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lpetrich
Contributor
With a different distance function, one gets the 
Thomson problem, for the distribution of charged particles on a sphere. I was working with sum of |xj - xi|, while he Thomson function features 1/|xj - xi|. But I get the same solutions for the named cases in Wikipedia's table, except for 11, where I am not sure where I do or not. The named ones are 1 - 12, 14, and 24.
I've written some Mathematica code that uses a distance power law, so I can use either distance or inverse distance, the Thomson case.
Mathematica's optimization algorithms come in two families, distinguished by what algorithms that they use. I call them families because they can do either maximum or minimum, and because they can return the optimum arg values, the optimum function value, or both.
The NMinimize family has methods "NelderMead", "DifferentialEvolution", "SimulatedAnnealing", and "RandomSearch".
The FindMinimum family has methods "ConjugateGradient", "PrincipalAxis", "LevenbergMarquardt", "Newton", "QuasiNewton", "InteriorPoint", and "LinearProgramming"
The NMinimize family does not need derivatives of the function to be optimized, while FindMinimum does use them in some of its algorithms. How many:
In NMinimize, only NelderMead is deterministic, though it may use random initial conditions.
Thomson problem, for the distribution of charged particles on a sphere. I was working with sum of |xj - xi|, while he Thomson function features 1/|xj - xi|. But I get the same solutions for the named cases in Wikipedia's table, except for 11, where I am not sure where I do or not. The named ones are 1 - 12, 14, and 24.I've written some Mathematica code that uses a distance power law, so I can use either distance or inverse distance, the Thomson case.
Mathematica's optimization algorithms come in two families, distinguished by what algorithms that they use. I call them families because they can do either maximum or minimum, and because they can return the optimum arg values, the optimum function value, or both.
The NMinimize family has methods "NelderMead", "DifferentialEvolution", "SimulatedAnnealing", and "RandomSearch".
The FindMinimum family has methods "ConjugateGradient", "PrincipalAxis", "LevenbergMarquardt", "Newton", "QuasiNewton", "InteriorPoint", and "LinearProgramming"
The NMinimize family does not need derivatives of the function to be optimized, while FindMinimum does use them in some of its algorithms. How many:
- 0: LinearProgramming (linear optimization only), PrincipalAxis
- 1: ConjugateGradient, QuasiNewton
- 2: LevenbergMarquardt, Newton, InteriorPoint
In NMinimize, only NelderMead is deterministic, though it may use random initial conditions.
lpetrich
Contributor
Back to the spherical point-spacing problem, one may generate solutions by tesselating polyhedra with all-triangular faces ("deltahedra").
For spherical topology, the number of vertices V, edges E, and faces F satisfies V - E + F = 2 (in general, the Euler characteristic of the surface, 2*(1-g) for genus g).
For an all-triangular polyhedron, E = (3/2)*F. That gives V = 2 + (1/2)*F.
Let us consider vertices with different numbers of edges/faces attached to them. I'll call that number the vertex order: V(k) for order k. Thus,
V = sum over k of V(k)
F = (1/3) * (sum over k of k*V(k))
giving us
sum over k of (1-k/6)*V(k) = 2
Note that V(6) drops out, and for each one alone, V(5) = 12, V(4) = 6, V(3) = 4, and V(2) = 3.
Most of the point-spacing solutions that I've found are deltahedra, but for at least 3 points, I've found only three that are not: the 8-point, 16-point, and 24-point cases. These are the 4-antiprism and the snub cube. The latter one has the property of not being reflection-symmetric, of being "chiral". That means that there are two mirror-mage snub cubes. That is also true of the snub dodecahedron, though not of the snub tetrahedron. That one is the regular icosahedron, and it is thus is mirror-symmetric.
I've checked that these high-symmetry deltahedron solutions are all maximal-separation solutions:
For tessellating a triangle with n triangles on each side, one gets n2 faces. But while some optimum solutions may be tessellations of smaller ones, that does not seem to be the case for the smaller tessellations.
For spherical topology, the number of vertices V, edges E, and faces F satisfies V - E + F = 2 (in general, the Euler characteristic of the surface, 2*(1-g) for genus g).
For an all-triangular polyhedron, E = (3/2)*F. That gives V = 2 + (1/2)*F.
Let us consider vertices with different numbers of edges/faces attached to them. I'll call that number the vertex order: V(k) for order k. Thus,
V = sum over k of V(k)
F = (1/3) * (sum over k of k*V(k))
giving us
sum over k of (1-k/6)*V(k) = 2
Note that V(6) drops out, and for each one alone, V(5) = 12, V(4) = 6, V(3) = 4, and V(2) = 3.
