* A construction is described completely, up to conformal transformations, by the Gram matrix of the vectors representing its elements. Express the constraints as fixed entries of the Gram matrix, and use numerical linear algebra techniques to find a list of vectors whose Gram matrix fits the bill. We tried the following techniques.
  - **LDL decomposition** *(`gram-test.sage`, `gram-test.jl`, `overlap-test.jl`).* Find a cluster of up to five elements whose Gram matrix is completely filled in by the constraints. Use LDL decomposition to find a list of vectors with that Gram matrix. This technique can be made algebraic, as seen in `overlap-test.jl`.
  - **Low-rank factorization** *(source files listed in findings section).* Write down a quadratic loss function that says how far a set of vectors is from meeting the Gram matrix constraints. Use a smooth optimization technique like Newton's method or gradient descent to find a zero of the loss function. In addition to the polished prototype described in the results section, we have an early prototype using an off-the-shelf factorization package (`low-rank-test.jl`) and an visualization of the loss function landscape near global minima (`basin-shapes.jl`).

  The [Gram matrix parameterization](wiki/Gram-matrix-parameterization) wiki page contains detailed notes on this approach.

### Findings

With the algebraic approach, we hit a performance wall pretty quickly as our constructions grew. It was often hard to find real solutions of the polynomial system, since the techniques we use work most naturally in the complex world.

With the Gram matrix approach, on the other hand, we could solve interesting problems in acceptably short times using the low-rank factorization technique. We put the optimization routine in its own module (`Engine.jl`) and used it to solve five example problems:
- `overlapping-pyramids.jl`
- `circles-in-triangle.jl`
- `sphere-in-tetrahedron.jl`
- `tetrahedron-radius-ratio.jl`
- `irisawa-hexlet.jl`

We plan to use low-rank factorization of the Gram matrix in our first app prototype.

### Visualizations

We used the visualizer in the `ganja-test` folder to visually check our low-rank factorization results. The visualizer runs [Ganja.js](https://enkimute.github.io/ganja.js/) in an Electron app, made with [Blink](https://github.com/JuliaGizmos/Blink.jl). Although Ganja.js makes beautiful pictures under most circumstances, we found two obstacles to using it in production.

- It seems to have precision problems with low-curvature spheres.
- We couldn't figure out how to customize its clipping and transparency settings, and the default settings often obscure construction details.

Co-authored-by: Aaron Fenyes <aaron.fenyes@fareycircles.ooo>
Co-authored-by: Glen Whitney <glen@studioinfinity.org>
Reviewed-on: #13
Co-authored-by: Vectornaut <vectornaut@nobody@nowhere.net>
Co-committed-by: Vectornaut <vectornaut@nobody@nowhere.net>
This commit is contained in:
Vectornaut 2024-10-21 03:18:47 +00:00 committed by Glen Whitney
parent c48d685ad6
commit b92be312e8
20 changed files with 3977 additions and 21 deletions

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module Viewer
using Blink
using Colors
using Printf
using Main.Engine
export ConstructionViewer, display!, opentools!, closetools!
# === Blink utilities ===
append_to_head!(w, type, content) = @js w begin
@var element = document.createElement($type)
element.appendChild(document.createTextNode($content))
document.head.appendChild(element)
end
style!(w, stylesheet) = append_to_head!(w, "style", stylesheet)
script!(w, code) = append_to_head!(w, "script", code)
# === construction viewer ===
mutable struct ConstructionViewer
win::Window
function ConstructionViewer()
# create window and open developer console
win = Window(Blink.Dict(:width => 620, :height => 830))
# set stylesheet
style!(win, """
body {
background-color: #ccc;
}
/* the maximum dimensions keep Ganja from blowing up the canvas */
#view {
display: block;
width: 600px;
height: 600px;
margin-top: 10px;
margin-left: 10px;
border-radius: 10px;
background-color: #f0f0f0;
}
#control-panel {
width: 600px;
height: 200px;
box-sizing: border-box;
padding: 5px 10px 5px 10px;
margin-top: 10px;
margin-left: 10px;
overflow-y: scroll;
border-radius: 10px;
background-color: #f0f0f0;
}
#control-panel > div {
margin-top: 5px;
padding: 4px;
border-radius: 5px;
border: solid;
font-family: monospace;
}
""")
# load Ganja.js. for an automatically updated web-hosted version, load from
#
# https://unpkg.com/ganja.js
#
# instead
loadjs!(win, "http://localhost:8000/ganja-1.0.204.js")
# create global functions and variables
script!(win, """
// create algebra
var CGA3 = Algebra(4, 1);
// initialize element list and palette
var elements = [];
var palette = [];
// declare handles for the view and its options
var view;
var viewOpt;
// declare handles for the controls
var controlPanel;
var visToggles;
// create scene function
function scene() {
commands = [];
for (let n = 0; n < elements.length; ++n) {
if (visToggles[n].checked) {
commands.push(palette[n], elements[n]);
}
}
return commands;
}
function updateView() {
requestAnimationFrame(view.update.bind(view, scene));
}
""")
@js win begin
# create view
viewOpt = Dict(
:conformal => true,
:gl => true,
:devicePixelRatio => window.devicePixelRatio
)
view = CGA3.graph(scene, viewOpt)
view.setAttribute(:id, "view")
view.removeAttribute(:style)
document.body.replaceChildren(view)
# create control panel
controlPanel = document.createElement(:div)
controlPanel.setAttribute(:id, "control-panel")
document.body.appendChild(controlPanel)
end
new(win)
end
end
mprod(v, w) =
v[1]*w[1] + v[2]*w[2] + v[3]*w[3] + v[4]*w[4] - v[5]*w[5]
function display!(viewer::ConstructionViewer, elements::Matrix)
# load elements
elements_full = []
for elt in eachcol(Engine.unmix * elements)
if mprod(elt, elt) < 0.5
elt_full = [0; elt; fill(0, 26)]
else
# `elt` is a spacelike vector, representing a generalized sphere, so we
# take its Hodge dual before passing it to Ganja.js. the dual represents
# the same generalized sphere, but Ganja.js only displays planes when
# they're represented by vectors in grade 4 rather than grade 1
elt_full = [fill(0, 26); -elt[5]; -elt[4]; elt[3]; -elt[2]; elt[1]; 0]
end
push!(elements_full, elt_full)
end
@js viewer.win elements = $elements_full.map((elt) -> @new CGA3(elt))
# generate palette. this is Gadfly's `default_discrete_colors` palette,
# available under the MIT license
palette = distinguishable_colors(
length(elements_full),
[LCHab(70, 60, 240)],
transform = c -> deuteranopic(c, 0.5),
lchoices = Float64[65, 70, 75, 80],
cchoices = Float64[0, 50, 60, 70],
hchoices = range(0, stop=330, length=24)
)
palette_packed = [RGB24(c).color for c in palette]
@js viewer.win palette = $palette_packed
# create visibility toggles
@js viewer.win begin
controlPanel.replaceChildren()
visToggles = []
end
for (elt, c) in zip(eachcol(elements), palette)
vec_str = join(map(t -> @sprintf("%.3f", t), elt), ", ")
color_str = "#$(hex(c))"
style_str = "background-color: $color_str; border-color: $color_str;"
@js viewer.win begin
@var toggle = document.createElement(:div)
toggle.setAttribute(:style, $style_str)
toggle.checked = true
toggle.addEventListener(
"click",
() -> begin
toggle.checked = !toggle.checked
toggle.style.backgroundColor = toggle.checked ? $color_str : "inherit";
updateView()
end
)
toggle.appendChild(document.createTextNode($vec_str))
visToggles.push(toggle);
controlPanel.appendChild(toggle);
end
end
# update view
@js viewer.win updateView()
end
function opentools!(viewer::ConstructionViewer)
size(viewer.win, 1240, 830)
opentools(viewer.win)
end
function closetools!(viewer::ConstructionViewer)
closetools(viewer.win)
size(viewer.win, 620, 830)
end
end
# ~~~ sandbox setup ~~~
elements = let
a = sqrt(BigFloat(3)/2)
sqrt(0.5) * BigFloat[
1 1 -1 -1 0
1 -1 1 -1 0
1 -1 -1 1 0
0.5 0.5 0.5 0.5 1+a
0.5 0.5 0.5 0.5 1-a
]
end
# show construction
viewer = Viewer.ConstructionViewer()
Viewer.display!(viewer, elements)

