Anahelixis

Analyticity and the Complex Logarithm We consider a complex-valued function $f$ of one complex variable $z$, and we write: \[ \int_{\mathrm{A}} f(z) d z=(\mathrm{A}, f(z) d z)\] interpreting it as being bilinear in the differential \( f(z) d z \) and the homological $1$-chain \( \mathrm {A} \). Then, if \( f \) is a continuous function of the complex variable \( z \) in the region $\mathbf R$ and if it holds: \[\int_{\mathrm{C}} f(z) d z=0\] about any closed chain $\mathrm {C}$ in \( \mathbf{R}, \) then \( f(z) \) represents an analytic function in $\mathbf {R}$. In practice, if \( f \) is analytic merely on a closed path \( \mathcal{C}, \) it does not follow that \( \int_{\mathrm{C}} f(z) d z=0 . \) In particular, \[\int_{\mathrm{C}} \frac{d z}{z}=2 \pi i\] where \( \mathcal{C} \) is a simple closed curve embracing the origin. This leads to the conclusion that it is precisely the complex logarithm function that characterizes the geometry of the complex plane in all its essential aspe...

Archimedean Area Metronomy by Descent The area of a  spherical dome is equal to the area of a disk, the radius of which is the length of the segment joining the top vertex of the dome to its boundary. Equivalently, if $S$ is this vertex of the spherical dome and $A$ any point on the boundary circle, then the area of the spherical dome is equal to the area of a disk with radius $SA$. If the spherical dome is the whole sphere, then $S$ and $A$ are antipodes, and thus the area of the sphere is $\pi (SA)^2=4 \pi \rho^2$, where $\rho=SA/2$. \[S C=h, O C=R-h, O A=R\] Hence, we have: \[A C^{2}=R^{2}-(R-h)^{2}=2 R h-h^{2}\] \[S A^{2}=A C^{2}+C S^{2}=2 R h\] Thus, we identify $SA$ with the geometric mean of $2SO$ and $SC$: \[S A=\sqrt{2 S O \cdot S C}\] We conclude that the area of a spherical dome is measured by descending to the area of the corresponding disk in a measure-preserving manner. Let \( A_{1}, A_{2} \) be two points on the half-sphere; we consider the domes delimited by the p...

Imaginary Circular Points and the Rectangular Hyperbola Consider the equation of the circle: \[(x-a)^{2}+(y-b)^{2}=r^{2}\] which can be written in the homogeneous form: \[(\xi-a \tau)^{2}+(\eta-b \tau)^{2}-r^{2} \tau^{2}=0\] The intersection with the line at infinity \( \tau=0 \) will thus be given by the equations: \[\xi^{2}+\eta^{2}=0, \quad \tau=0\] The constants \( a, b, \) and \( r, \) which characterise the present circle, do not appear at all. Hence, every circle cuts the line at infinity in the same two fixed points: \[\xi: \eta=\pm i, \quad \tau=0\] which are called the imaginary circular points. The converse is also true: Every curve of second degree, which passes through the imaginary circular points in its plane is a circle. The distance from the origin to the imaginary circular points is not definitely infinite, as might perhaps at first be thought of. Instead, this distance has the form:  \[ \sqrt{x^{2}+y^{2}}=\sqrt{\xi^{2}+\eta^{2}} / \tau=0 / 0 \]  and is con...

Spiral Bridge of Rectification Archimedes thinks of the spiral in terms of the combined mechanical motion of a point undergoing two distinct uniform motions: A uniform motion in a fixed linear direction with constant velocity, and a uniform motion in a circle with constant velocity as well. Both uniform motions start at the same point, which is considered as the pole of the non-uniform combined motion traced by the emerging spiral. The spiral is depicted together with its tangent attached to each point of the motion it traces. After a complete turn, the area of the circle is rectified, through the spiral bridge, in terms of the area of the adjacent to the spiral triangle emerging by means of the attached tangent, and thus the quadrature of the circle follows. The perimeter of the circle is metronomically unrolled to the side of the orthogonal triangle opposite to the angle of the tangent, whose other side is the radius of the circle.

