- In what sense are these six specifications?
- In the context of the Riemann sphere, with a point at infinity, how are reflection and inversion related?
- How is inversion (polarity?) in projective geometry related to inversion in conformal geometry?
- What is inversion in affine geometry?
- What is inversion in symplectic geometry?
Literature
From Wikipedia: Geometric transformation:
- displacements preserve distances and oriented angles;
- isometries preserve angles and distances;
- similarities preserve angles and ratios between distances;
- affine transformations preserve parallelism;
- projective transformations preserve collinearity;
Each of these classes contains the previous one.
Möbius transformation
Möbius transformation of the complex plane is a rational function of the form
{$$f(z) = \frac{a z + b}{c z + d}$$}
of one complex variable {$z$}; here the coefficients {$a, b, c, d$} are complex numbers satisfying {$ad − bc ≠ 0$}.
Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group {$PGL(2,\mathbb{C})$}. The projective linear group is the general linear group (of invertible matrices) modded out by the center, which are the nonzero scalar transformations. The projective linear group has trivial center.
Over the real and complex numbers, the projective special linear groups are the minimal (centerless) Lie group realizations for the special linear Lie algebra {$\mathfrak{sl}(n)$}: every connected Lie group whose Lie algebra is {$\mathfrak{sl}(n)$} is a cover of {$PSL(n, F)$}. Conversely, its universal covering group is the maximal (simply connected) element, and the intermediary realizations form a lattice of covering groups.
For example, {$SL(2, R)$} has center {$\{±1\}$} and fundamental group {$\mathbb{Z}$}, and thus has universal cover {$SL(2, R)$} and covers the centerless {$PSL(2, R)$}.
The cross-ratio is invariant under the projective transformations of the line.
Classification of Möbius transformations
Non-identity Möbius transformations are commonly classified into parabolic, elliptic and loxodromic (which include hyperbolic).
Parabolic transformations are translations. They are conjugates of
{$$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$}
A shear transformation (a horizontal shear) maps {$(x,y)$} to {$(x+my,y)$}
{$$\begin{pmatrix} 1 & m \\ 0 & 1 \end{pmatrix}$$}
Elliptic transformations are rotations. They are conjugates of
{$$\begin{pmatrix} e^{i\alpha} & 0 \\ 0 & e^{-i\alpha} \end{pmatrix}$$}
{$$\begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$$}
An example of a hyperbolic transform, where {$\theta\in\mathbb{R}$}, is
{$$\begin{pmatrix} e^{\theta} & 0 \\ 0 & e^{-\theta} \end{pmatrix}$$}
More generally, for {$\lambda\in\mathbb{C}$}, we have a loxodromic transform
{$$\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix}$$}
Summary
{$SL(2,\mathbb{R})$} is the special linear group
Reflection | {$(x)\rightarrow (-x)$} | Defining distance. Movement between two points within a single axis-dimension. | hyperbolic element of {$SL(2,\mathbb{R})$} with {$a=-1$} |
Shear | {$(x,y)\rightarrow (x+ay,y)$} | Defining angle. Movement of one axis parallel with regard to another axis. | parabolic element of {$SL(2,\mathbb{R})$} |
Rotation | {$(x,y)\rightarrow (-y,x)$} | Defining angle. Movement of one axis with regard to another axis sharing an origin. | elliptic element of {$SL(2,\mathbb{R})$} and also simple Möbius transformation with {$c=0,b=0,d=1$} and {$|a|=1$}. |
Dilation | {$(x,y)\rightarrow (ax,ay)$} | Defining area. Equal movement on two axes with regard to an absolute reference frame. | simple Möbius transformation with {$c=0,b=0,d=1$} |
Squeeze | {$(x,y)\rightarrow (ax,y/a)$} | Defining area. Compensatory movement on two axes with regard to an absolute reference frame. | hyperbolic element of {$SL(2,\mathbb{R})$} |
Translation | {$(x,y)\rightarrow (x+a,f(x+a))$} or {$z\rightarrow z+b$} | Defining area. Movement on one axis synchronized to another axis by way of an absolute reference frame. | simple Möbius transformation with {$c=0,a=1,d=1$} |
Also consider inversion as in inversive geometry.
Consider how any Moebius transformation is composed of simple ones.
Formulations
Reflections
Given a hyperplane defined by its normal vector {$a$}, reflection is defined by:
{$\operatorname{Ref}_a(v) = v - 2\frac{v\cdot a}{a\cdot a}a$}
But more relevant is simply the one-dimensional transformation {$v\rightarrow -v$}. It determines a length with regard to a zero.
