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ENH: Remove template Group<> and Cosets<>

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This commit is contained in:
David Allemang
2023-02-06 09:42:47 -05:00
parent cb758166fa
commit 84a47db73b
11 changed files with 73 additions and 188 deletions
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135
tc/include/tc/core.hpp
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@@ -11,18 +11,9 @@ namespace tc {
using Mult = std::uint16_t; using Mult = std::uint16_t;
constexpr Mult FREE = 0; constexpr Mult FREE = 0;
/**
* @brief Mapping from "global" generator names or objects to indexes used for value lookup.
* @tparam Gen_
*/
template<typename Gen_=void>
struct Index;
/** /**
* @brief Complete representation of a quotient group. Describes the action of each generator on each coset. * @brief Complete representation of a quotient group. Describes the action of each generator on each coset.
* @tparam Gen_
*/ */
template<typename Gen_=void>
struct Cosets; struct Cosets;
/** /**
@@ -42,7 +33,6 @@ namespace tc {
* @see * @see
* <a href="https://en.wikipedia.org/wiki/Coxeter_group#Definition">Coxeter Group (Wikipedia)</a> * <a href="https://en.wikipedia.org/wiki/Coxeter_group#Definition">Coxeter Group (Wikipedia)</a>
*/ */
template<typename Gen_=void>
struct Group; struct Group;
/** /**
@@ -52,33 +42,7 @@ namespace tc {
template<typename Gen_=void> template<typename Gen_=void>
struct Path; struct Path;
template<> struct Cosets {
struct Index<> {
size_t operator()(size_t const &idx) const {
return idx;
}
};
template<typename Gen_>
struct Index {
using Gen = Gen_;
std::vector<Gen> _gens{};
explicit Index(std::vector<Gen> gens) : _gens(gens) {}
template<typename It>
Index(It begin, It end) : _gens(begin, end) {}
size_t operator()(Gen const &gen) const {
auto it = std::find(_gens.begin(), _gens.end(), gen);
assert(it != _gens.end());
return it - _gens.begin();
}
};
template<>
struct Cosets<> {
static constexpr size_t UNSET = std::numeric_limits<size_t>::max(); static constexpr size_t UNSET = std::numeric_limits<size_t>::max();
private: private:
@@ -108,7 +72,7 @@ namespace tc {
[[nodiscard]] size_t size() const; [[nodiscard]] size_t size() const;
friend Group<>; // only constructible via Group<>::solve friend Group; // only constructible via Group<>::solve
private: private:
explicit Cosets(size_t rank); explicit Cosets(size_t rank);
@@ -122,8 +86,7 @@ namespace tc {
[[nodiscard]] bool isset(size_t idx) const; [[nodiscard]] bool isset(size_t idx) const;
}; };
template<> struct Group {
struct Group<> {
using Rel = std::tuple<size_t, size_t, Mult>; using Rel = std::tuple<size_t, size_t, Mult>;
private: private:
@@ -147,7 +110,7 @@ namespace tc {
[[nodiscard]] Group sub(std::vector<size_t> const &idxs) const; [[nodiscard]] Group sub(std::vector<size_t> const &idxs) const;
[[nodiscard]] Cosets<> solve(std::vector<size_t> const &idxs = {}, size_t bound = SIZE_MAX) const; [[nodiscard]] Cosets solve(std::vector<size_t> const &idxs = {}, size_t bound = SIZE_MAX) const;
}; };
template<> template<>
@@ -160,7 +123,7 @@ namespace tc {
public: public:
// todo be smarter about move semantics // todo be smarter about move semantics
explicit Path(Cosets<> const &cosets) : _data() { explicit Path(Cosets const &cosets) : _data() {
_data.resize(cosets.order()); _data.resize(cosets.order());
std::vector<bool> complete(cosets.order(), false); std::vector<bool> complete(cosets.order(), false);
@@ -206,101 +169,21 @@ namespace tc {
} }
}; };
template<typename Gen_>
struct Cosets : public Cosets<> {
using Gen = Gen_;
private:
Index<Gen> _index;
friend Path<Gen>;
public:
Cosets(Cosets<> g, std::vector<Gen> gens) : Cosets<>(g), _index(gens) {}
void set(size_t coset, Gen const &gen, size_t target) {
Cosets<>::set(coset, _index(gen), target);
}
[[nodiscard]] size_t get(size_t coset, Gen const &gen) const {
return Cosets<>::get(coset, _index(gen));
}
[[nodiscard]] bool isset(size_t coset, Gen const &gen) const {
return Cosets<>::isset(coset, _index(gen));
}
[[nodiscard]] std::vector<Gen> gens() const {
return _index._gens;
}
private:
Cosets(size_t rank, std::vector<Gen> gens) : Cosets<>(rank), _index(gens) {}
};
template<typename Gen_>
struct Group : public Group<> {
using Gen = Gen_;
using Rel = std::tuple<Gen, Gen, Mult>;
private:
Index<Gen> _index;
public:
Group(Group const &) = default;
Group(Group &&) noexcept = default;
Group(Group<> g, std::vector<Gen> gens) : Group<>(g), _index(gens) {}
Group(size_t rank, std::vector<Gen> gens) : Group<>(rank), _index(gens) {}
~Group() = default;
void set(Gen const &u, Gen const &v, Mult m) {
Group<>::set(_index(u), _index(v), m);
}
[[nodiscard]] Mult get(Gen const &u, Gen const &v) const {
return Group<>::get(_index(u), _index(v));
}
[[nodiscard]] std::vector<Gen> gens() const {
return _index._gens;
}
[[nodiscard]] Group sub(std::vector<Gen> const &gens) const {
std::vector<size_t> idxs(gens.size());
std::transform(gens.begin(), gens.end(), idxs.begin(), _index);
return Group(Group<>::sub(idxs), gens);
}
[[nodiscard]] Cosets<Gen> solve(std::vector<Gen> const &gens = {}, size_t bound = SIZE_MAX) const {
std::vector<size_t> idxs(gens.size());
std::transform(gens.begin(), gens.end(), idxs.begin(), _index);
return Cosets<Gen>(Group<>::solve(idxs, bound), _index._gens);
}
};
template<typename Gen_> template<typename Gen_>
struct Path : public Path<> { struct Path : public Path<> {
using Gen = Gen_; using Gen = Gen_;
private: private:
Index<Gen> _index; std::vector<Gen> _index;
public: public:
// todo be smarter about move semantics
explicit Path(Cosets<Gen> const &cosets) : Path<>(cosets), _index(cosets._index) {}
template<typename T> template<typename T>
Path(Cosets<> const &cosets, T const &gens) Path(Cosets const &cosets, T &&gens)
: Path<>(cosets), _index(gens.begin(), gens.end()) {} : Path<>(cosets), _index(std::forward<T>(gens).begin(), std::forward<T>(gens).end()) {}
template<typename Elem, typename BinaryOp, std::random_access_iterator It> template<typename Elem, typename BinaryOp, std::random_access_iterator It>
void walk(Elem const &start, BinaryOp op, It out) { void walk(Elem const &start, BinaryOp op, It out) {
Path<>::walk(start, op, out, _index._gens.begin()); Path<>::walk(start, op, out, _index.begin());
} }
}; };
} }