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module Algebraic
export
codimension, dimension,
Construction, realize,
Element, Point, Sphere,
Relation, LiesOn, AlignsWithBy, mprod
import Subscripts
using LinearAlgebra
using AbstractAlgebra
using Groebner
using ...HittingSet
# --- commutative algebra ---
# as of version 0.36.6, AbstractAlgebra only supports ideals in multivariate
# polynomial rings when the coefficients are integers. we use Groebner to extend
# support to rationals and to finite fields of prime order
Generic.reduce_gens(I::Generic.Ideal{U}) where {T <: FieldElement, U <: MPolyRingElem{T}} =
Generic.Ideal{U}(base_ring(I), groebner(gens(I)))
function codimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}}
leading = [exponent_vector(f, 1) for f in gens(I)]
targets = [Set(findall(.!iszero.(exp_vec))) for exp_vec in leading]
length(HittingSet.solve(HittingSetProblem(targets), maxdepth))
end
dimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}} =
length(gens(base_ring(I))) - codimension(I, maxdepth)
# --- primitve elements ---
abstract type Element{T} end
mutable struct Point{T} <: Element{T}
coords::Vector{MPolyRingElem{T}}
vec::Union{Vector{MPolyRingElem{T}}, Nothing}
rel::Nothing
## [to do] constructor argument never needed?
Point{T}(
coords::Vector{MPolyRingElem{T}} = MPolyRingElem{T}[],
vec::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing
) where T = new(coords, vec, nothing)
end
function buildvec!(pt::Point)
coordring = parent(pt.coords[1])
pt.vec = [one(coordring), dot(pt.coords, pt.coords), pt.coords...]
end
mutable struct Sphere{T} <: Element{T}
coords::Vector{MPolyRingElem{T}}
vec::Union{Vector{MPolyRingElem{T}}, Nothing}
rel::Union{MPolyRingElem{T}, Nothing}
## [to do] constructor argument never needed?
Sphere{T}(
coords::Vector{MPolyRingElem{T}} = MPolyRingElem{T}[],
vec::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing,
rel::Union{MPolyRingElem{T}, Nothing} = nothing
) where T = new(coords, vec, rel)
end
function buildvec!(sph::Sphere)
coordring = parent(sph.coords[1])
sph.vec = sph.coords
sph.rel = mprod(sph.coords, sph.coords) + one(coordring)
end
const coordnames = IdDict{Symbol, Vector{Union{Symbol, Nothing}}}(
nameof(Point) => [nothing, nothing, :xₚ, :yₚ, :zₚ],
nameof(Sphere) => [:rₛ, :sₛ, :xₛ, :yₛ, :zₛ]
)
coordname(elt::Element, index) = coordnames[nameof(typeof(elt))][index]
function pushcoordname!(coordnamelist, indexed_elt::Tuple{Any, Element}, coordindex)
eltindex, elt = indexed_elt
name = coordname(elt, coordindex)
if !isnothing(name)
subscript = Subscripts.sub(string(eltindex))
push!(coordnamelist, Symbol(name, subscript))
end
end
function takecoord!(coordlist, indexed_elt::Tuple{Any, Element}, coordindex)
elt = indexed_elt[2]
if !isnothing(coordname(elt, coordindex))
push!(elt.coords, popfirst!(coordlist))
end
end
# --- primitive relations ---
abstract type Relation{T} end
mprod(v, w) = (v[1]*w[2] + w[1]*v[2]) / 2 - dot(v[3:end], w[3:end])
# elements: point, sphere
struct LiesOn{T} <: Relation{T}
elements::Vector{Element{T}}
LiesOn{T}(pt::Point{T}, sph::Sphere{T}) where T = new{T}([pt, sph])
end
equation(rel::LiesOn) = mprod(rel.elements[1].vec, rel.elements[2].vec)
# elements: sphere, sphere
struct AlignsWithBy{T} <: Relation{T}
elements::Vector{Element{T}}
cos_angle::T
AlignsWithBy{T}(sph1::Sphere{T}, sph2::Sphere{T}, cos_angle::T) where T = new{T}([sph1, sph2], cos_angle)
end
equation(rel::AlignsWithBy) = mprod(rel.elements[1].vec, rel.elements[2].vec) - rel.cos_angle
# --- constructions ---
mutable struct Construction{T}
points::Vector{Point{T}}
spheres::Vector{Sphere{T}}
relations::Vector{Relation{T}}
function Construction{T}(; elements = Vector{Element{T}}(), relations = Vector{Relation{T}}()) where T
allelements = union(elements, (rel.elements for rel in relations)...)
new{T}(
filter(elt -> isa(elt, Point), allelements),
filter(elt -> isa(elt, Sphere), allelements),
relations
)
end
end
function Base.push!(ctx::Construction{T}, elt::Point{T}) where T
push!(ctx.points, elt)
end
function Base.push!(ctx::Construction{T}, elt::Sphere{T}) where T
push!(ctx.spheres, elt)
end
function Base.push!(ctx::Construction{T}, rel::Relation{T}) where T
push!(ctx.relations, rel)
for elt in rel.elements
push!(ctx, elt)
end
end
function realize(ctx::Construction{T}) where T
# collect coordinate names
coordnamelist = Symbol[]
eltenum = enumerate(Iterators.flatten((ctx.spheres, ctx.points)))
for coordindex in 1:5
for indexed_elt in eltenum
pushcoordname!(coordnamelist, indexed_elt, coordindex)
end
end
# construct coordinate ring
coordring, coordqueue = polynomial_ring(parent_type(T)(), coordnamelist, ordering = :degrevlex)
# retrieve coordinates
for (_, elt) in eltenum
empty!(elt.coords)
end
for coordindex in 1:5
for indexed_elt in eltenum
takecoord!(coordqueue, indexed_elt, coordindex)
end
end
# construct coordinate vectors
for (_, elt) in eltenum
buildvec!(elt)
end
# turn relations into equations
eqns = vcat(
equation.(ctx.relations),
[elt.rel for (_, elt) in eltenum if !isnothing(elt.rel)]
)
# add relations to center, orient, and scale the construction
# [to do] the scaling constraint, as written, can be impossible to satisfy
# when all of the spheres have to go through the origin
if !isempty(ctx.points)
append!(eqns, [sum(pt.coords[k] for pt in ctx.points) for k in 1:3])
end
if !isempty(ctx.spheres)
append!(eqns, [sum(sph.coords[k] for sph in ctx.spheres) for k in 3:4])
end
n_elts = length(ctx.points) + length(ctx.spheres)
if n_elts > 0
push!(eqns, sum(elt.vec[2] for elt in Iterators.flatten((ctx.points, ctx.spheres))) - n_elts)
end
(Generic.Ideal(coordring, eqns), eqns)
end
end

