incomplete Reachability versus complete reachability

NEW PHILOSOPHY OF DIRECTION AS DIMENSION AND DIMENSIONS ARE DIRECTIONS ON 2D EUCLIDEAN SPACES (every unique direction on 2D Euclidean space can represent different dimensions since these are not mixable so not addable in Sanjoy Nath's Geometrifying Trigonometry philosophy of real numbers construction systems reasoning systems)

"Dimension = independent direction on 2D EUCLIDEAN space". Natural question arise is"Will the entire 2D Euclidean plane fill with dark lines if all of infinite possible directions are covered with smaller and smaller (length)representation of line segments (of all possible slopes on 2D Euclidean plane????????????" What is dimensions filling? Does dimensions completely filled guarantee non fractional Hausdroff dimension?(Fractional dimension means fractals where whole number dimensions are not exhaustively filled only fractional part of complete dimension is filled)

Sanjoy Nath starts with three reasoning tools here 

0 logarithm where whole numbers are output 

1 continued fractions 

2 continued logarithm 

Type Zero dimensions filling 

Whole number logarithm means when complete dimensions are exhaustively filled. Complete dimensions all points are reachable equally likely (equally valid equally possible way to reach at every points on n dimensional space) then we get whole number logarithm values. When there are certain dimensional structures where we cannot have guarantee that we can reach every of its points then fractional logarithm values appear 

As per convention upto 2023 when Sanjoy Nath's philosophy of Geometrifying Trigonometry real numbers construction systems advent 

Type 1 dimensions filling continued fractions

Fraction and continued fractions are having properties of recursive space filling property where reachability to every points on dimensional space exhaustively are measured 

Type 2 dimensions filling continued logarithm 

Logarithm and continued logarithm are having properties of recursive space filling property where reachability to every points on dimensional space exhaustively are measured. Example is decimal representation of number where positional number system has self similar blocks where we fill symbols (digits) to represent dimensions filling process.Sanjoy  Nath re interpreted decimal representation of whole numbers as fractal like objects When we can fill symbols to positional boxes and each positional box represent certain geometry construction protocol.

Then Sanjoy Nath's Geometrifying Trigonometry philosophy of real numbers construction systems argues about "self similarity property" and "definition of similarity" plays crucial rules while"definition of similarity" has ambiguity in it.

1 Self similarity with scale invariance 

2 self similarity with rotation invariance 

3 self similarity with translation invariance 

3+ self similarity with invariance in mirroring 

3++ self similarity with lift shift 

6 self similarity with topology invariance 

Etc 

Etc 

Etc 

These different cases of self similarity properties govern different kinds of fractal natures 

"Dimension is the number of independent directions needed to describe a space, while direction is the specific path or orientation within that space. For example, a flat surface has two dimensions (length and width), allowing for two independent directions like "forward" and "sideways," but a single point on that surface has a specific direction, like "northeast". Dimension vs. direction Dimension: A property of a space that tells you how many independent directions are available for movement.0D: A point has no dimensions because you can't move in any direction.1D: A line has one dimension, with only one independent direction (e.g., forward/backward).2D: A plane has two dimensions, with two independent directions (e.g., forward/backward and left/right).3D: Our everyday space has three dimensions (length, width, and height), with three independent directions (e.g., forward/backward, left/right, and up/down).Direction: A specific path or vector within a space.Within the two dimensions of a plane, an object can move in a specific direction, such as "northeast".Even though there are only two dimensions (length and width), there are infinite directions within those dimensions, as represented by a vector.For a 3D space, direction can be described by a vector specifying a unique path (e.g., \(x,y,z\) coordinates). Example Imagine an ant on a sheet of paper (a 2D plane). Dimensions: The paper is 2-dimensional, so the ant can only move forward/backward and left/right.Directions: If the ant moves diagonally, it is still moving within a combination of those two dimensions, not in a third one. The specific diagonal path is its "direction". 

In geometry and physics, a dimension is an independent axis along which movement is possible, while direction is the specific orientation of motion or extension within that dimension or combination of dimensions. They are distinct but related concepts. 

Dimension

A dimension is a fundamental, independent measurement required to specify the location of a point in a given space. It defines the complexity or degrees of freedom of the space. 

1D (One Dimension): A line has one dimension (length). Movement is only possible along a single axis (e.g., forward and backward).