8
tc/include/tc/groups.hpp
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@@ -10,14 +10,14 @@ namespace tc {
* Construct a group from a (simplified) Schlafli Symbol of the form [a, b, ..., c] * Construct a group from a (simplified) Schlafli Symbol of the form [a, b, ..., c]
* @param mults: The sequence of multiplicites between adjacent generators. * @param mults: The sequence of multiplicites between adjacent generators.
*/ */
Group<> schlafli(const std::vector<unsigned int> &mults); Group schlafli(const std::vector<unsigned int> &mults);
Group<> coxeter(const std::string &symbol); Group coxeter(const std::string &symbol);
Group<> vcoxeter(const std::string &symbol, const std::vector<unsigned int> &values); Group vcoxeter(const std::string &symbol, const std::vector<unsigned int> &values);
template<typename ...Args> template<typename ...Args>
Group<> coxeter(const std::string &symbol, const Args &... args) { Group coxeter(const std::string &symbol, const Args &... args) {
std::vector<unsigned int> values = {{args...}}; std::vector<unsigned int> values = {{args...}};
return vcoxeter(symbol, values); return vcoxeter(symbol, values);
} }

24
tc/src/cosets.cpp
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@@ -1,54 +1,54 @@
#include <tc/core.hpp> #include <tc/core.hpp>
namespace tc { namespace tc {
Cosets<>::Cosets(size_t rank) Cosets::Cosets(size_t rank)
: _rank(rank), _order(0), _complete(false), _data() {} : _rank(rank), _order(0), _complete(false), _data() {}
void Cosets<>::set(size_t coset, size_t gen, size_t target) { void Cosets::set(size_t coset, size_t gen, size_t target) {
set(coset * rank() + gen, target); set(coset * rank() + gen, target);
} }
[[nodiscard]] size_t Cosets<>::get(size_t coset, size_t gen) const { [[nodiscard]] size_t Cosets::get(size_t coset, size_t gen) const {
return get(coset * rank() + gen); return get(coset * rank() + gen);
} }
[[nodiscard]] bool Cosets<>::isset(size_t coset, size_t gen) const { [[nodiscard]] bool Cosets::isset(size_t coset, size_t gen) const {
return isset(coset * rank() + gen); return isset(coset * rank() + gen);
} }
[[nodiscard]] size_t Cosets<>::rank() const { [[nodiscard]] size_t Cosets::rank() const {
return _rank; return _rank;
} }
[[nodiscard]] size_t Cosets<>::order() const { [[nodiscard]] size_t Cosets::order() const {
return _order; return _order;
} }
[[nodiscard]] bool Cosets<>::complete() const { [[nodiscard]] bool Cosets::complete() const {
return _complete; return _complete;
} }
[[nodiscard]] size_t Cosets<>::size() const { [[nodiscard]] size_t Cosets::size() const {
return _data.size(); return _data.size();
} }
void Cosets<>::add_row() { void Cosets::add_row() {
_data.resize(_data.size() + rank(), UNSET); _data.resize(_data.size() + rank(), UNSET);
_order++; _order++;
} }
void Cosets<>::set(size_t idx, size_t target) { void Cosets::set(size_t idx, size_t target) {
size_t coset = idx / rank(); size_t coset = idx / rank();
size_t gen = idx % rank(); size_t gen = idx % rank();
_data[idx] = target; _data[idx] = target;
_data[target * rank() + gen] = coset; _data[target * rank() + gen] = coset;
} }
[[nodiscard]] size_t Cosets<>::get(size_t idx) const { [[nodiscard]] size_t Cosets::get(size_t idx) const {
return _data[idx]; return _data[idx];
} }
[[nodiscard]] bool Cosets<>::isset(size_t idx) const { [[nodiscard]] bool Cosets::isset(size_t idx) const {
return get(idx) != UNSET; return get(idx) != UNSET;
} }

12
tc/src/group.cpp
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@@ -3,13 +3,13 @@
#include <cassert> #include <cassert>
namespace tc { namespace tc {
Group<>::Group(size_t rank) : _rank(rank), _mults(_rank * _rank, 2) { Group::Group(size_t rank) : _rank(rank), _mults(_rank * _rank, 2) {
for (int idx = 0; idx < rank; ++idx) { for (int idx = 0; idx < rank; ++idx) {
set(idx, idx, 1); set(idx, idx, 1);
} }
} }
void Group<>::set(size_t u, size_t v, Mult m) { void Group::set(size_t u, size_t v, Mult m) {
assert(u < rank()); assert(u < rank());
assert(v < rank()); assert(v < rank());
@@ -17,19 +17,19 @@ namespace tc {
_mults[v * rank() + u] = m; _mults[v * rank() + u] = m;
} }
[[nodiscard]] Mult Group<>::get(size_t u, size_t v) const { [[nodiscard]] Mult Group::get(size_t u, size_t v) const {
assert(u < rank()); assert(u < rank());
assert(v < rank()); assert(v < rank());
return _mults[u * rank() + v]; return _mults[u * rank() + v];
} }
[[nodiscard]] size_t Group<>::rank() const { [[nodiscard]] size_t Group::rank() const {
return _rank; return _rank;
} }
[[nodiscard]] Group<> Group<>::sub(std::vector<size_t> const &idxs) const { [[nodiscard]] Group Group::sub(std::vector<size_t> const &idxs) const {
Group<> res(idxs.size()); Group res(idxs.size());
for (int i = 0; i < idxs.size(); ++i) { for (int i = 0; i < idxs.size(); ++i) {
for (int j = i; j < idxs.size(); ++j) { for (int j = i; j < idxs.size(); ++j) {