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module Numerical
using Random: default_rng
using LinearAlgebra
using AbstractAlgebra
using HomotopyContinuation:
Variable, Expression, AbstractSystem, System, LinearSubspace,
nvariables, isreal, witness_set, results
import GLMakie
using ..Algebraic
# --- polynomial conversion ---
# hat tip Sascha Timme
# https://github.com/JuliaHomotopyContinuation/HomotopyContinuation.jl/issues/520#issuecomment-1317681521
function Base.convert(::Type{Expression}, f::MPolyRingElem)
variables = Variable.(symbols(parent(f)))
f_data = zip(coefficients(f), exponent_vectors(f))
sum(cf * prod(variables .^ exp_vec) for (cf, exp_vec) in f_data)
end
# create a ModelKit.System from an ideal in a multivariate polynomial ring. the
# variable ordering is taken from the polynomial ring
function System(I::Generic.Ideal)
eqns = Expression.(gens(I))
variables = Variable.(symbols(base_ring(I)))
System(eqns, variables = variables)
end
# --- sampling ---
function real_samples(F::AbstractSystem, dim; rng = default_rng())
# choose a random real hyperplane of codimension `dim` by intersecting
# hyperplanes whose normal vectors are uniformly distributed over the unit
# sphere
# [to do] guard against the unlikely event that one of the normals is zero
normals = transpose(hcat(
(normalize(randn(rng, nvariables(F))) for _ in 1:dim)...
))
cut = LinearSubspace(normals, fill(0., dim))
filter(isreal, results(witness_set(F, cut, seed = 0x1974abba)))
end
AbstractAlgebra.evaluate(pt::Point, vals::Vector{<:RingElement}) =
GLMakie.Point3f([evaluate(u, vals) for u in pt.coords])
function AbstractAlgebra.evaluate(sph::Sphere, vals::Vector{<:RingElement})
radius = 1 / evaluate(sph.coords[1], vals)
center = radius * [evaluate(u, vals) for u in sph.coords[3:end]]
GLMakie.Sphere(GLMakie.Point3f(center), radius)
end
end