Shear transformation
Shear transformations are expressed two-dimensionally. They determine an angle. They translate one dimension but fix the other. They map {$(x,y)$} to {$(x+my,y)$}
{$$\begin{pmatrix} 1 & m \\ 0 & 1 \end{pmatrix}$$}
Rotations
Rotations are expressed two-dimensionally. They determine an angle. They swap dimensions, reflecting the second one.
{$$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -y \\ x \end{pmatrix}$$}
{$$\begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$$}
Dilations
A dilation determines a two-dimensional area by relating it to another area by way of an additional dimension.
{$(x,y)\mapsto (ax,ay)$}
This can be expressed in terms of an exponential factor such as {$e^{\theta}$} and, in the opposite direction, {$e^{-\theta}$}.
Squeeze
{$(x,y)\mapsto (ax,y/a)$}
A squeeze transformation equates two different areas.
A bilinear form is invariant under this mapping.
It is natural to think of the squeeze mapping as a hyperbolic rotation since {$\{(u,v)\,:\,uv=\mathrm {constant} \}$} is a hyperbola.
Translation
A translation maps {$(x,f(x))\rightarrow (x+a,f(x+a))$}
A translation decomposes an area into a length and a width, as with the Fundamental Theorem of Calculus. For example, it decomposes the dimensions of mood and location in a journey, which can then be multiplied together.
A translation thus decomposes dimensions.
Geometry, Geometries
- Relate "love is a journey" with not just 12 circumstances but also 6 geometric transformations.
Pairs of Geometries
Each of the four geometries would serve to define what we mean by perspective, but especially, how a view from outside of a system (from a higher dimension) and a view inside of a system (a lower dimension) can be considered one and the same. In general, I am thinking that geometry can be thought of as the ways of embedding one space into another space, that is, a lower dimensional space into a higher dimensional space. I imagine that tensors are important as the trivial, "plain vanilla" version of this.
Think of pairs of geometries as defining equivalence classes variously. Equivalence classes are related to actions of symmetry groups.
6 Specifications
The 6 specifications between 4 geometries are transformations which make one geometry more specific than another geometry by introducing orientation, angles and areas. This also makes distance more sophisticated, allowing for negative (oriented) numbers, rational (angular) numbers, and real (continuous) numbers.
Sources to think about
- Sylvain Poirer's list of permutations which I used.
- Grothendieck's six operations:
- pushforward along a morphism and its left adjoint
- compactly supported pushforward and its right adjoint
- tensor product and its adjoint internal hom
- The various ways that we interpret multiplication in arithmetic.
- Möbius transformation combines translation, inversion, reflection, rotation, homothety. See the classification of Moebius transformations. Note also that the classification of elements of SL2(R) includes elliptic (conjugate to a rotation), parabolic (shear) and hyperbolic (squeeze).
- The six transformations in the anharmonic group of the cross-ratio. If ratio is affine invariant, and cross-ratio is projective invariant, what kinds of ratio are conformal invariant or symplectic invariant?
- The 6 specifications can be compared with cinematographic movements of a camera. But I don't know how to think of shear or squeeze mappings in terms of a camera. However, consider what a camera would do to a tiled floor. Shear? Squeeze: the camera looks out onto the horizon?
- Reflection: a camera in a mirror, a frame within a frame...
- Rotation: a camera swivels from left to right, makes a choice, like turning one's head
- Dilation: a camera zooms for the desired composition.
- Translation: a camera moves around.
- Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern.
Ideas about transformations
- Reflection introduces the action of Z2. It is the reflection across the boundary of self and world. (We can later also think of reflection across the horizon around us, as inversion.) This is the parity of multisets (element or not an element). And that circle S02 is then referenced by rotations and shear mapping and all work with angles. And then the relationship between two dimensions is given perhaps by Z2 x S02, the relationship between two axes: x vs. x (dilation), x vs. 1/x (squeeze) and x vs. y (translation).
- Squeeze specification draws a hyperbola (x vs. 1/x). Dilation draws a line (x vs. x). Are there specifications that draw circles (rotation?), ellipses? parabolas?
- Transformacijos sieja nepriklausomus matus.
Reflection
- Flip around our search, turn vector around: (reflection)
Shear
- Shear map takes parallelogram to square, preserves area
- Turn a corner into another dimension
Rotation
- Harmonic analysis, periodic functions, circle are rotation.
