= Geometry
{wiki}
= Minimum bounding box
{parent=Geometry}
{wiki}
= Bounding box
{parent=Geometry}
{wiki}
= Fractal
{parent=Geometry}
{wiki}
= Point
{disambiguate=geometry}
{parent=Geometry}
{wiki}
= Point
{synonym}
= Line
{disambiguate=geometry}
{parent=Point (geometry)}
{wiki}
= Line
{synonym}
= Hyperplane
{parent=Point (geometry)}
{wiki}
Generalization of a for any number of dimensions.
Kind of the opposite of a line: the line has dimension 1, and the plane has dimension D-1.
In $D=2$, both happen to coincide, a boring example of an .
= Plane
{disambiguate=geometry}
{parent=Hyperplane}
{wiki}
= Plane
{synonym}
= n-sphere
{parent=Geometry}
{title2=$S^n$}
{wiki}
= Antipodal point
{parent=n-sphere}
{wiki}
= Diameter
{parent=n-sphere}
{wiki}
= Radius
{parent=Diameter}
{wiki}
= Circle
{parent=n-sphere}
{title2=$S^1$}
{wiki}
= 1-sphere
{synonym}
{title2}
= Squaring the circle
{parent=Circle}
{wiki}
= Tarski's circle-squaring problem
{c}
{parent=Circle}
{title2=Cut a circle into square}
{wiki}
Does not require straight line cuts.
= Sphere
{parent=n-sphere}
{title2=$S^2$}
{wiki}
= 2-sphere
{synonym}
{title2}
= Great circle
{parent=Sphere}
{wiki}
= 3-sphere
{parent=n-sphere}
{title2=$S^3$}
{wiki}
Diffeomorphic to .
= Projective geometry
{parent=Geometry}
{wiki}
= Projective space
{parent=Projective geometry}
{title2=$\projectiveSpace(V)$}
{wiki}
A projective space can be defined for any .
The projective space associated with a given $V$ is denoted $\projectiveSpace(V)$.
The definition is to take the vector space, remove the zero element, and identify all elements that lie on the same line, i.e. $\vec{v} = \lambda \vec{w}$
The most important initial example to study is the .
= Projective plane
{parent=Projective space}
{wiki}
= Real projective space
{parent=Projective geometry}
{title2=$RP^n$}
{title2=$\projectiveSpace(\R^{n+1})$}
In those cases at least, it is possible to add a to the spaces, leading to .
= Real projective line
{parent=Real projective space}
{title2=$RP^1$}
{title2=$\projectiveSpace(\R^2)$}
{wiki}
Just a .
Take $\R^2$ with a line at $x = 0$. Identify all the points that an observer
= Real projective plane
{parent=Real projective space}
{title2=$RP^2$}
{title2=$\projectiveSpace(\R^3)$}
{wiki}
For some reason, is mildly obsessed with understanding and visualizing the real projective plane.
To see why this is called a plane, move he center of the sphere to $z=1$, and project each line passing on the center of the sphere on the x-y plane. This works for all points of the sphere, except those at the equator $z=1$. Those are the . Note that there is one such point at infinity for each direction in the x-y plane.
= Synthetic geometry of the real projective plane
{parent=Real projective plane}
It good to think about how look like in the real projective plane:
* two parallel lines on the plane meet at a point on the sphere!
Since there is one point of infinity for each direction, there is one such point for every direction the two parallel lines might be at. The does not hold, and is replaced with a simpler more elegant version: every two lines meet at exactly one point.
One thing to note however is that ther does not have defined on it by definition. Those can be defined, forming through the , but we can interpret the "parallel lines" as "two lines that meet at a point at infinity"
* points in the real projective plane are lines in <\R^3>
* lines in the real projective plane are planes in <\R^3>.
For every two projective points there is a single projective line that passes through them.
Since it is a plane in <\R^3>, it always intersects the real plane at a line.
Note however that not all lines in the real plane correspond to a projective line: only lines tangent to a circle at zero do.
Unlike the which is to the , the is not to the .
The difference bewteen the and the is that for the all those points in the x-y circle are identified to a single point.
One more generalized argument of this is the , in which the is a with a hole cut and one glued in.
= Model of the real projective plane
{parent=Real projective plane}
= Lines through origin model of the real projective plane
{parent=Model of the real projective plane}
This is the standard model.
= Spherical cap model of the real projective plane
{parent=Model of the real projective plane}
's preferred visualization of the real projective plane is a small variant of the standard "lines through origin in <\R^3>".
Take a open half e.g. a sphere but only the points with $z > 0$.
Each point in the half sphere identifies a unique line through the origin.
Then, the only lines missing are the lines in the x-y plane itself.