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include("HittingSet.jl")
module Engine
include("Engine.Algebraic.jl")
include("Engine.Numerical.jl")
using .Algebraic
using .Numerical
export Construction, mprod, codimension, dimension
end
# ~~~ sandbox setup ~~~
using Random
using Distributions
using LinearAlgebra
using AbstractAlgebra
using HomotopyContinuation
using GLMakie
CoeffType = Rational{Int64}
spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
tangencies = [
Engine.AlignsWithBy{CoeffType}(
spheres[n],
spheres[mod1(n+1, length(spheres))],
CoeffType(1)
)
for n in 1:3
]
ctx_tan_sph = Engine.Construction{CoeffType}(elements = spheres, relations = tangencies)
ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
freedom = Engine.dimension(ideal_tan_sph)
println("Three mutually tangent spheres: $freedom degrees of freedom")
# --- test rational cut ---
coordring = base_ring(ideal_tan_sph)
vbls = Variable.(symbols(coordring))
# test a random witness set
system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
norm2 = vec -> real(dot(conj.(vec), vec))
rng = MersenneTwister(6071)
n_planes = 6
samples = []
for _ in 1:n_planes
real_solns = solution.(Engine.Numerical.real_samples(system, freedom, rng = rng))
for soln in real_solns
if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
push!(samples, soln)
end
end
end
println("Found $(length(samples)) sample solutions")
# show a sample solution
function show_solution(ctx, vals)
# evaluate elements
real_vals = real.(vals)
disp_points = [Engine.Numerical.evaluate(pt, real_vals) for pt in ctx.points]
disp_spheres = [Engine.Numerical.evaluate(sph, real_vals) for sph in ctx.spheres]
# create scene
scene = Scene()
cam3d!(scene)
scatter!(scene, disp_points, color = :green)
for sph in disp_spheres
mesh!(scene, sph, color = :gray)
end
scene
end

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module HittingSet
export HittingSetProblem, solve
HittingSetProblem{T} = Pair{Set{T}, Vector{Pair{T, Set{Set{T}}}}}
# `targets` should be a collection of Set objects
function HittingSetProblem(targets, chosen = Set())
wholeset = union(targets...)
T = eltype(wholeset)
unsorted_moves = [
elt => Set(filter(s -> elt s, targets))
for elt in wholeset
]
moves = sort(unsorted_moves, by = pair -> length(pair.second))
Set{T}(chosen) => moves
end
function Base.display(problem::HittingSetProblem{T}) where T
println("HittingSetProblem{$T}")
chosen = problem.first
println(" {", join(string.(chosen), ", "), "}")
moves = problem.second
for (choice, missed) in moves
println(" | ", choice)
for s in missed
println(" | | {", join(string.(s), ", "), "}")
end
end
println()
end
function solve(pblm::HittingSetProblem{T}, maxdepth = Inf) where T
problems = Dict(pblm)
while length(first(problems).first) < maxdepth
subproblems = typeof(problems)()
for (chosen, moves) in problems
if isempty(moves)
return chosen
else
for (choice, missed) in moves
to_be_chosen = union(chosen, Set([choice]))
if isempty(missed)
return to_be_chosen
elseif !haskey(subproblems, to_be_chosen)
push!(subproblems, HittingSetProblem(missed, to_be_chosen))
end
end
end
end
problems = subproblems
end
problems
end
function test(n = 1)
T = [Int64, Int64, Symbol, Symbol][n]
targets = Set{T}.([
[
[1, 3, 5],
[2, 3, 4],
[1, 4],
[2, 3, 4, 5],
[4, 5]
],
# example from Amit Chakrabarti's graduate-level algorithms class (CS 105)
# notes by Valika K. Wan and Khanh Do Ba, Winter 2005
# https://www.cs.dartmouth.edu/~ac/Teach/CS105-Winter05/
[
[1, 3], [1, 4], [1, 5],
[1, 3], [1, 2, 4], [1, 2, 5],
[4, 3], [ 2, 4], [ 2, 5],
[6, 3], [6, 4], [ 5]
],
[
[:w, :x, :y],
[:x, :y, :z],
[:w, :z],
[:x, :y]
],
# Wikipedia showcases this as an example of a problem where the greedy
# algorithm performs especially poorly
[
[:a, :x, :t1],
[:a, :y, :t2],
[:a, :y, :t3],
[:a, :z, :t4],
[:a, :z, :t5],
[:a, :z, :t6],
[:a, :z, :t7],
[:b, :x, :t8],
[:b, :y, :t9],
[:b, :y, :t10],
[:b, :z, :t11],
[:b, :z, :t12],
[:b, :z, :t13],
[:b, :z, :t14]
]
][n])
problem = HittingSetProblem(targets)
if isa(problem, HittingSetProblem{T})
println("Correct type")
else
println("Wrong type: ", typeof(problem))
end
problem
end
end