Most of the point-spacing solutions that I've found are deltahedra, but for at least 3 points, I've found only three that are not: the 8-point, 16-point, and 24-point cases. These are the 4-antiprism and the snub cube. The latter one has the property of not being reflection-symmetric, of being "chiral". That means that there are two mirror-mage snub cubes. That is also true of the snub dodecahedron, though not of the snub tetrahedron. That one is the regular icosahedron, and it is thus is mirror-symmetric.
I've checked that these high-symmetry deltahedron solutions are all maximal-separation solutions:
- 4 - tetrahedron
- 6 - octahedron
- 12 - icosahedron
- 32 - pyramid-face dodecahedron
- 72 - pyramid-on-pentagon snub dodecahedron
For tessellating a triangle with n triangles on each side, one gets n2 faces. But while some optimum solutions may be tessellations of smaller ones, that does not seem to be the case for the smaller tessellations.
lpetrich
Contributor
Abstract algebra, Algebraic structure, Outline of algebraic structures -- Wikipedia has a listbox with group-like, ring-like, lattice-like, module-like, and algebra-like algebraic structures. These are abstractions based on familiar arithmetic operations, and mathematicians like creating abstractions.These pages have a listbox with these expandable entries: Group-like, Ring-like, Lattice-like, Module-like, Algebra-like
The article "Algebraic structure" mentions all these and some additional ones. An algebraic structure has a set S and operations that go from S to S, the closure property. All valid inputs must be in S, and all possible outputs must also be in S.
The additional ones include a set with no operations, a degenerate case, and a set with a unary operation.
If a unary operation always undoes itself, an operation f that satisfies f(f(x)) = x for all x in S, then it is an "involution".
If f is a bijection, then f can be interpreted as making a permutation of S.
If S is finite, then applying f repeatedly to some element of S will yield a limit cycle, a cycle a1 - a2 - a3 - ... - a1 again. A limit cycle can have a length of 1: a1 - a1 - a1 - ...
lpetrich
Contributor
For group-like entities, we start with a magma or groupoid, one with a binary operation f(S,S) = S or S*S = S or S+S = S.
If an element x satisfies x*x = x, then it is "idempotent". If that is true for every element, then the operation is also called idempotent.
An element e is a "left identity" if e*x = x for all x in S. It is a right identity if x*e = x for all x in S. If there is both a left identity and a right identity, then they are equal and that identity is unique. There can be more than one left identity if there are no right ones, and vice versa.
An element z is a "left zero" if z*x = z for all x in S. It is a right zero if x*z = z for all x in S. As with left and right identities, if there is both a left zero and a right zero, they are equal and unique. There can be multiple ones only if they are all on the same side.
A set J, subset of S, is a left ideal if S*J = J, and a right ideal if J*S = J. If it is both, then it is two-sided. Here is a simple kind of ideal: for integers under multiplication, the multiples of some integer form an ideal.
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A magma with associativity, (a*b)*c = a*(b*c), is a semigroup.
A semigroup with an identity is a monoid.
A monoid with inverses, a*inv(a) = inv(a)*a = e, is a group.
A commutative group is an abelian group.
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A magma where division is always possible is a quasigroup.
For every a and b, there are some x and y that satisfy
x*a = b
a*y = b
This makes the operation table a "Latin square", where every row and column is some permutation of S. Here are the smallest Latin squares:
The second to fourth ones of the 4-square are equivalent, as is evident by renaming 2, 3, and 4 to some permutation of them and then doing suitable row and column permutations.
A loop is a quasigroup with identity.
A group is an associative loop.
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A semilattice is a commutative semigroup with its operation being idempotent.
If an element x satisfies x*x = x, then it is "idempotent". If that is true for every element, then the operation is also called idempotent.
An element e is a "left identity" if e*x = x for all x in S. It is a right identity if x*e = x for all x in S. If there is both a left identity and a right identity, then they are equal and that identity is unique. There can be more than one left identity if there are no right ones, and vice versa.
An element z is a "left zero" if z*x = z for all x in S. It is a right zero if x*z = z for all x in S. As with left and right identities, if there is both a left zero and a right zero, they are equal and unique. There can be multiple ones only if they are all on the same side.
A set J, subset of S, is a left ideal if S*J = J, and a right ideal if J*S = J. If it is both, then it is two-sided. Here is a simple kind of ideal: for integers under multiplication, the multiples of some integer form an ideal.
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A magma with associativity, (a*b)*c = a*(b*c), is a semigroup.
A semigroup with an identity is a monoid.