- Rotations are multiplicative but not additive. This brings to mind the field with one element.
Dilation
- Dilation (scaling) including negative (flipping). Dilations add absolutely and multiply relatively.
Complex number dilation (rotating).
- Homothety is related to dilation. In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic in
- https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends {\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}}, in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.
- Dilation brings to mind the Cartesian product A x B. There is also the inner (direct) product A + B. How is it related to the disjoint union? And there is the tensor product which I think is like an expansion in terms of A.B and so is like multiplication.
Squeeze
- Squeeze mapping
- Squeeze transformacija trijuose matuose: a b c = 1. Tai simetrinė funkcija.
Translation
- Homotopy is translation.
- Sweep a new dimension in terms of an old dimension (translation)
- Translation - does not affect vectors
Other transformations
- Special conformal is reflection and inversion
- Isometry
- Homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation.
- Affine transformation
- Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (repère mobile).[1] The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand. For instance, in relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Cartan connection.
Liouville's theorem on conformal mappings any smooth conformal mapping on a domain of Rn, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are Möbius transformations (in n dimensions).[2][3] This theorem severely limits the variety of possible conformal mappings in R3 and higher-dimensional spaces. By contrast, conformal mappings in R2 can be much more complicated – for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem.
Similarities In Euclidean geometry, two objects are similar if they have the same shape, or if one is the mirror shape of the other. One can be obtained from the other by scaling and possibly translating, rotating and reflecting.
An inversion models God going beyond himself and likewise us going beyond ourselves. Thus it is very meaningful that all Mobius transformations can be generated by involutions.
- Inversive geometry in higher dimensions
- The transformation by inversion in hyperplanes or hyperspheres in En can be used to generate dilations, translations, or rotations. Indeed, two concentric hyperspheres, used to produce successive inversions, result in a dilation or contraction on the hyperspheres' center. Such a mapping is called a similarity.
- When two parallel hyperplanes are used to produce successive reflections, the result is a translation. When two hyperplanes intersect in an (n–2)-flat, successive reflections produce a rotation where every point of the (n–2)-flat is a fixed point of each reflection and thus of the composition.
- All of these are conformal maps, and in fact, where the space has three or more dimensions, the mappings generated by inversion are the only conformal mappings. Liouville's theorem is a classical theorem of conformal geometry.
- Related to the idea that reflections are inversions: The addition of a point at infinity to the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive geometry is frequently studied then in the presumed context of an n-sphere as the base space.
- The transformations of inversive geometry are often referred to as Möbius transformations. Inversive geometry has been applied to the study of colorings, or partitionings, of an n-sphere.
Thus conformal geometry is the most basic, the plain vanilla, based on the complex numbers, the root system {$A_n$}.
We can get a different geometry by degeneracy of complex numbers into real numbers. This yields a geometry of hyperplanes rather than hyperspheres. There should be two different geometries corresponding to odd and even dimension. What happens to rotations?
And we can also get a different geometry by "folding" ("doubling") the complex numbers to get quaternions. This should add the notion of time and motion, velocity and momentum.
Möbius transformations
Basics of complex analysis
Complex analysis
Wikipedia: Complex analysis
Visual Complex Analysis by Tristan Needham
Harmonic function
Complex algebra
- I dreamed of the complex numbers as a line that curls, winds, rolls up in one way on one end, and in the mirror opposite way on the other end, like rolling up a carpet from both ends. Like a scroll.
- How is the Riemann sheet, winding around, going to a different Riemann sheet, related to the winding number? and the roots of polynomials?
Complex analysis
- Differentiability of a complex function means that it can be written as an infinite power series. So differentiation reduces complex functions to infinite power series. This is analogous to evolution abstracting the "real world" to a representation of it. And differentiation is relevant as a shift from the known to focus on the unknown - the change.
Analytic continuation
- Understand analytic continuation. Can we think of it as cutting the plane into a spiral of width {$e^n-e^{n-1}$}?
- Learn how to extend the Gamma function to the complex numbers.
Riemann Surface
Classification of Riemann surfaces
Schwarz lemma is a result in complex analysis about holomorphic functions from the open unit disk to itself.
Riemann mapping theorem If {$U$} is a non-empty simply connected open subset of the complex number plane {$C$} which is not all of {$C$}, then there exists a biholomorphic mapping {$f$} (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from {$U$} onto the open unit disk {$D=\{z\in \mathbf {C} :|z|<1\}$}.