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<!DOCTYPE html>
<html>
<head>
<style>
body {
background-color: #ffe0f0;
}
/* needed to keep Ganja canvas from blowing up */
canvas {
min-width: 600px;
max-width: 600px;
min-height: 600px;
max-height: 600px;
}
</style>
<script src="https://unpkg.com/ganja.js"></script>
</head>
<body>
<p><button onclick="flip()">Flip</button></p>
<script>
// in the default view, e4 + e5 is the point at infinity
let CGA3 = Algebra(4, 1);
let elements = [
CGA3.inline(() => Math.sqrt(0.5)*( 1e1 + 1e2 + 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*( 1e1 - 1e2 - 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*(-1e1 + 1e2 - 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*(-1e1 - 1e2 + 1e3 + 1e5))(),
CGA3.inline(() => -Math.sqrt(3)*1e4 + Math.sqrt(2)*1e5)()
];
/*
these blocks of commented-out code can be used to confirm that a spacelike
vector and its Hodge dual represent the same generalized sphere
*/
/*let elements = [
CGA3.inline(() => Math.sqrt(0.5)*!( 1e1 + 1e2 + 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*!( 1e1 - 1e2 - 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*!(-1e1 + 1e2 - 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*!(-1e1 - 1e2 + 1e3 + 1e5))(),
CGA3.inline(() => !(-Math.sqrt(3)*1e4 + Math.sqrt(2)*1e5))()
];*/
/*let elements = [
CGA3.inline(() => 1e1 + 1e5)(),
CGA3.inline(() => 1e2 + 1e5)(),
CGA3.inline(() => 1e3 + 1e5)(),
CGA3.inline(() => -1e4 + 1e5)(),
CGA3.inline(() => Math.sqrt(0.5)*(1e1 + 1e2 + 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*!(1e1 + 1e2 + 1e3 - 0.01e4 + 1e5))()
];*/
// set up palette
var colorIndex;
var palette = [0xff00b0, 0x00ffb0, 0x00b0ff, 0x8040ff, 0xc0c0c0];
function nextColor() {
colorIndex = (colorIndex + 1) % palette.length;
return palette[colorIndex];
}
function resetColorCycle() {
colorIndex = palette.length - 1;
}
resetColorCycle();
// create scene function
function scene() {
commands = [];
resetColorCycle();
elements.forEach((elt) => commands.push(nextColor(), elt));
return commands;
}
// initialize graph
let graph = CGA3.graph(
scene,
{
conformal: true, gl: true, grid: true
}
)
document.body.appendChild(graph);
function flip() {
let last = elements.length - 1;
for (let n = 0; n < last; ++n) {
// reflect
elements[n] = CGA3.Mul(CGA3.Mul(elements[last], elements[n]), elements[last]);
// de-noise
for (let k = 6; k < elements[n].length; ++k) {
/*for (let k = 0; k < 26; ++k) {*/
elements[n][k] = 0;
}
}
requestAnimationFrame(graph.update.bind(graph, scene));
}
</script>
</body>
</html>

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@ -0,0 +1,127 @@
using Blink
using Colors
# === utilities ===
append_to_head!(w, type, content) = @js w begin
@var element = document.createElement($type)
element.appendChild(document.createTextNode($content))
document.head.appendChild(element)
end
style!(w, stylesheet) = append_to_head!(w, "style", stylesheet)
script!(w, code) = append_to_head!(w, "script", code)
function add_element!(vec)
# add element
full_vec = [0; vec; fill(0, 26)]
n = @js win elements.push(@new CGA3($full_vec))
# generate palette. this is Gadfly's `default_discrete_colors` palette,
# available under the MIT license
palette = distinguishable_colors(
n,
[LCHab(70, 60, 240)],
transform = c -> deuteranopic(c, 0.5),
lchoices = Float64[65, 70, 75, 80],
cchoices = Float64[0, 50, 60, 70],
hchoices = range(0, stop=330, length=24)
)
palette_packed = [RGB24(c).color for c in palette]
@js win palette = $palette_packed
end
# === build page ===
# create window and open developer console
win = Window()
opentools(win)
# set stylesheet
style!(win, """
body {
background-color: #ffe0f0;
}
/* needed to keep Ganja canvas from blowing up */
canvas {
min-width: 600px;
max-width: 600px;
min-height: 600px;
max-height: 600px;
}
""")
# load Ganja.js
loadjs!(win, "https://unpkg.com/ganja.js")
# create global functions and variables
script!(win, """
// create algebra
var CGA3 = Algebra(4, 1);
// initialize element list and palette
var elements = [];
var palette = [];
// declare visualization handle
var graph;
// create scene function
function scene() {
commands = [];
for (let n = 0; n < elements.length; ++n) {
commands.push(palette[n], elements[n]);
}
return commands;
}
function flip() {
let last = elements.length - 1;
for (let n = 0; n < last; ++n) {
// reflect
elements[n] = CGA3.Mul(CGA3.Mul(elements[last], elements[n]), elements[last]);
// de-noise
for (let k = 6; k < elements[n].length; ++k) {
elements[n][k] = 0;
}
}
requestAnimationFrame(graph.update.bind(graph, scene));
}
""")
# set up controls
body!(win, """
<p><button id="flip-button" onclick="flip()">Flip</button></p>
""", async = false)
# === set up visualization ===
# list elements. in the default view, e4 + e5 is the point at infinity
elements = sqrt(0.5) * BigFloat[
1 1 -1 -1 0;
1 -1 1 -1 0;
1 -1 -1 1 0;
0 0 0 0 -sqrt(6);
1 1 1 1 2
]
# load elements
for vec in eachcol(elements)
add_element!(vec)
end
# initialize visualization
@js win begin
graph = CGA3.graph(
scene,
Dict(
"conformal" => true,
"gl" => true,
"grid" => true
)
)
document.body.appendChild(graph)
end