6
tc/src/groups.cpp
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@@ -5,15 +5,15 @@
#include <numeric> #include <numeric>
namespace tc { namespace tc {
Group<> schlafli(const std::vector<unsigned int> &mults) { Group schlafli(const std::vector<unsigned int> &mults) {
Group<> res(mults.size() + 1); Group res(mults.size() + 1);
for (size_t i = 0; i < mults.size(); ++i) { for (size_t i = 0; i < mults.size(); ++i) {
res.set(i, i + 1, mults[i]); res.set(i, i + 1, mults[i]);
} }
return res; return res;
} }
Group<> vcoxeter(const std::string &symbol, const std::vector<unsigned int> &values) { Group vcoxeter(const std::string &symbol, const std::vector<unsigned int> &values) {
fmt::dynamic_format_arg_store<fmt::format_context> ds; fmt::dynamic_format_arg_store<fmt::format_context> ds;
for (const auto &value: values) { for (const auto &value: values) {

6
tc/src/lang.cpp
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@@ -14,7 +14,7 @@
struct Graph { struct Graph {
size_t rank{}; size_t rank{};
std::vector<tc::Group<>::Rel> edges{}; std::vector<tc::Group::Rel> edges{};
}; };
struct Op { struct Op {
@@ -287,10 +287,10 @@ Graph eval(const std::vector<Op> &ops) {
} }
namespace tc { namespace tc {
Group<> coxeter(const std::string &symbol) { Group coxeter(const std::string &symbol) {
auto ops = compile(symbol); auto ops = compile(symbol);
auto diagram = eval(ops); auto diagram = eval(ops);
Group<> res(diagram.rank); Group res(diagram.rank);
for (const auto &[i, j, m]: diagram.edges) { for (const auto &[i, j, m]: diagram.edges) {
res.set(i, j, m); res.set(i, j, m);
} }

10
tc/src/solve.cpp
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@@ -20,10 +20,10 @@ namespace tc {
}; };
struct Tables { struct Tables {
std::vector<Group<>::Rel> rels; std::vector<Group::Rel> rels;
std::vector<std::vector<Row>> rows; std::vector<std::vector<Row>> rows;
explicit Tables(std::vector<Group<>::Rel> rels) explicit Tables(std::vector<Group::Rel> rels)
: rels(std::move(rels)), rows() { : rels(std::move(rels)), rows() {
} }
@@ -36,9 +36,9 @@ namespace tc {
} }
}; };
[[nodiscard]] Cosets<> Group<>::solve(std::vector<size_t> const &idxs, size_t bound) const { [[nodiscard]] Cosets Group::solve(std::vector<size_t> const &idxs, size_t bound) const {
// region Initialize Cosets Table // region Initialize Cosets Table
Cosets<> cosets(rank()); Cosets cosets(rank());
cosets.add_row(); cosets.add_row();
if (rank() == 0) { if (rank() == 0) {
@@ -53,7 +53,7 @@ namespace tc {
// endregion // endregion
// region Initialize Relation Tables // region Initialize Relation Tables
std::vector<Group<>::Rel> rels; std::vector<Group::Rel> rels;
for (int i = 0; i < rank(); ++i) { for (int i = 0; i < rank(); ++i) {
for (int j = i + 1; j < rank(); ++j) { for (int j = i + 1; j < rank(); ++j) {
// The algorithm only works for Coxeter groups; multiplicities m_ii=1 are assumed. Relation tables // The algorithm only works for Coxeter groups; multiplicities m_ii=1 are assumed. Relation tables