This mapping is known as a Riemann mapping. The existence of f is equivalent to the existence of a Green’s function.
The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.
(Poincaré–Koebe) Uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.
- The Riemann sphere {$\widehat{\mathbb{C}} \equiv \mathbb {C} ∪ { ∞ }$}, which is isomorphic to the {$\mathbf {P} ^{1}(\mathbf {C} )$}
- The complex plane {$\mathbf {C}$}
- The open disk {$\mathbf {D} \equiv \{z\in \mathbf {C} :|z|<1\}$} which is isomorphic to the upper half-plane {$\mathbf {H} \equiv \{z\in \mathbf {C} :\mathrm {Im} (z)>0\}$}
Consequently, every Riemann surface admits a Riemannian metric of constant curvature.
A Riemann surface is elliptic, parabolic or hyperbolic according to whether its universal cover is isomorphic to {$\mathbf {P} ^{1}(\mathbf {C} )$}, {$\mathbf {C}$} or {$\mathbf {D} $}.
Note that if we start with the open disk and then distinguish z into two conjugates, topside and bottomside, then we get the sphere minus the circle.
Thoughts
- Path-preserving is related to homotopy?
- Length-preserving is related to the unit spheres and rotations in Lie groups. And thus to the real numbers.
- Angle-preserving (conformal) is related to being holomorphic, related to the complex variable.
- Volume-preserving is related to the determinant being constant.
The family of conformal maps in three dimensions is very poor, and essentially contains only Möbius transformations. (Riemann mapping theorem
Literature
Wikipedia
Notes
- Given z1, z2, z3 in projective complex plane, there exists a unique Mobius transformation such that f(z1)=0, f(z2)=1, f(z3)=infinity. Note that there is a fourth symbol z, and they get paired: 0 and infinity, 1 and z.
{$$f(z)=\frac{z-z_1}{z-z_3} \cdot \frac{z_2-z_3}{z_2-z_1} $$}
- The Mobius transformation f(z) which sends f(0)=p, f(1)=r, f(infinity)=s is given by:
{$$f(z)=\frac{zs(p-r) + p(r-s)}{z(p-r) + (r-s)} $$}
- 1/z swaps inside and outside as conjugates, just as it swaps counterclockwise rotations i with counterclockwise rotations -i.
- Give a geometrical interpretation of e.
- SL(2,C) models the transformation in the relationship between two wills: human's and God's. The complex variable describes the will.
- The squeeze function defines area.
- W: Mobius transfomation important examples
- SL(2,C) lines (plus infinity) become circles. Do linear equations become circular equations? What does that mean? Are SL(2,C) circular equations related to the continuum?
- Intuit SL(2,C) as three-dimensional in C (because ad-bc=1 so we lose one complex dimension - intuit that). And in what sense is that different from ad-bc=0 (a line? a one-dimensional subspace?)
- SL(2,C) is the spin relativistic group.
- Investigate: In what sense do the properties of being an inverse (Cramer's rule) dictate the symmetry of SL(2,C)? Because the inverse has to maintain the same form. So why do the b, c switch sign and the a, d switch places? What imposition is made on duality?
- J-invariant is related to SL(2,Z) and monstrous moonshine.
- Why these three structures? How do they relate to the Moebius transformations? And how do these structures relate to the classical Lie families?
- Functions of One Complex Variable John Conway
- Understand the relations between U(1) and electromagnetism, SU(2) and the weak force, SU(3) and the strong force.
- Standard Model
- Electroweak interaction
- Mobius transformations can be composed from translations, dilations, inversions. But dilations (by complex numbers) could be understood as dilations (in positive reals), reflections, and rotations.
- Reconsider what Shu-Hong's thesis has to say about fractions of differences, and how they relate to the Mobius group.
Note that the Mobius transformations classify into types which accord with my six transformations:
- reflection = circular
- shear = parabolic
- rotation = elliptic
- dilation = hyperbolic
- squeeze = internal of hyperbolic (e^t e^{-t}=1)
- translation = internal of parabolic
Need to define "internal". Also, note that reflection is like rotation but more specific. Similarly, is translation like shear, but more specific? Analyze the Mobius group in terms of what it does to circles and lines, and analyze the transformations likewise.
Cross-ratio
- Relate the levels of the foursome with the cross-ratio (for example, how-what are the two points within a circle, and why-whether are the two points outside.) And likewise relate the six pairs of four levels with geometric concepts, the six lines that relate the four points of an inscribed quadrilateral, or the six possible values of the cross-ratio upon permuting its elements.