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@ -0,0 +1,450 @@
module Engine
using LinearAlgebra
using GenericLinearAlgebra
using SparseArrays
using Random
using Optim
export
rand_on_shell, Q, DescentHistory,
realize_gram_gradient, realize_gram_newton, realize_gram_optim, realize_gram
# === guessing ===
sconh(t, u) = 0.5*(exp(t) + u*exp(-t))
function rand_on_sphere(rng::AbstractRNG, ::Type{T}, n) where T
out = randn(rng, T, n)
tries_left = 2
while dot(out, out) < 1e-6 && tries_left > 0
out = randn(rng, T, n)
tries_left -= 1
end
normalize(out)
end
##[TO DO] write a test to confirm that the outputs are on the correct shells
function rand_on_shell(rng::AbstractRNG, shell::T) where T <: Number
space_part = rand_on_sphere(rng, T, 4)
rapidity = randn(rng, T)
sig = sign(shell)
nullmix * [sconh(rapidity, sig)*space_part; sconh(rapidity, -sig)]
end
rand_on_shell(rng::AbstractRNG, shells::Array{T}) where T <: Number =
hcat([rand_on_shell(rng, sh) for sh in shells]...)
rand_on_shell(shells::Array{<:Number}) = rand_on_shell(Random.default_rng(), shells)
# === elements ===
point(pos) = [pos; 0.5; 0.5 * dot(pos, pos)]
plane(normal, offset) = [-normal; 0; -offset]
function sphere(center, radius)
dist_sq = dot(center, center)
[
center / radius;
0.5 / radius;
0.5 * (dist_sq / radius - radius)
]
end
# === Gram matrix realization ===
# basis changes
nullmix = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [-1 1; 1 1]//2]
unmix = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [-1 1; 1 1]]
# the Lorentz form
Q = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [0 -2; -2 0]]
# project a matrix onto the subspace of matrices whose entries vanish away from
# the given indices
function proj_to_entries(mat, indices)
result = zeros(size(mat))
for (j, k) in indices
result[j, k] = mat[j, k]
end
result
end
# the difference between the matrices `target` and `attempt`, projected onto the
# subspace of matrices whose entries vanish at each empty index of `target`
function proj_diff(target::SparseMatrixCSC{T, <:Any}, attempt::Matrix{T}) where T
J, K, values = findnz(target)
result = zeros(size(target))
for (j, k, val) in zip(J, K, values)
result[j, k] = val - attempt[j, k]
end
result
end
# a type for keeping track of gradient descent history
struct DescentHistory{T}
scaled_loss::Array{T}
neg_grad::Array{Matrix{T}}
base_step::Array{Matrix{T}}
hess::Array{Hermitian{T, Matrix{T}}}
slope::Array{T}
stepsize::Array{T}
positive::Array{Bool}
backoff_steps::Array{Int64}
last_line_L::Array{Matrix{T}}
last_line_loss::Array{T}
function DescentHistory{T}(
scaled_loss = Array{T}(undef, 0),
neg_grad = Array{Matrix{T}}(undef, 0),
hess = Array{Hermitian{T, Matrix{T}}}(undef, 0),
base_step = Array{Matrix{T}}(undef, 0),
slope = Array{T}(undef, 0),
stepsize = Array{T}(undef, 0),
positive = Bool[],
backoff_steps = Int64[],
last_line_L = Array{Matrix{T}}(undef, 0),
last_line_loss = Array{T}(undef, 0)
) where T
new(scaled_loss, neg_grad, hess, base_step, slope, stepsize, positive, backoff_steps, last_line_L, last_line_loss)
end
end
# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every
# explicit entry of `gram`. use gradient descent starting from `guess`
function realize_gram_gradient(
gram::SparseMatrixCSC{T, <:Any},
guess::Matrix{T};
scaled_tol = 1e-30,
min_efficiency = 0.5,
init_stepsize = 1.0,
backoff = 0.9,
max_descent_steps = 600,
max_backoff_steps = 110
) where T <: Number
# start history
history = DescentHistory{T}()
# scale tolerance
scale_adjustment = sqrt(T(nnz(gram)))
tol = scale_adjustment * scaled_tol
# initialize variables
stepsize = init_stepsize
L = copy(guess)
# do gradient descent
Δ_proj = proj_diff(gram, L'*Q*L)
loss = dot(Δ_proj, Δ_proj)
for _ in 1:max_descent_steps
# stop if the loss is tolerably low
if loss < tol
break
end
# find negative gradient of loss function
neg_grad = 4*Q*L*Δ_proj
slope = norm(neg_grad)
dir = neg_grad / slope
# store current position, loss, and slope
L_last = L
loss_last = loss
push!(history.scaled_loss, loss / scale_adjustment)
push!(history.neg_grad, neg_grad)
push!(history.slope, slope)
# find a good step size using backtracking line search
push!(history.stepsize, 0)
push!(history.backoff_steps, max_backoff_steps)
empty!(history.last_line_L)
empty!(history.last_line_loss)
for backoff_steps in 0:max_backoff_steps
history.stepsize[end] = stepsize
L = L_last + stepsize * dir
Δ_proj = proj_diff(gram, L'*Q*L)
loss = dot(Δ_proj, Δ_proj)
improvement = loss_last - loss
push!(history.last_line_L, L)
push!(history.last_line_loss, loss / scale_adjustment)
if improvement >= min_efficiency * stepsize * slope
history.backoff_steps[end] = backoff_steps
break
end
stepsize *= backoff
end
# [DEBUG] if we've hit a wall, quit
if history.backoff_steps[end] == max_backoff_steps
break
end
end
# return the factorization and its history
push!(history.scaled_loss, loss / scale_adjustment)
L, history
end
function basis_matrix(::Type{T}, j, k, dims) where T
result = zeros(T, dims)
result[j, k] = one(T)
result
end
# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every
# explicit entry of `gram`. use Newton's method starting from `guess`
function realize_gram_newton(
gram::SparseMatrixCSC{T, <:Any},
guess::Matrix{T};
scaled_tol = 1e-30,
rate = 1,
max_steps = 100
) where T <: Number
# start history
history = DescentHistory{T}()
# find the dimension of the search space
dims = size(guess)
element_dim, construction_dim = dims
total_dim = element_dim * construction_dim
# list the constrained entries of the gram matrix
J, K, _ = findnz(gram)
constrained = zip(J, K)
# scale the tolerance
scale_adjustment = sqrt(T(length(constrained)))
tol = scale_adjustment * scaled_tol
# use Newton's method
L = copy(guess)
for step in 0:max_steps
# evaluate the loss function
Δ_proj = proj_diff(gram, L'*Q*L)
loss = dot(Δ_proj, Δ_proj)
# store the current loss
push!(history.scaled_loss, loss / scale_adjustment)