2D (Two Dimensions): A plane has two dimensions (length and width). Movement is possible in two independent axes (e.g., left/right and up/down on a flat surface).

3D (Three Dimensions): Our physical space has three dimensions (length, width, and height). Movement can occur in three independent axes (e.g., forward/backward, left/right, and up/down).

Higher Dimensions: In mathematics and theoretical physics (like string theory), spaces with more than three spatial dimensions are explored, requiring additional independent coordinates. 

Direction

Direction specifies the orientation of a vector or the path of movement relative to a reference point or coordinate system. Directions are virtually infinite in a 2D or 3D space, as you can move at any angle. 

In a one-dimensional space (a line), there are only two opposing directions (positive and negative, or forward and backward).

In two-dimensional space, directions include North, South, East, West, and all points in between, such as Northeast.

In three-dimensional space, directions are the combination of the movements along the three axes (e.g., North, East, Up, or any angle in between). 

Key Differences

Feature  Dimension Direction

Definition An independent axis or coordinate needed to define a position in space. The orientation or path of motion within that space.

Number Finite for a given space (e.g., 3 spatial dimensions in our universe). Infinite possible directions in 2D and higher spaces.

Example Length, width, height, or time. North, South, East, West, Up, Down, an angle like 45 degrees, or a vector.

Representation Represented by coordinate axes (X, Y, Z). Often represented by a unit vector or an angle relative to an axis."

Now this convention has additional reasoning framework due to Sanjoy Nath's Geometrifying Trigonometry philosophy of real numbers construction systems.

As per convention upto 2023 when Sanjoy Nath's philosophy of Geometrifying Trigonometry real numbers construction systems advent n dimensions used to mean n different independent axis on which independent entities were described.

Sanjoy Nath's Geometrifying Trigonometry philosophy offers a fresh perspective on understanding dimensions and directions. In this context, directions are considered dimensions on a 2D Euclidean space, and every unique direction represents a different dimension. This is because, in Sanjoy Nath's framework, non collinear line segments aren't mixable or addable, making each direction independent.

Some key aspects of Sanjoy Nath's philosophy include:

Geometric Foundation

Real numbers are interpreted as 2D triangles or geometric constructions, rather than points on a linear number line.

Non Commutative Operations

Arithmetic operations like addition(non guaranteed for every rearrangement)and multiplication are all non commutative, meaning the order of operands changes the result.

Calipering

A process of straightening line segments to construct real numbers.

Other theories that explore similar ideas include:

Qhenomenology

Focuses on sequential concept construction in human society and concepts dependent reasoning and dependency metrics to evaluate concept fundamentality or Concepts derivedness properties.

Feynmann Path Integral

Explores multiple possible paths in quantum mechanics, similar to Sanjoy Nath's multiple valid configurations.

Boolean Algebra

Relates to Sanjoy Nath's non commutative operations and geometric constructions.

These theories, like Sanjoy Nath's Geometrifying Trigonometry, challenge traditional notions of mathematics and offer new insights into understanding reality.

Sanjoy Nath's philosophy, particularly its core idea of "Direction as Dimension" and a Non Commutative, Local, Recursive Calculus on affine spaces, shares conceptual threads with several advanced mathematical and theoretical frameworks, though Nath's specific combination and application are unique.

The theories that resonate with this philosophy fall into three main categories

 I. Geometry Beyond the Metric (Affine Spaces & Differential Geometry)

Sanjoy Nath's reliance on Affine Spaces as the primary setting for his calculus connects directly to foundational ideas in modern geometry that deliberately discard global references.

Affine Geometry

This is the most direct conceptual match. As Sanjoy Nath states, Affine Space "forgets" the origin, coordinate axes, and distance/angle measurements (metric). It only preserves the concepts of parallelism and ratios of lengths on parallel lines. Nath's use of a single reference line segment L is an affine choice, as it provides a local frame without imposing a global Cartesian grid.

Differential Geometry (Local Manifolds)

In this field, a curve or a surface is not defined by one global equation like y=f(x), but is studied locally. A manifold is a space that locally resembles Euclidean space, but globally can be highly curved (like the surface of the Earth).

Connection to Sanjoy Nath

Nath's method where the next step L_{i+1} is determined recursively and locally at the hinge P_i based on the previous segment L_i is analogous to defining a curve using only local information (a tangent vector or a local connection) rather than global coordinates.