For those sphere points in the on the x-y plane, you should think of them as magic poins that are identified with the corresponding , also on the x-y, but on the other side of the origin. So basically you you can teleport from one of those to the other side, and you are still in the same point.
Ciro likes this model because then all the magic is confined just to the $z=0$ part of the model, and everything else looks exactly like the sphere.
It is useful to contrast this with the sphere itself. In the sphere, all points in the circle $z = 0$ are the same point. But this is not the case for the . You cannot instantly go to any other point on the $z=0$ by just moving a little bit, you have to walk around that circle.
\Image[https://raw.githubusercontent.com/cirosantilli/media/master/spherical-cap-model-of-the-real-projective-plane.svg]
{title=Spherical cap model of the real projective plane}
{description=On the x-y plane, you can magically travel immediately between such as A/A', B/B' and C/C'. Or equivalently, those pairs are the same point. Every other point outside the x-y plane is just a regular point like a normal .}
= The real projective plane is not simply connected
{parent=Real projective plane}
To see that the is not , considering the , take a that starts at $(1, 0, 0)$ and moves along the $y=0$ ends at $(-1, 0, 0)$.
Note that both of those points are the same, so we have a loop.
Now try to shrink it to a point.
There's just no way!
= Point at infinity
{parent=Real projective plane}
{wiki}
= Points at infinity
{synonym}
= Homogenous coordinates
{parent=Real projective plane}
{wiki}
= Polytope
{parent=Geometry}
{wiki}
A is a 2-dimensional , is a 3-dimensional .
= Convex polytope
{parent=Polytope}
{wiki}
= Convex
{synonym}
= Regular polytope
{parent=Polytope}
{wiki}
TODO understand and explain definition.
= Classification of regular polytopes
{parent=Regular polytope}
{wiki=Regular_polytope#Classification_and_description}
The 3D regular convex polyhedrons are super famous, have the name: , and have been known since antiquity. In particular, there are only 5 of them.
The counts per dimension are:
\Table[
|| Dimension
|| Count
| 2
| Infinite
| 3
| 5
| 4
| 6
| >4
| 3
]
{title=Number of regular polytopes per dimension}
The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:
*
*
*
Then, the 2 3D missing ones have 4D analogues and the sixth one in 4D does not have a 3D analogue: https://en.wikipedia.org/wiki/24-cell[the 24-cell]. Yes, this is the kind of irregular stuff lives [for].
= Simplex
{parent=Classification of regular polytopes}
{wiki}
, .
The name does not imply regular by default. For regular ones, you should say "regular polytope".
Non-regular description: take convex hull take D + 1 vertices that are not on a single D-plan.
= Hypercube
{parent=Classification of regular polytopes}
{wiki}
, cube. 4D case known as .
Convex hull of all $\{-1, 1\}^D$ ( power) D-tuples, e.g. in <3D>:
``
( 1, 1, 1)
( 1, 1, -1)
( 1, -1, 1)
( 1, -1, -1)
(-1, 1, 1)
(-1, 1, -1)
(-1, -1, 1)
(-1, -1, -1)
``
From this we see that there are $2^D$ .
Two are linked iff they differ by a single number. So each vertex has D neighbors.
= Hyperrectangle
{parent=Hypercube}
{wiki}
The [non-regular] version of the .
= Cross polytope
{parent=Classification of regular polytopes}
{wiki}
Examples: , .
Take $(0, 0, 0, \dots, 0)$ and flip one of 0's to $\pm 1$. Therefore has $2 \times D$ .
Each edge E is linked to every other edge, except it's opposite -E.
= Polygon
{parent=Polytope}
{wiki}
= Quadrilateral
{parent=Polygon}
{wiki}
= Rectangle
{parent=Quadrilateral}
{wiki}
= Parallelogram
{parent=Polygon}
{wiki}
= Parallelepiped
{parent=Parallelogram}
{wiki}
<3D> .
= Volume of the parallelepiped
{parent=Parallelepiped}
= Volume of a parallelepiped
{synonym}
= Regular polygon
{parent=Polygon}
{wiki}
= Regular convex polygon
{parent=Regular polygon}
= Triangle
{parent=Regular convex polygon}
{wiki}
= Square
{parent=Regular convex polygon}
{tag=Rectangle}
{wiki}
= Pentagon
{parent=Regular convex polygon}
{wiki}
= Hexagon
{parent=Regular convex polygon}
{wiki}
= Octagon
{parent=Regular convex polygon}
{wiki}
= Polyhedron
{parent=Polytope}
{wiki}
= Polyhedra
{synonym}
= Tetrahedron
{parent=Polyhedron}
{wiki}
= Octahedron
{parent=Polyhedron}
{wiki}
= Regular polyhedron
{parent=Polytope}
{wiki}
= Platonic solid
{c}
{parent=Regular polyhedron}
{wiki}
A .