A monoid with inverses, a*inv(a) = inv(a)*a = e, is a group.
A commutative group is an abelian group.
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A magma where division is always possible is a quasigroup.
For every a and b, there are some x and y that satisfy
x*a = b
a*y = b
This makes the operation table a "Latin square", where every row and column is some permutation of S. Here are the smallest Latin squares:
Code:
1
1 2
2 1
1 2 3
2 3 1
3 1 2
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
2 1 4 3 2 1 4 3 2 3 4 1 2 4 1 3
3 4 1 2 3 4 2 1 3 4 1 2 3 1 4 2
4 3 2 1 4 3 1 2 4 1 2 3 4 3 2 1A loop is a quasigroup with identity.
A group is an associative loop.
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A semilattice is a commutative semigroup with its operation being idempotent.
lpetrich
Contributor
Now to ring-like entities or ringoids. They have two binary operations, usually called multiplication and addition (*,+), with multiplication being distributive over addition.
A semiring is a ringoid where each operation is a monoid over the ringoid's set. Addition is usually assumed to be commutative.
A near-ring is a semiring where the addition operation is a group that is not necessarily abelian.
A ring is a semiring where the addition operation is an abelian group.
a rng or rung or pseudo-ring is a ring where the multiplication operation does not necessarily have an identity.
A commutative ring has commutative multiplication as well as addition. Addition and multiplication of numbers form a commutative ring, while these operations on matrices do not.
An integral domain has (nonzero)*(nonzero) = (nonzero), where "zero" (0) is the ring's additive identity.
A division ring or a skew field has division by every element but 0: every nonzero element has a multiplicative inverse or reciprocal.
A field is a commutative ring where the nonzero elements form a group under multiplication.
A semiring is a ringoid where each operation is a monoid over the ringoid's set. Addition is usually assumed to be commutative.
A near-ring is a semiring where the addition operation is a group that is not necessarily abelian.
A ring is a semiring where the addition operation is an abelian group.
a rng or rung or pseudo-ring is a ring where the multiplication operation does not necessarily have an identity.
A commutative ring has commutative multiplication as well as addition. Addition and multiplication of numbers form a commutative ring, while these operations on matrices do not.
An integral domain has (nonzero)*(nonzero) = (nonzero), where "zero" (0) is the ring's additive identity.
A division ring or a skew field has division by every element but 0: every nonzero element has a multiplicative inverse or reciprocal.
A field is a commutative ring where the nonzero elements form a group under multiplication.
lpetrich
Contributor
Now,lattice-like entities.
A lattice has two binary operations, "meet" and "join", operations that are associative, commutative, and idempotent.
For the real numbers or some subset of them, meet = minimum or infimum, and join = maximum or supremum.
For a set of subsets of some set, meet = intersection and join = union.
For the set of divisors of some positive integer, meet = greatest common denominator and join = least common multiple.
So the lattice operations impose an ordering on the lattice's set. If (a meet b) == a and (a join b) == b, then a <= b.
A lattice has an upper bound ub if for all a in its set, (a join ub) == ub and (a meet ub) == a. Likewise, a lattice has a lower bound lb if (a meet lb) == lb and (a join lb) == a.
A modular lattice has (x <= y) implying x join (y meet a) == y meet (x join a) for all a.
A distributive lattice has meet distributive over join and also join distributive over meet.
A complemented lattice is a bounded lattice with a complement operation: a meet (cmpl a) = lb, a join (cmpl a) = ub
Such a lattice is orthocomplemented if complementation is an involution (it undoes itself):
cmpl (cmpl a) = a
and if complementation reverses order:
a <= b is equivalent to (cmpl b) <= (cmpl a)
An orthomodular lattice is one that is orthocomplemented and modular.
For numbers, complementation only exists if there are two of them, while for subsets, cmpl (subset) is (original set) - (subset), the complement operation for sets. The upper bound is the original set and the lower bound the empty set.
A lattice is totally ordered if its partial ordering satisfies:
Totality: at least one of a <= b and b <= a is true.
Transitivity: a <= b and b <= c implies a <= c.
Antisymmetry: a <= b and b <= a implies a == b (equality).
Thus, real numbers are totally ordered, while subsets need not be.
A Boolean algebra is a bounded lattice that is distributive and that has a complement operation.
For numbers, the only nontrivial Boolean algebra has two of them. This is the familiar one.
The power set (set of all subsets of some set) with set operations is a Boolean algebra.
The set of divisors of a square-free positive integer with gcd and lcm is a Boolean algebra.
The number one is equivalent to the power set of a one-element set, while the divisor ones are equivalent to power sets more generally.