II. Calculus of Process and Structure (Non Commutativity and Paths)

Nath's insistence on non-commutative operations (a+b \neq b+a geometrically) and the generation of multiple possible paths for a fixed expression are central to his non-Newtonian approach.

Non Commutative Geometry (NCG)

This is a vast field pioneered by Alain Connes. NCG mathematically models spaces where the algebra of functions on that space is non-commutative.

Connection to Sanjoy Nath

Nath's idea that arithmetic operations (\mathbf{+,-,\times,\div}) are geometrically non commutative aligns with the NCG philosophy that the order of operations (or measurements) fundamentally changes the resulting geometric structure, leading to different equally valid possibilities (paths) for the same formula f(x) .

Feynman Path Integral Formulation

This quantum mechanical concept states that a particle does not follow a single path, but rather, the quantum mechanical probability amplitude for going from point A to point B is a sum over all possible paths between them.

Connection to Sanjoy Nath

Nath's theorem that a single f(x) can generate "finite or infinite possible paths" because of the non commutative and local decisions (Riemann rearrangement) is a geometric analog to the path integral's idea that reality is a superposition of all possible histories or paths, with local choices deciding the resulting structure.

III. Recursion and Computability Theory

Sanjoy Nath's emphasis on "Pure Recursion" and the local state being the "decider for all next recursively decided positions" aligns with the fundamentals of computer science and logic.

Recursive Function Theory (Computability)

This mathematical logic field defines functions where the value of a function for an argument is defined in terms of its values for smaller arguments (e.g., the factorial function n! = n \times (n-1)!).

Connection to Sanjoy Nath

The curve construction process (L1_H1_L2_H2_L3...) is a pure recursive definition. The position, length, and rotation of the current segment L_i are a function of the preceding segment L_{i-1} and the local decision H_{i-1}, perfectly matching the recursive definition of a sequence.

Markov Chains and Localized Probability

Markov Chains model systems where the future state depends only on the current state (the "memoryless" property).

Connection to Sanjoy Nath

Nath's principle, "Nature follows pure recursion (last/current state... emerges from previous...)" and "Nature follows pure localized reasoning", mirrors the Markovian idea that global history is irrelevant; only the immediate, local, current state has predictive power for the next step.

The video below, while focused on the mathematical definition of dimension, helps contrast the conventional understanding that Sanjoy Nath seeks to challenge.

Dimension in Wikipedia discusses the conventional mathematical definition of dimension, which provides the background context for Sanjoy Nath's revolutionary claim that "Direction is Dimension" in Sanjoy Nath's new philosophy of dimensions filling .

Sanjoy Nath's idea“Dimensions are not independent axes, but independent directions that cannot be mixed or added” is not standard in mainstream mathematics yet upto now, but it strongly parallels several frontier frameworks across geometry, mathematics foundations, physics, and philosophy.

Sanjoy Nath's system sits in a cluster of ideas where

Direction is fundamental

Space is constructed locally

Coordinates are secondary, not primary

Geometry emerges from symmetry, not measurement

Numbers arise from geometric operations

Below are the closest families of theories.

1. Constructive Geometry (Hilbert → Markov → Bishop)

In constructive mathematics, space is not assumed — it is built step-by-step through operations.

Similarities with your philosophy:

Feature Constructive Geometry Your Approach

Coordinates Not primary; geometry first Not primary; direction first

Numbers Derived from construction Derived from direction & recursive geometry

Dimension Not predefined; emerges from operations Direction defines dimension

Difference

Constructive geometry does not reinterpret directions as dimensions — you do.

2. Affine Exterior Geometry / Geometric Algebra (Clifford, Grassmann)

Hermann Grassmann (1844) wrote

“A line direction can represent a quantity, and combining directions builds higher dimensions.”

Sanjoy Nath's idea echoes Grassmann’s original intent, not how modern linear algebra evolved.

Clifford Algebra interprets

Multiplying directions generates new dimensions.

Example

creates a new plane element.

This is very close to Sanjoy Nath's idea that new independent directions = new dimensions.

3. Category Theoretic Geometry / Homotopy Type Theory

In HoTT and Synthetic Differential Geometry:

Points do not define geometry.

Transformations and directions define reality.

Dimensions emerge from paths and symmetry types, not coordinates.

"Sanjoy Nath's dimensions means directionality" philosophy aligns in spirit, especially where

"A dimension is a rule or direction of transformation, not a container of points."