# stop if the loss is tolerably low
if loss < tol || step > max_steps
break
end
# find the negative gradient of loss function
neg_grad = 4*Q*L*Δ_proj
# find the negative Hessian of the loss function
hess = Matrix{T}(undef, total_dim, total_dim)
indices = [(j, k) for k in 1:construction_dim for j in 1:element_dim]
for (j, k) in indices
basis_mat = basis_matrix(T, j, k, dims)
neg_dΔ = basis_mat'*Q*L + L'*Q*basis_mat
neg_dΔ_proj = proj_to_entries(neg_dΔ, constrained)
deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj)
hess[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim)
end
hess = Hermitian(hess)
push!(history.hess, hess)
# compute the Newton step
step = hess \ reshape(neg_grad, total_dim)
L += rate * reshape(step, dims)
end
# return the factorization and its history
L, history
end
LinearAlgebra.eigen!(A::Symmetric{BigFloat, Matrix{BigFloat}}; sortby::Nothing) =
eigen!(Hermitian(A))
function convertnz(type, mat)
J, K, values = findnz(mat)
sparse(J, K, type.(values))
end
function realize_gram_optim(
gram::SparseMatrixCSC{T, <:Any},
guess::Matrix{T}
) where T <: Number
# find the dimension of the search space
dims = size(guess)
element_dim, construction_dim = dims
total_dim = element_dim * construction_dim
# list the constrained entries of the gram matrix
J, K, _ = findnz(gram)
constrained = zip(J, K)
# scale the loss function
scale_adjustment = length(constrained)
function loss(L_vec)
L = reshape(L_vec, dims)
Δ_proj = proj_diff(gram, L'*Q*L)
dot(Δ_proj, Δ_proj) / scale_adjustment
end
function loss_grad!(storage, L_vec)
L = reshape(L_vec, dims)
Δ_proj = proj_diff(gram, L'*Q*L)
storage .= reshape(-4*Q*L*Δ_proj, total_dim) / scale_adjustment
end
function loss_hess!(storage, L_vec)
L = reshape(L_vec, dims)
Δ_proj = proj_diff(gram, L'*Q*L)
indices = [(j, k) for k in 1:construction_dim for j in 1:element_dim]
for (j, k) in indices
basis_mat = basis_matrix(T, j, k, dims)
neg_dΔ = basis_mat'*Q*L + L'*Q*basis_mat
neg_dΔ_proj = proj_to_entries(neg_dΔ, constrained)
deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj) / scale_adjustment
storage[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim)
end
end
optimize(
loss, loss_grad!, loss_hess!,
reshape(guess, total_dim),
Newton()
)
end
# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every
# explicit entry of `gram`. use gradient descent starting from `guess`
function realize_gram(
gram::SparseMatrixCSC{T, <:Any},
guess::Matrix{T},
frozen = nothing;
scaled_tol = 1e-30,
min_efficiency = 0.5,
init_rate = 1.0,
backoff = 0.9,
reg_scale = 1.1,
max_descent_steps = 200,
max_backoff_steps = 110
) where T <: Number
# start history
history = DescentHistory{T}()
# find the dimension of the search space
dims = size(guess)
element_dim, construction_dim = dims
total_dim = element_dim * construction_dim
# list the constrained entries of the gram matrix
J, K, _ = findnz(gram)
constrained = zip(J, K)
# scale the tolerance
scale_adjustment = sqrt(T(length(constrained)))
tol = scale_adjustment * scaled_tol
# list the un-frozen indices
has_frozen = !isnothing(frozen)
if has_frozen
is_unfrozen = fill(true, size(guess))
is_unfrozen[frozen] .= false
unfrozen = findall(is_unfrozen)
unfrozen_stacked = reshape(is_unfrozen, total_dim)
end
# initialize variables
grad_rate = init_rate
L = copy(guess)
# use Newton's method with backtracking and gradient descent backup
Δ_proj = proj_diff(gram, L'*Q*L)
loss = dot(Δ_proj, Δ_proj)
for step in 1:max_descent_steps
# stop if the loss is tolerably low
if loss < tol
break
end
# find the negative gradient of loss function
neg_grad = 4*Q*L*Δ_proj
# find the negative Hessian of the loss function
hess = Matrix{T}(undef, total_dim, total_dim)
indices = [(j, k) for k in 1:construction_dim for j in 1:element_dim]
for (j, k) in indices
basis_mat = basis_matrix(T, j, k, dims)
neg_dΔ = basis_mat'*Q*L + L'*Q*basis_mat
neg_dΔ_proj = proj_to_entries(neg_dΔ, constrained)
deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj)
hess[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim)
end
hess = Hermitian(hess)
push!(history.hess, hess)
# regularize the Hessian
min_eigval = minimum(eigvals(hess))
push!(history.positive, min_eigval > 0)
if min_eigval <= 0
hess -= reg_scale * min_eigval * I
end
# compute the Newton step
neg_grad_stacked = reshape(neg_grad, total_dim)
if has_frozen
hess = hess[unfrozen_stacked, unfrozen_stacked]
neg_grad_compressed = neg_grad_stacked[unfrozen_stacked]
else
neg_grad_compressed = neg_grad_stacked
end
base_step_compressed = hess \ neg_grad_compressed
if has_frozen
base_step_stacked = zeros(total_dim)
base_step_stacked[unfrozen_stacked] .= base_step_compressed
else
base_step_stacked = base_step_compressed
end
base_step = reshape(base_step_stacked, dims)
push!(history.base_step, base_step)
# store the current position, loss, and slope
L_last = L
loss_last = loss
push!(history.scaled_loss, loss / scale_adjustment)
push!(history.neg_grad, neg_grad)
push!(history.slope, norm(neg_grad))
# find a good step size using backtracking line search
push!(history.stepsize, 0)
push!(history.backoff_steps, max_backoff_steps)
empty!(history.last_line_L)
empty!(history.last_line_loss)
rate = one(T)
step_success = false
for backoff_steps in 0:max_backoff_steps
history.stepsize[end] = rate
L = L_last + rate * base_step
Δ_proj = proj_diff(gram, L'*Q*L)
loss = dot(Δ_proj, Δ_proj)
improvement = loss_last - loss
push!(history.last_line_L, L)
push!(history.last_line_loss, loss / scale_adjustment)
if improvement >= min_efficiency * rate * dot(neg_grad, base_step)
history.backoff_steps[end] = backoff_steps
step_success = true
break
end
rate *= backoff
end
# if we've hit a wall, quit
if !step_success
return L_last, false, history
end
end
# return the factorization and its history
push!(history.scaled_loss, loss / scale_adjustment)
L, loss < tol, history
end
end