Their [beauty is a classification type result] as stated at .
https://en.wikipedia.org/wiki/Platonic_solid#Topological_proof
= 4-polytope
{parent=Polytope}
{wiki}
= Regular 4-polytope
{parent=4-polytope}
{wiki}
= Tesseract
{parent=Regular 4-polytope}
{wiki}
= Differential geometry
{parent=Geometry}
Bibliography:
* https://maths-people.anu.edu.au/~andrews/DG/ Lectures on Differential Geometry by Ben Andrews
= Lie group
{c}
{parent=Differential geometry}
{wiki}
The key and central motivation for studying Lie groups and their appears to be to characterize in through , just start from there.
Notably appear to map to forces, and local means "around the identity", notably: .
More precisely: .
TODO really wants to understand what all the fuss is about:
* https://math.stackexchange.com/questions/1322206/lie-groups-lie-algebra-applications
* https://mathoverflow.net/questions/58696/why-study-lie-algebras
* https://math.stackexchange.com/questions/405406/definition-of-lie-algebra
Oh, there is a low dimensional classification! Ciro is [a sucker for classification theorems]! https://en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras
The fact that there are elements arbitrarily close to the identity, which is only possible due to the group being continuous, is the key factor that simplifies the treatment of Lie groups, and follows the philosophy of .
Bibliography:
* https://youtu.be/kpeP3ioiHcw?t=2655 "Particle Physics Topic 6: Lie Groups and Lie Algebras" by Alex Flournoy (2016). Good [SO(3)] explicit exponential expansion example. Then next lecture shows why SU(2) is the representation of SO(3). Next ones appear to eventually get to the physical usefulness of the thing, but I lost patience. Not too far out though.
* https://www.youtube.com/playlist?list=PLRlVmXqzHjURZO0fviJuyikvKlGS6rXrb "Lie Groups and Lie Algebras" playlist by XylyXylyX (2018). Tutorial with infinitely many hours
* http://www.staff.science.uu.nl/~hooft101/lectures/lieg07.pdf
* http://www.physics.drexel.edu/~bob/LieGroups.html
\Video[https://www.youtube.com/watch?v=ZRca3Ggpy_g]
{title=What is Lie theory? by Mathemaniac 2023}
= Lie derivative
{c}
{parent=Lie group}
{wiki}
Bibliography:
* https://takeshimg92.github.io/posts/lie_derivatives.html
= Applications of Lie groups to differential equations
{parent=Lie group}
= How to use Lie Groups to solve differential equations
{synonym}
{title2}
Solving was apparently Lie's original motivation for developing . It is therefore likely one of the most understandable ways to approach it.
It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like had done for . Both approaches use as the key tool.
* https://www.researchgate.net/profile/Michael_Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differential_Equations/links/548cbf250cf214269f20e267/Lie-Groups-as-a-Tool-for-Solving-Differential-Equations.pdf Lie-Groups as a Tool for Solving Differential Equations by Michael Frewer. Slides with good examples.
= Lie algebra
{c}
{parent=Lie group}
{wiki}
Like everything else in , first start with the as discussed at {full}.
Intuitively, a is a simpler object than a . Without any extra structure, groups can be very complicated non-linear objects. But a is just an , and one with a restricted called the , that has to also be [alternating] and satisfy the .
Another important way to think about Lie algebras, is as .
Because of the , we know that there is almost a between each and the corresponding . So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how are a simpler representation of .
To make things even simpler, because , the only things we need to specify a through a are:
* the dimension
* the
Note that the can look different under different basis of the however. This is shown for example at page 71 for the .
As mentioned at Chapter 4 "Lie Algebras", taking the around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.
Bibliography:
* https://physicstravelguide.com/advanced_tools/group_theory/lie_algebras#tab__concrete on
* http://jakobschwichtenberg.com/lie-algebra-able-describe-group/ by
= Infinitesimal generator
{parent=Lie algebra}
Elements of a can (should!) be seen a continuous analogue to the in finite groups.
For continuous groups however, we can't have a finite generating set in the strict sense, as a finite set won't ever cover every possible point.
But the can be finite.
And just like in finite groups, where you can specify the full group by specifying only the relationships between generating elements, in the Lie algebra you can almost specify the full group by specifying the relationships between the elements of a .
This "specification of a relation" is done by defining the .
The reason why the algebra works out well for continuous stuff is that by definition an is a with some extra structure, and we know very well how to make infinitesimal elements in a vector space: just multiply its vectors by a constant $c$ that cana be arbitrarily small.
= Lie group-Lie algebra correspondence
{c}
{parent=Lie algebra}
{wiki=Lie_group–Lie_algebra_correspondence}
Every corresponds to a single .