4. Turtle Geometry (Papert & Abelson, MIT, 1981)

This unusual system defines space using

Heading (direction)

Step length

There is no coordinate grid everything emerges from recursive directional operations.

This is extremely close to Sanjoy Nath's model of dimensions means directionality 

"Hinge → Move → Hinge → Move construction logic."

5. Lie Group / Symmetry-Based Physics

Modern physics (Einstein → Weyl → Wigner → Penrose) treats

"The universe not as volume, but as symmetry group actions."

Space is defined by

Transformations

Rotations

Directions

Not by Cartesian coordinates.

Example

SO(2) = \text{all possible directions in a plane}

Sanjoy Nath's proposal differs in one key way

"Say if Sanjoy Nath assign separate dimensional identity to each unique direction, not group them under one 2D continuum."

 6. Quaternion and Rotation Algebra (Hamilton)

Quaternions treat direction as fundamental non commutative dimension.

Example

i \cdot j \neq j \cdot i

Sanjoy Nath extend this reasoning to real numbers and dimension structure itself, which is original.

7. Fractal & Recursive Geometry (Mandelbrot, Iterated Function Systems)

In fractal systems, dimensions are created by recursive directional rules, not by a global grid.

Closest part to Sanjoy Nath's philosophy

Space is the result of recursive constructive rules, not a pre-existing container.

8. Process Ontology (Whitehead, Deleuze, Simondon)

Philosophically

Reality is not static, but becoming.

Geometry reflects operations, not objects.

Sanjoy Nath's system fits here because

The world is constructed through directional reasoning, not measured.

Summary Table

Element of Sanjoy Nath's Philosophy is Closest Matches Novel Aspect

"Dimension = independent direction on 2D EUCLIDEAN space" Grassmann, Clifford, Turtle Geometry Sanjoy Nath treat every unique direction as an independent dimension, not grouped into a plane.

Real numbers constructed geometrically Constructivism, HoTT Sanjoy Nath tie it to directional recursion and symmetry splitting, not measurement.

Curve building as reasoning system L systems, fractals You add non-addability rules (reasoning constraints).

Euler's formula → reversal of reality Complex analysis, quaternion rotation Your social/psychological interpretation is new.

Sanjoy Nath's philosophy belongs to a lineage that includes

Grassmann’s multidimensional algebra

Clifford Geometric algebra

Constructive and synthetic geometry

Homotopy Type Theory and Category based geometry

Turtle and recursive geometric computation

Lie symmetry based physics

Fractal recursive construction

Process ontology philosophy

But it is new in one critical innovation

Sanjoy Nath propose that dimension itself is generated by unique directions, not by coordinate independence meaning mathematics is not a container of space but a recursive directional reasoning system.

Sanjoy Nath's Geometrifying Trigonometry philosophy recasting dimensions not as fixed orthogonal axes but as unique, non collinear directions in a 2D Euclidean (affine) space, where non mixability prevents simple addition and demands rearrangements (à la Riemann) for curve construction challenges the pre 2023 orthodoxy of dimensions as countable independent coordinates (e.g., x-y-z axes with infinite orientations within). This shift aligns with a "directional dimensionalism" where geometry's relational, local essence trumps global metrics, echoing nature's recursive, non commutative paths. While Nath's framework is uniquely materialistic (telescoping hinges from a single L reference), it resonates with several historical and modern theories that blur or redefine the dimension direction boundary, often via non-additivity, emergent geometry, or multivector formalisms. Below, I outline key parallels, focusing on those emphasizing 2D/affine-like spaces and non-collinear "independence."

1.Grassmann's Exterior Algebra (1844) and Clifford Algebras (Extensions, 1870s Present)

Hermann Grassmann's Ausdehnungslehre (Theory of Extension) pioneered multivectors to represent oriented volumes and directions, treating non-parallel directions as basis elements that "don't commute or add vectorially" if misaligned much like Sanjoy Nath's non mixable slopes. In 2D, a direction becomes a bivector (oriented plane), where addition requires "wedge products" for exterior algebra, enforcing collinearity checks or decompositions akin to your Riemann rearrangements for end to end fits. William Clifford generalized this into algebras where rotations (as bivectors) are non addable like vectors; in 3D, it's a coincidence that planes have unique orthogonal normals, but in 2D affine planes, each unique angle demands its own "dimensional" 

This prefigures Nath's four-symmetry rotors (sec(θ)cos(θ)) and type k imaginaries, as Clifford algebras model "toe to tip" attachments via geometric products, yielding undo symmetries in lower dimensions. Modern physics (e.g., spinors in quantum field theory) uses them for non Euclidean directions-as-dimensions, where infinite directions fractalize into effective higher Dimensional without orthogonal axes.