The Spider Web Exponential Epimorphism  The exponential homomorphism \[\exp : \mathbf{C} \rightarrow \mathbf{C}^{\mathbf{x}}, \quad \boldsymbol{z} \mapsto \exp z \] is clearly an epimorphism, that is, surjective, but it is not injective. From the complex exponential group epimorphism and the fact that \( \operatorname{\mid exp} z \mid=1 \text { holds if and only if } z \in \mathbb{R} i \) it can be deduced that \( \exp (i \mathbb{R})=S^{1}, \) where \( S^{1} \) denotes the multiplicative circle group. Hence, we obtain the following  polar spider web epimorphism: The morphism \( p: \mathbb{R} \rightarrow S^{1}, \varphi \mapsto e^{i \varphi} \) is a group epimorphism whose kernel is the group \( 2 \pi \mathbb{Z}, \) and we have: \( p\left(\frac{\pi}{2}\right)=i .\) Note that for \( \varphi, \psi \in \mathbb{R} \) we have: \( p(\varphi+\psi)=\exp (i \varphi+i \psi)=(\exp i \varphi)(\exp i \psi)= \) \( p(\varphi) p(\psi) . \) Thus, \( p \) is an epimorphism, since \( p(\mathbb{R...

Exponential Group Homomorphism - The \( \pi \)-Tuning The morphism of groups: \[\exp : \mathbf{C} \rightarrow \mathbf{C}^{\mathbf{x}}, \quad \boldsymbol{z} \mapsto \exp z \] is a homomorphism of the additive group \( \mathrm{C} \) into the multiplicative group \( \mathrm{C}^{\times}. \) As in the case of a general group homomorphism \( \sigma: G \rightarrow H \) we should consider the image group \( \sigma(G):=\exp (\mathrm{C}), \) and the kernel: \[\text { Ker } \sigma:=\{g \in G: \sigma(g)=\text { neutral element of } H\}\] For the exponential group homomorphism we obtain: \[\exp (\mathrm{C})=\mathrm{C}^{\times}, \quad \operatorname{Ker}(\mathrm{exp})=2 \pi i \mathbb{Z}.\] We conclude that there is a uniquely defined real number \( \pi>0 \), such that the numbers \( 2 n \pi i, n \in \mathbb{Z}, \) constitute the set of numbers mapped on to 1 by the exponential mapping exp \( z \); Equivalently there is a unique tuning real number number \( \pi \) with the property that: \[\{w \i...

The transparent spider web Independent variables are no measurable quantities, they are a cognitive spider web of coordinates arbitrarily spread out over the world. The dependence of a physical quantity on these variables can therefore not be controlled by measurement; only if several physical quantities are in play, one can arrive at relations between the observable quantities by elimination of the independent variables. -H, Weyl

Uniformization  A Schottky group is a group \( \Gamma \) with generators \( \gamma_{1}, \ldots, \gamma_{p}, p \geq 1, \) such that there exist \( 2 p \) disjoint Jordan curves \( l_{1}, l_{1}^{\prime}, \ldots, l_{p}, l_{p}^{\prime} \) bounding a \( 2 p- \) connected region \( D \) for which \( \gamma_{j}(D) \cap D=\varnothing \) and \( \gamma_{j}\left(l_{j}\right)=l_{j}^{\prime}, j=1, \ldots, p . \) The group \( \Gamma \) is necessarily free and purely loxodromic, i.e., all its elements \( \gamma \in \Gamma \backslash\{I\} \) are loxodromic or hyperbolic. The factor \( \Omega(\Gamma) / \Gamma \) is a closed surface of genus \( p . \) The geometric approach to Schottky groups is based on the concept of a fundamental region. A fundamental region of a discontinuous group \( \Gamma \) is defined to be a set \( F \subset \Omega(\Gamma) \) containing one point from each orbit \( \Gamma z_{0}, z_{0} \in \Omega(\Gamma), \) and such that each nonempty component \( F \cap \Omega \) of it ...

Homology Relation Each diagram is a network or linear graph in which each vertex represents a group, and each oriented edge represents a homomorphism connecting the groups at its two ends. A directed path in the network represents the homomorphism which is the composition of the homomorphisms assigned to its edges. Two paths connecting the same pair of vertices usually give the same homomorphism. This is called a commutativity relation. The combinatorially minded individual can regard it as a homology relation due to the presence of 2-dimensional cells adjoined to the graph. If, at some vertex of the graph, two abutting edges are in line, one oriented toward the vertex and the other away, it is frequently the case that the image of the incoming homomorphism coincides with the kernel of the outgoing homomorphism. This property is called exactness. It asserts that the group at the vertex is determined, up to a group extension, by the two neighboring groups, the kernel of the incoming hom...