22
tc/test/test_solve.cpp
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@@ -11,7 +11,7 @@ testing::AssertionResult AssertSolveOrder(
const char *group_expr, const char *group_expr,
const char *sub_gens_expr, const char *sub_gens_expr,
const char *expected_order_expr, const char *expected_order_expr,
const tc::Group<> &group, const tc::Group &group,
const std::vector<size_t> &sub_gens, const std::vector<size_t> &sub_gens,
size_t expected_order size_t expected_order
) { ) {
@@ -50,43 +50,43 @@ testing::AssertionResult AssertSolveOrder(
using v = std::vector<size_t>; using v = std::vector<size_t>;
tc::Group<> A(unsigned int n) { tc::Group A(unsigned int n) {
return tc::vcoxeter("3 * {}", {n - 1}); return tc::vcoxeter("3 * {}", {n - 1});
} }
tc::Group<> B(unsigned int n) { tc::Group B(unsigned int n) {
return tc::vcoxeter("4 3 * {}", {n - 2}); return tc::vcoxeter("4 3 * {}", {n - 2});
} }
tc::Group<> D(unsigned int n) { tc::Group D(unsigned int n) {
return tc::vcoxeter("3 * [1 1 {}]", {n - 3}); return tc::vcoxeter("3 * [1 1 {}]", {n - 3});
} }
tc::Group<> E(unsigned int n) { tc::Group E(unsigned int n) {
return tc::vcoxeter("3 * [1 2 {}]", {n - 4}); return tc::vcoxeter("3 * [1 2 {}]", {n - 4});
} }
tc::Group<> F4() { tc::Group F4() {
return tc::coxeter("3 4 3"); return tc::coxeter("3 4 3");
} }
tc::Group<> G2() { tc::Group G2() {
return tc::coxeter("6"); return tc::coxeter("6");
} }
tc::Group<> H(unsigned int n) { tc::Group H(unsigned int n) {
return tc::vcoxeter("5 3 * {}", {n - 2}); return tc::vcoxeter("5 3 * {}", {n - 2});
} }
tc::Group<> I2(unsigned int n) { tc::Group I2(unsigned int n) {
return tc::vcoxeter("{}", {n}); return tc::vcoxeter("{}", {n});
} }
tc::Group<> T(unsigned int m, unsigned int n) { tc::Group T(unsigned int m, unsigned int n) {
return tc::vcoxeter("{} 2 {}", {m, n}); return tc::vcoxeter("{} 2 {}", {m, n});
} }
tc::Group<> T(unsigned int n) { tc::Group T(unsigned int n) {
return T(n, n); return T(n, n);
} }

2
vis/include/mirror.hpp
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@@ -9,7 +9,7 @@
#include <geometry.hpp> #include <geometry.hpp>
template<unsigned int N> template<unsigned int N>
Eigen::Matrix<float, N, N> mirror(const tc::Group<> &group) { Eigen::Matrix<float, N, N> mirror(const tc::Group &group) {
Eigen::Matrix<float, N, N> res; Eigen::Matrix<float, N, N> res;
res.setZero(); res.setZero();