A lattice has two binary operations, "meet" and "join", operations that are associative, commutative, and idempotent.
For the real numbers or some subset of them, meet = minimum or infimum, and join = maximum or supremum.
For a set of subsets of some set, meet = intersection and join = union.
For the set of divisors of some positive integer, meet = greatest common denominator and join = least common multiple.
So the lattice operations impose an ordering on the lattice's set. If (a meet b) == a and (a join b) == b, then a <= b.
A lattice has an upper bound ub if for all a in its set, (a join ub) == ub and (a meet ub) == a. Likewise, a lattice has a lower bound lb if (a meet lb) == lb and (a join lb) == a.
A modular lattice has (x <= y) implying x join (y meet a) == y meet (x join a) for all a.
A distributive lattice has meet distributive over join and also join distributive over meet.
A complemented lattice is a bounded lattice with a complement operation: a meet (cmpl a) = lb, a join (cmpl a) = ub
Such a lattice is orthocomplemented if complementation is an involution (it undoes itself):
cmpl (cmpl a) = a
and if complementation reverses order:
a <= b is equivalent to (cmpl b) <= (cmpl a)
An orthomodular lattice is one that is orthocomplemented and modular.
For numbers, complementation only exists if there are two of them, while for subsets, cmpl (subset) is (original set) - (subset), the complement operation for sets. The upper bound is the original set and the lower bound the empty set.
A lattice is totally ordered if its partial ordering satisfies:
Totality: at least one of a <= b and b <= a is true.
Transitivity: a <= b and b <= c implies a <= c.
Antisymmetry: a <= b and b <= a implies a == b (equality).
Thus, real numbers are totally ordered, while subsets need not be.
A Boolean algebra is a bounded lattice that is distributive and that has a complement operation.
For numbers, the only nontrivial Boolean algebra has two of them. This is the familiar one.
The power set (set of all subsets of some set) with set operations is a Boolean algebra.
The set of divisors of a square-free positive integer with gcd and lcm is a Boolean algebra.
The number one is equivalent to the power set of a one-element set, while the divisor ones are equivalent to power sets more generally.
lpetrich
Contributor
I now turn to module-like entities.
First, vector spaces. x = {x(1), x(2), ..., x
} for an n-dimensional vector, where x(1), x(2), ..., x are in some ring R.
Addition is component-by-component: x + y = {x(1)+y(1), x(2)+y(2), ..., x+y} and scalar multiplication is likewise component-by-component: a*x = {a*x(1), a*x(2), ..., a*x} and x*a = {x(1)*a, x(2)*a, ..., x*a} where a is also in R.
One can interpret a vector as a function of its index variable, and then generalize that index variable beyond 1, 2, ..., n.
It is evident that vectors form an abelian group under vector addition.
Modules generalize vector spaces. A left module is defined as follows. For an abelian group M and a ring R, and an operation * with R*M = M,
a*(x + y) = (a*x) + (a*y)
(a + b)*x = (a*x) + (b*x)
(a*b)*x = a*(b*x)
(R multiplicative identity: 1)*x = x
where a and b are in R and x and y are in M.
Right modules and two-sided modules can be defined similarly.
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Going into algebra-like entities, an algebra over a field is a vector space with a bilinear operator (vector) # (vector) -> (vector). It is distributive on both sides and it has scalar multiplication:
(x + y) # z = (x # z) + (y # z)
x # (y + z) = (x # y) + (x # z)
(a*x) # (b*y) = (a*b)*(x # y)
An important one is a Lie algebra, where # is antisymmetric and non-associative, though it nevertheless satisfies the Jacobi identity.
(b # a) = - (a # b)
(a # b) # c + (b # c) # a + (c # a) # b = 0
Both identities are readily verified for # being a matrix commutator: a # b = a.b - b.a
Another non-associative one is the 3-vector cross product. It is equivalent to the Lie algebra SO(3).
There are also some associative ones, like complex-number and quaternion multiplication.
First, vector spaces. x = {x(1), x(2), ..., x
} for an n-dimensional vector, where x(1), x(2), ..., x are in some ring R.Addition is component-by-component: x + y = {x(1)+y(1), x(2)+y(2), ..., x
+y} and scalar multiplication is likewise component-by-component: a*x = {a*x(1), a*x(2), ..., a*x} and x*a = {x(1)*a, x(2)*a, ..., x*a} where a is also in R.One can interpret a vector as a function of its index variable, and then generalize that index variable beyond 1, 2, ..., n.
It is evident that vectors form an abelian group under vector addition.