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include("Engine.jl")
using LinearAlgebra
using SparseArrays
function sphere_in_tetrahedron_shape()
# initialize the partial gram matrix for a sphere inscribed in a regular
# tetrahedron
J = Int64[]
K = Int64[]
values = BigFloat[]
for j in 1:5
for k in 1:5
push!(J, j)
push!(K, k)
if j == k
push!(values, 1)
elseif (j <= 4 && k <= 4)
push!(values, -1/BigFloat(3))
else
push!(values, -1)
end
end
end
gram = sparse(J, K, values)
# plot loss along a slice
loss_lin = []
loss_sq = []
mesh = range(0.9, 1.1, 101)
for t in mesh
L = hcat(
Engine.plane(normalize(BigFloat[ 1, 1, 1]), BigFloat(1)),
Engine.plane(normalize(BigFloat[ 1, -1, -1]), BigFloat(1)),
Engine.plane(normalize(BigFloat[-1, 1, -1]), BigFloat(1)),
Engine.plane(normalize(BigFloat[-1, -1, 1]), BigFloat(1)),
Engine.sphere(BigFloat[0, 0, 0], BigFloat(t))
)
Δ_proj = Engine.proj_diff(gram, L'*Engine.Q*L)
push!(loss_lin, norm(Δ_proj))
push!(loss_sq, dot(Δ_proj, Δ_proj))
end
mesh, loss_lin, loss_sq
end
function circles_in_triangle_shape()
# initialize the partial gram matrix for a sphere inscribed in a regular
# tetrahedron
J = Int64[]
K = Int64[]
values = BigFloat[]
for j in 1:8
for k in 1:8
filled = false
if j == k
push!(values, 1)
filled = true
elseif (j == 1 || k == 1)
push!(values, 0)
filled = true
elseif (j == 2 || k == 2)
push!(values, -1)
filled = true
end
#=elseif (j <= 5 && j != 2 && k == 9 || k == 9 && k <= 5 && k != 2)
push!(values, 0)
filled = true
end=#
if filled
push!(J, j)
push!(K, k)
end
end
end
append!(J, [6, 4, 6, 5, 7, 5, 7, 3, 8, 3, 8, 4])
append!(K, [4, 6, 5, 6, 5, 7, 3, 7, 3, 8, 4, 8])
append!(values, fill(-1, 12))
# plot loss along a slice
loss_lin = []
loss_sq = []
mesh = range(0.99, 1.01, 101)
for t in mesh
L = hcat(
Engine.plane(BigFloat[0, 0, 1], BigFloat(0)),
Engine.sphere(BigFloat[0, 0, 0], BigFloat(t)),
Engine.plane(BigFloat[1, 0, 0], BigFloat(1)),
Engine.plane(BigFloat[cos(2pi/3), sin(2pi/3), 0], BigFloat(1)),
Engine.plane(BigFloat[cos(-2pi/3), sin(-2pi/3), 0], BigFloat(1)),
Engine.sphere(4//3*BigFloat[-1, 0, 0], BigFloat(1//3)),
Engine.sphere(4//3*BigFloat[cos(-pi/3), sin(-pi/3), 0], BigFloat(1//3)),
Engine.sphere(4//3*BigFloat[cos(pi/3), sin(pi/3), 0], BigFloat(1//3))
)
Δ_proj = Engine.proj_diff(gram, L'*Engine.Q*L)
push!(loss_lin, norm(Δ_proj))
push!(loss_sq, dot(Δ_proj, Δ_proj))
end
mesh, loss_lin, loss_sq
end

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include("Engine.jl")
using SparseArrays
using Random
# initialize the partial gram matrix for a sphere inscribed in a regular
# tetrahedron
J = Int64[]
K = Int64[]
values = BigFloat[]
for j in 1:9
for k in 1:9
filled = false
if j == 9
if k <= 5 && k != 2
push!(values, 0)
filled = true
end
elseif k == 9
if j <= 5 && j != 2
push!(values, 0)
filled = true
end
elseif j == k
push!(values, 1)
filled = true
elseif j == 1 || k == 1
push!(values, 0)
filled = true
elseif j == 2 || k == 2