28
vis/include/solver.hpp
View File

@@ -7,6 +7,9 @@
#include <iostream> #include <iostream>
#include "combo_iterator.hpp" #include "combo_iterator.hpp"
#include "fmt/core.h"
#include "fmt/ranges.h"
/** /**
* An primitive stage N indices. * An primitive stage N indices.
* @tparam N * @tparam N
@@ -28,7 +31,7 @@ struct Primitive {
~Primitive() = default; ~Primitive() = default;
void apply(const tc::Cosets<> &table, int gen) { void apply(const tc::Cosets &table, int gen) {
for (auto &ind: indices) { for (auto &ind: indices) {
ind = table.get(ind, gen); ind = table.get(ind, gen);
} }
@@ -38,7 +41,7 @@ struct Primitive {
/** /**
* Produce a list of all generators for the group context. The range [0..group.ngens). * Produce a list of all generators for the group context. The range [0..group.ngens).
*/ */
std::vector<size_t> generators(const tc::Group<> &context) { std::vector<size_t> generators(const tc::Group &context) {
std::vector<size_t> g_gens(context.rank()); std::vector<size_t> g_gens(context.rank());
std::iota(g_gens.begin(), g_gens.end(), 0); std::iota(g_gens.begin(), g_gens.end(), 0);
return g_gens; return g_gens;
@@ -65,7 +68,7 @@ std::vector<size_t> recontext_gens(
* Apply some context transformation to all primitives of this mesh. * Apply some context transformation to all primitives of this mesh.
*/ */
template<unsigned N> template<unsigned N>
std::vector<Primitive<N>> apply(std::vector<Primitive<N>> prims, const tc::Cosets<> &table, int gen) { std::vector<Primitive<N>> apply(std::vector<Primitive<N>> prims, const tc::Cosets &table, int gen) {
for (auto &prim: prims) { for (auto &prim: prims) {
prim.apply(table, gen); prim.apply(table, gen);
} }
@@ -79,7 +82,7 @@ template<unsigned N>
[[nodiscard]] [[nodiscard]]
std::vector<Primitive<N>> recontext( std::vector<Primitive<N>> recontext(
std::vector<Primitive<N>> prims, std::vector<Primitive<N>> prims,
const tc::Group<> &context, const tc::Group &context,
const std::vector<size_t> &g_gens, const std::vector<size_t> &g_gens,
const std::vector<size_t> &sg_gens const std::vector<size_t> &sg_gens
) { ) {
@@ -127,18 +130,17 @@ template<unsigned N>
[[nodiscard]] [[nodiscard]]
std::vector<std::vector<Primitive<N>>> each_tile( std::vector<std::vector<Primitive<N>>> each_tile(
std::vector<Primitive<N>> prims, std::vector<Primitive<N>> prims,
const tc::Group<> &context, const tc::Group &context,
const std::vector<size_t> &g_gens, const std::vector<size_t> &g_gens,
const std::vector<size_t> &sg_gens const std::vector<size_t> &sg_gens
) { ) {
std::vector<Primitive<N>> base = recontext(prims, context, g_gens, sg_gens); std::vector<Primitive<N>> base = recontext(prims, context, g_gens, sg_gens);
const auto proper_sg_gens = recontext_gens(g_gens, sg_gens); const auto proper_sg_gens = recontext_gens(g_gens, sg_gens);
const auto table = context.sub(g_gens).solve({}); const auto &table = context.sub(g_gens).solve({});
const auto &cosets = context.sub(g_gens).solve(proper_sg_gens);
const tc::Cosets<> &cosets = context.sub(g_gens).solve(proper_sg_gens); tc::Path<> path(cosets);
tc::Path path(cosets);
std::vector<std::vector<Primitive<N>>> res(path.order()); std::vector<std::vector<Primitive<N>>> res(path.order());
path.walk(base, [&](auto from, auto to) { path.walk(base, [&](auto from, auto to) {
@@ -152,7 +154,7 @@ template<unsigned N>
[[nodiscard]] [[nodiscard]]
std::vector<Primitive<N>> tile( std::vector<Primitive<N>> tile(
std::vector<Primitive<N>> prims, std::vector<Primitive<N>> prims,
const tc::Group<> &context, const tc::Group &context,
const std::vector<size_t> &g_gens, const std::vector<size_t> &g_gens,
const std::vector<size_t> &sg_gens const std::vector<size_t> &sg_gens
) { ) {
@@ -183,7 +185,7 @@ std::vector<Primitive<N + 1>> fan(std::vector<Primitive<N>> prims, size_t root)
*/ */
template<unsigned N> template<unsigned N>
std::vector<Primitive<N>> triangulate( std::vector<Primitive<N>> triangulate(
const tc::Group<> &context, const tc::Group &context,
const std::vector<size_t> &g_gens const std::vector<size_t> &g_gens
) { ) {
if (g_gens.size() + 1 != N) { if (g_gens.size() + 1 != N) {
@@ -207,7 +209,7 @@ std::vector<Primitive<N>> triangulate(
// Single-index primitives should not be further triangulated. // Single-index primitives should not be further triangulated.
template<> template<>
std::vector<Primitive<1>> triangulate( std::vector<Primitive<1>> triangulate(
const tc::Group<> &, const tc::Group &,
const std::vector<size_t> &g_gens const std::vector<size_t> &g_gens
) { ) {
if (not g_gens.empty()) { if (not g_gens.empty()) {
@@ -220,7 +222,7 @@ std::vector<Primitive<1>> triangulate(
} }
template<unsigned N, class T> template<unsigned N, class T>
auto hull(const tc::Group<> &group, T all_sg_gens, const std::vector<std::vector<size_t>> &exclude) { auto hull(const tc::Group &group, T all_sg_gens, const std::vector<std::vector<size_t>> &exclude) {
std::vector<std::vector<Primitive<N>>> parts; std::vector<std::vector<Primitive<N>>> parts;
auto g_gens = generators(group); auto g_gens = generators(group);

2
vis/src/comps.hpp
View File

@@ -19,7 +19,7 @@
namespace vis { namespace vis {
struct Group { struct Group {
tc::Group<> group; tc::Group group;
vec5 root; vec5 root;
vec3 color; vec3 color;