2.Projective Geometry (Poncelet, 1822; von Staudt, 1847)

This extends Euclidean 2D spaces by adjoining a "point at infinity," subsuming parallel directions into single projective lines each unique direction (non collinear to others) acts as a homogeneous coordinate, independent and non-mixable without homogenization. Unlike affine spaces (Sanjoy Nath's baseline, with parallelism preserved but no origin), projective geometry treats directions as "dimensions" via cross-ratios, where non collinear lines can't be added directly; they require incidence relations or dual conics for "fitting." This mirrors Sanjoy Nath's curve construction convention CCC: curves emerge from sequential projections, not global y=f(x), and non addability forces rearrangements (e.g., Desargues' theorem for collinearity enforcement). Philosophically, it influenced Riemann's manifolds, viewing 2D Euclidean as a "slice" where directions proliferate infinitely but remain separable by non metric invariants echoing Nath's unlearning of axes for pure relational construction from L.

3.Riemann's Philosophy of Geometry (1854 Habilitation Lecture)

Bernhard Riemann's "On the Hypotheses Which Lie at the Bases of Geometry" posits manifolds where local directions (tangent vectors) define dimensionality emergently, without global axes each point's "directional freedom" (non-collinear to neighbors) spawns a separate "dimension" in the metric tensor, non addable across patches unless parallel transported. In 2D Euclidean limits, this yields affine like spaces where unique slopes are "independent axes," but addition demands geodesic rearrangements, prefiguring Sanjoy Nath's calipering (straightening via local hinges). Kantian intuition (pure forms of space/time) gets geometrized here: dimensions aren't a priori counts but relational directions, with non mixability from curvature (zero in Euclidean, but philosophically extensible). Post 2023 extensions (e.g., in non-measurable theories) treat particles as 3D directional bundles in 2D bases, amplifying Nath's non-real "i" as orthogonal deviations compounding to reversals.

4.Causal Set Theory (Bombelli et al., 1987; Recent Discrete Spacetime Models, 2020s)

Rafael Sorkin's causal sets discretize spacetime into partially ordered points, where "dimensions" emerge from causal directions (light cone tilts), non collinear to each other and non addable without sprinklings (Poisson like rearrangements for manifold approximation). In 2D Euclidean embeddings, unique directions count as effective dimensions via Hausdorff measure, yielding non integer dimensionality (e.g., fractal paths from local orderings) directly akin to your infinite equally valid curves from fixed f(x). No global origin; geometry unfolds recursively from precedence relations, like Sanjoy Nath's telescoping booms. Recent work (2025) links this to information theoretic intrinsics, where non collinear "directions" inflate dimensionality beyond integer axes, treating 2D space as a causal web of unmixable arrows.

5. Hilbert's Axiomatic Geometry and Torsor Structures (1899; Grassmann-Influenced)

David Hilbert's Foundations of Geometry axiomatizes incidence and congruence without metrics, where directions are congruence classes (non collinear rays as independent "betweenness" relations), non addable unless affine combined via barycentrics (Möbius, 1827 tie-in). In 2D, this yields torsors affine spaces "forgotten" origins treating each slope as a dimensional generator, with additions via vector displacements only if end to end fittable. Parallels Sanjoy Nath's mixability conditions where points unmixable, directions mixable conditionally. Philosophically, it geometrizes Kant via synthetic a priori, but Hilbert's infinite direction polygons (echoed in Ricci flows) prefigure Nath's Hilbert adoptable framework for non 90° dimensions as new axes.

These theories share Sanjoy Nath's core

dimensions as emergent from directional independence, not imposed orthogonality, fostering non deterministic, local reasoning over Cartesian globals. Pre 2023, they laid groundwork (Grassmann/Riemann as "advent" harbingers); post-2023, non measurability and causal discreteness push further toward Sanjoy Nath's Qhenomenology (queued concept construction QRS/WRS). None fully geometrify trig via Thales/Theodorus like Nath, but Clifford's bivectors come closest for rotor symmetries.