Quadratrix Synchronization The quadratrix curve is described as follows: ABCD is a square and BED is a quadrantπ of a circle with centre A and radius AB. As the radius AB rotates about A to move to the position AD then the line BC translates at the same rate parallel to itself (for instance, B ′ C′) to terminate at AD. Then. the locus of the point of intersection F of the rotating radius AB (for instance, AE=AB) and the parallel translating line BC is the quadratrix. The quadratrix curve derives its name from the fact that its characteristic property, called its symptoma, encodes a specific relation that decodes into the rectification of the circle. This simply means that the quadratrix curve serves as a bidirectional bridge to square the circle. Originally, it was designed to construct angle division in any given ratio. The rectification property appears to have been discovered afterwards. The curve is generated via synchronized motions, and the ratio of the speeds involved implies π,...

Conformality and Uniformization By an isogonal representation of two areas on one another we mean a one-one, continuous, and continuously differentiable representation of the areas, which is such that two curves of the first area which intersect at an angle \( \alpha \) are transformed into two curves intersecting at the same angle \( \alpha . \) If the sense of rotation of a tangent is preserved, an isogonal transformation is called conformal. Disregarding as trivial the Euclidean magnification of the plane, we may say that the oldest known transformation of this kind is the stereographic projection of the sphere, which was devised by Ptolemy for the representation of the celestial sphere; it transforms the sphere conformally into a plane. A quite different conformal representation of the sphere on a plane area is given by Mercator's Projection; in this the spherical earth, cut along a meridian circle, is conformally represented on a plane strip. Uniformisation Theorem: The un...

Gnomon This discovery has ancient letters: neuter in gender, the word "gnomon", which in the Greek language designated the sundial's axis, signified "that which understands, decides, judges, distinguishes, interprets, yes, that which knows"; as if a thing, already, knew. Intercepting the sunlight, its shadow writes, on the dial itself, a few events of the sky and of the Earth, the solstice, the equinox and the latitude of the site. It functions automatically. "Automatic" means: without the intervention of intention, which is subjective and cognitive. It can be said of the gnomon that it knows the way it is said that it rains. The gnomon looks like a stylus, but no one holds it in their hand. Some things of the world give themselves to be seen to an object that shows them: entirely objective, theory does without any subject. A thing, the gnomon, intervenes in the world, and the world reads on itself the writing drawn by it. This type of intrahardware so...

Analysis-Resolution-Synthesis Assume that the problem is solvable. 1. Analysis: Apagoge/Epagoge. Systematically employ strategies of decomposition, of reduction, and of transformation (via auxiliary constructions, or completion of patterns), follow up until you arrive at a situation/configuration you can control. 2. Resolution: Epilysis. Show that the desired "end configuration" of the Apagoge is attainable without the initial assumption, determine what is "given" in the sense of being constructible from the original information without the Analysis-hypothesis, and express the conditions for solvability (Diorismos). 3. Synthesis: Proof. A classical Apodeixis, usually starting with a construction (Kataskeue) that matches, often even reiterates, the steps of the Resolution. -Pappus

Complexification-Quantum Mechanics-Cosmology The quantum mechanical amplitudes are given by Fourier sums or series of the form \( \sum_{n} a_{n} e^{i t} \) where \( a_{n} \) are complex numbers, and \( t \) is time, whereas probabilities in classical statistic descriptions are given by the similar sums with real \( a_{n}, \) and \( i t \) replaced by inverse (also real) temperature \( -1 / T \).  In this sense, quantum mechanics is a complexification of Ptolemy's epicycles. In the currently acceptable picture, our evolving Universe can be dissected into "space sections" corresponding to the values of global cosmological time (e.g. in the so called Bianchi cosmological models) to each of which a specific temperature of background cosmic radiation can be ascribed. Going back in time, our Universe becomes hotter, so that at the moment of the Big Bang (time \( =0 \) ) its temperature becomes infinite. This provides a highly romantic interpretation of the correspondence \( ...

Mathematical Forms-Abstraction-Axiomatics From the axiomatic point of view, mathematics appears thus as a storehouse of abstract forms-the mathematical structures; and it so happens-without our knowing why-that certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation. Of course, it cannot be denied that most of these forms had originally a very definite intuitive content; but, it is exactly by deliberately throwing out this content, that it has been possible to give these forms all the power which they were capable of displaying and to prepare them for new interpretations and for the development of their full power. It is only in this sense of the word "form" that one can call the axiomatic method a "formalism". The unity which it gives to mathematics is not the armor of formal logic, the unity of a lifeless skeleton; it is the nutritive fluid of an organism at the height of its development, the supple and fertil...