The basically defines how to map an algebra to the group.
Bibliography:
* Chapter 7 "EXPonentiation"
= Lie algebra exponential covering problem
{c}
{parent=Lie group-Lie algebra correspondence}
7.2 "The covering problem" gives some amazing intuition on the subject as usual.
= A single exponential map is not enough to recover a simple Lie group from its algebra
{parent=Lie algebra exponential covering problem}
Example at: Chapter 7 "EXPonentiation".
= The product of a exponential of the compact algebra with that of the non-compact algebra recovers a simple Lie from its algebra
{parent=Lie algebra exponential covering problem}
Example at: Chapter 7 "EXPonentiation".
Furthermore, the non- part is always to <\R^n>, only the non-compact part can have more interesting structure.
= Two different Lie groups can have the same Lie algebra
{parent=Lie group-Lie algebra correspondence}
The most important example is perhaps and , both of which have the same , but are not isomorphic.
= Every Lie algebra has a unique single corresponding simply connected Lie group
{parent=Two different Lie groups can have the same Lie algebra}
This is called the .
E.g. in the case of and , is , but is not.
= Universal covering group
{parent=Every Lie algebra has a unique single corresponding simply connected Lie group}
The group referred to at: .
= Every Lie group that has a given Lie algebra is the image of an homomorphism from the universal cover group
{parent=Two different Lie groups can have the same Lie algebra}
= Lie bracket
{c}
{parent=Lie algebra}
= Exponential map
{parent=Lie algebra}
{wiki}
Most commonly refers to: .
= Exponential map
{disambiguate=Lie theory}
{parent=Exponential map}
{wiki}
Like everything else in theory, you should first look at the version of this operation: the .
The links small transformations around the origin (infinitely small) back to larger finite transformations, and small transformations around the origin are something we can deal with a , so this map links the two worlds.
The idea is that we can decompose a finite transformation into infinitely arbitrarily small around the origin, and proceed just like the .
The definition of the exponential map is simply the same as that of the regular exponential function as given at , except that the argument $x$ can now be an operator instead of just a number.
Examples:
*
= Baker-Campbell-Hausdorff formula
{c}
{parent=Lie algebra}
{title2=BCH formula}
{wiki=Baker–Campbell–Hausdorff formula}
Solution $Z$ for given $X$ and $Y$ of:
$$
e^Z = e^X e^Y
$$
where $e$ is the .
If we consider just , $Z = X + Y$, but when X and Y are , things are not so simple.
Furthermore, TODO confirm it is possible that a solution does not exist at all if $X$ and $Y$ aren't sufficiently small.
This formula is likely the basis for the . With it, we express the actual in terms of the Lie algebra operations.
Notably, remember that a is just a with one extra product operation defined.
Vector spaces are simple because , so besides the dimension, once we define a , we also define the corresponding .
Since a group is basically defined by what the group operation does to two arbitrary elements, once we have that defined via the , we are basically done defining the group in terms of the algebra.
= Generator of a Lie algebra
{parent=Lie algebra}
= Generators of a Lie algebra
{parent=Lie algebra}
= Generator of the Lie algebra
{synonym}
Cardinality $\leq$ dimension of the vector space.
= Continuous symmetry
{parent=Lie group}
{wiki}
Basically a synonym for which is the way of modelling them.
= Local symmetry
{parent=Continuous symmetry}
{wiki}
Local symmetries appear to be a synonym to , see description at: {full}.
As mentioned at , local symmetries map to forces in the .
Appears to be a synonym for: .
A local symmetry is a transformation that you apply a different transformation for each point, instead of a single transformation for every point.
TODO what's the point of a local symmetry?
Bibliography:
*
* https://physics.stackexchange.com/questions/48188/local-and-global-symmetries
* https://www.physics.rutgers.edu/grad/618/lects/localsym.pdf by Joel Shapiro gives one nice high level intuitive idea:
\Q[In relativistic physics, global objects are awkward because the finite velocity with which effects can propagate is expressed naturally in terms of local objects. For this reason high energy physics is expressed in terms of a field theory.]
* :
* https://www.quora.com/What-does-a-local-symmetry-mean-in-physics
* https://www.quora.com/What-is-the-difference-between-local-and-global-symmetries-in-physics
* https://www.quora.com/What-is-the-difference-between-global-and-local-gauge-symmetry
= Local symmetries of the Lagrangian imply conserved currents
{parent=Local symmetry}
TODO. I think this is the key point. Notably, __ symmetry implies .
More precisely, each [generator of the corresponding Lie algebra] leads to one separate conserved current, such that a single symmetry can lead to multiple conserved currents.
This is basically the version of .
Then to maintain charge conservation, we have to maintain , which in turn means we have to add a as shown at __