Modules generalize vector spaces. A left module is defined as follows. For an abelian group M and a ring R, and an operation * with R*M = M,
a*(x + y) = (a*x) + (a*y)
(a + b)*x = (a*x) + (b*x)
(a*b)*x = a*(b*x)
(R multiplicative identity: 1)*x = x
where a and b are in R and x and y are in M.
Right modules and two-sided modules can be defined similarly.
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Going into algebra-like entities, an algebra over a field is a vector space with a bilinear operator (vector) # (vector) -> (vector). It is distributive on both sides and it has scalar multiplication:
(x + y) # z = (x # z) + (y # z)
x # (y + z) = (x # y) + (x # z)
(a*x) # (b*y) = (a*b)*(x # y)
An important one is a Lie algebra, where # is antisymmetric and non-associative, though it nevertheless satisfies the Jacobi identity.
(b # a) = - (a # b)
(a # b) # c + (b # c) # a + (c # a) # b = 0
Both identities are readily verified for # being a matrix commutator: a # b = a.b - b.a
Another non-associative one is the 3-vector cross product. It is equivalent to the Lie algebra SO(3).
There are also some associative ones, like complex-number and quaternion multiplication.
lpetrich
Contributor
Kharakov, how did you get that animated graphic?
As to finding the number of inflection points per unit length of the Mandelbrot set's boundary, my best guess is to find some sets of points that converge to the Mandelbrot set and do the calculation for them. One might be able to find some trends in the numbers.
I'll do that for the Koch snowflake, since one can easily get the intermediate cases.
Initial state: a triangle:
Side length = 1
Number of sides = 3
Number of outward vertices = 3
Number of inward vertices = 0
Next state:
Side length /= 3
Number of sides *= 4
Number of outward vertices += (previous number of sides)
Number of inward vertices += 2*(previous number of sides)
Side length = 3-n
Number of sides = 3*4n
Number of {outward,inward} vertices = {3,0} + {1,2}*(1 + 4 + ... + 4n-1) = {3,0} + {1,2}*(4n - 1)/3
So the Koch snowflake's perimeter increases as (4/3)n, and its number of vertices as 4n -- Thus, (number of vertices) / (perimeter) ~ 3n, which tends to infinity as n -> infinity.
As to finding the number of inflection points per unit length of the Mandelbrot set's boundary, my best guess is to find some sets of points that converge to the Mandelbrot set and do the calculation for them. One might be able to find some trends in the numbers.
I'll do that for the Koch snowflake, since one can easily get the intermediate cases.
Initial state: a triangle:
Side length = 1
Number of sides = 3
Number of outward vertices = 3
Number of inward vertices = 0
Next state:
Side length /= 3
Number of sides *= 4
Number of outward vertices += (previous number of sides)
Number of inward vertices += 2*(previous number of sides)
Side length = 3-n
Number of sides = 3*4n
Number of {outward,inward} vertices = {3,0} + {1,2}*(1 + 4 + ... + 4n-1) = {3,0} + {1,2}*(4n - 1)/3
So the Koch snowflake's perimeter increases as (4/3)n, and its number of vertices as 4n -- Thus, (number of vertices) / (perimeter) ~ 3n, which tends to infinity as n -> infinity.
Kharakov
Quantum Hot Dog
Fragmentarium (Render animation- saves to whatever format you want, and it's easy to code math in it)
VirtualDub: File--> Export--> create .gif,
Google photos (image size limit here is 500k, which is low for an animated gif)
Kharakov
Quantum Hot Dog
Shake
Senior Member
Not sure if anyone will be a fan of the music, but this video reminds me of those fractal creations:
[YOUTUBE]https://www.youtube.com/watch?v=oFiDcazicdk[/YOUTUBE]
Several sections of zooming out to see the overall structure. I'd love to see a "making of" on this video.
[YOUTUBE]https://www.youtube.com/watch?v=oFiDcazicdk[/YOUTUBE]
Several sections of zooming out to see the overall structure. I'd love to see a "making of" on this video.
Kharakov
Quantum Hot Dog
I think Julius Horsthuis did that one. If you're into fractals, you could join up with the Mandelbulb Maniacs on FB.
I was thinking about metal for that one, but it's old metal
I was thinking about metal for that one, but it's old metal
Last edited:
Phil Scott
Member
Dimension? Specifically, I might say that as vector spaces, they have 3 and 2 element minimal bases respectively.
beero1000
Veteran Member
This would be my choice, as long as the two things we are comparing are finite-dimensional vector spaces over the same field (which is true for \(\mathbb{R}^2\) and \(\mathbb{R}^3\) as vector spaces over \(\mathbb{R}\)).