# A Geometric System with Infinity at the Origin

## Abstract
This article presents a geometric framework where infinity resides at the origin, an infinitesimal scale defines the system's boundary, and zero is explicitly excluded. Designed for robustness and practical utility, this system reimagines the Cartesian plane and extends to configuration spaces for multi-point systems, integrating scale-shape decomposition and time transformations to facilitate advanced mathematical analysis and computational efficiency. We establish a unified axiomatic foundation that seamlessly connects individual points, configurations, and dynamic behaviors, deriving key properties and demonstrating applications in asymptotic analysis, numerical computation, differential equations, and multi-body dynamics. This transformative tool offers new perspectives for mathematicians and scientists across diverse fields.

## Introduction
Traditional geometry positions zero at the origin and extends outward to infinity, a setup that often complicates analyses at extreme scales—whether approaching infinity or zero. Limits grow unwieldy, unbounded domains challenge numerical methods, and singularities disrupt continuity. Here, we propose an alternative: a plane where infinity anchors the origin, an infinitesimal boundary encircles the system, and zero is absent. For systems of multiple points, we incorporate a scale-shape decomposition to separate overall size from relative configuration and introduce flexible scale mappings and time transformations to simplify dynamics at extreme scales. This unified framework promises theoretical elegance and practical advantages, streamlining computations and illuminating behaviors across scales. Built on a rigorous axiomatic base, this system is both unassailable and versatile, with applications spanning pure mathematics, physics, and computational science.

## Axiomatic Foundation
The system is defined over the hyperreal numbers $*\mathbb{R}$, encompassing finite reals, infinitesimals (positive numbers smaller than any positive real), and infinite numbers (larger than any real). This framework ensures precision in managing infinite and infinitesimal quantities. The foundation consists of a tightly integrated set of axioms that build a cohesive structure for individual points, configurations, and dynamic systems.

1. **The τ-Plane**
The system operates on the τ-plane, defined as $(*\mathbb{R})^2$, the set of all ordered pairs $(\tau_x, \tau_y)$ where $\tau_x, \tau_y \in *\mathbb{R}$.

2. **Mapping to the x-Plane**
For each point $(\tau_x, \tau_y)$ with $\tau_x \neq 0$ and $\tau_y \neq 0$, there corresponds a point $(x, y) = \left( \frac{1}{\tau_x}, \frac{1}{\tau_y} \right)$ in the x-plane, defined as $(*\mathbb{R} \setminus \{0\})^2$, excluding points where $x = 0$ or $y = 0$.

3. **Origin as Infinity**
The origin $(0,0)$ in the τ-plane represents infinite scale in the x-plane. If $\tau_x$ and $\tau_y$ are infinitesimal, then $x = \frac{1}{\tau_x}$ and $y = \frac{1}{\tau_y}$ are infinite hyperreals, with signs determined by $\tau_x$ and $\tau_y$.

4. **Infinitesimal Boundary**
Points where $|\tau| = \sqrt{\tau_x^2 + \tau_y^2}$ is infinite in the hyperreal sense correspond to points in the x-plane where $x$ and $y$ are infinitesimal, bounding the system at the smallest scales.

5. **Scale Circles**
For each hyperreal $\rho > 0$, the circle $|\tau| = \rho$ in the τ-plane defines a locus where $|x|$ and $|y|$ are of order $\frac{1}{\rho}$. When $\rho$ is infinitesimal, $|x|$ and $|y|$ are infinite; when $\rho$ is infinite, $|x|$ and $|y|$ are infinitesimal.

6. **Scaling Identity**
For an infinitesimal $\delta > 0$ and an infinite hyperreal $H = \frac{1}{\delta}$, the product $H \cdot \delta = 1$, formalizing the reciprocal link between infinite and infinitesimal scales.

7. **Directional Continuity**
The mapping preserves directional behavior: as $\tau_x$ transitions from positive to negative through infinitesimals around zero, $x = \frac{1}{\tau_x}$ moves from positive infinity to negative infinity, and similarly for $\tau_y$ and $y$.

8. **Configuration Space for Multiple Points**
For a system of $(n)$ points, each point $(i)$ is associated with a vector of properties $(f({p}_i) = (p_{i1}, p_{i2}, \dots, p_{ik}))$, where each $( p_{ij})$ represents a specific attribute relevant to the system (e.g., mass, charge, wealth, population size). The positions of the points are denoted by $(f({r}_1), \dots, f({r}_n) \in \mathbb{R}^d )$. Depending on the context, additional constraints (such as fixing a central tendency) may be applied.

9. **Scale Factor**
Define the scale factor $s = \sqrt{\frac{\sum_{i=1}^n m_i |\mathbf{r}_i|^2}{M}}$, where $M = \sum_{i=1}^n m_i$, representing the overall size of the configuration.

10. **Flexible Scale Coordinate Mapping**
Introduce the flexible scale coordinate $\sigma = \log s$, where $\sigma \in (-\infty, \infty)$. This maps large separations ($s \to \infty$) to $\sigma \to \infty$ and small separations ($s \to 0$) to $\sigma \to -\infty$.

11. **Shape and Orientation Coordinates**
Define shape coordinates $\theta$, which are scale-invariant and describe the relative configuration, and orientation coordinates $\phi$, which account for rotation. For three bodies in 2D, $\theta$ may be coordinates on the shape sphere, and $\phi$ a rotation angle.

12. **Full Parameterization of Configurations**
Parameterize the configuration space by $(\sigma, \phi, \theta)$, where $\sigma \in (-\infty, \infty)$, $\phi$ lies in the rotation group, and $\theta$ resides in the shape space.

13. **Behavior at Configuration Extremes**
As $\sigma \to \infty$, configurations approach infinite separation ($s \to \infty$); as $\sigma \to -\infty$, they approach collisions or tight clustering ($s \to 0$), with shape determined by $\theta$.

14. **Interaction with Individual τ-Mappings**
Each point’s position $\mathbf{r}_i$ can be individually mapped to $\mathbf{\tau}_i = \left( \frac{1}{x_i}, \frac{1}{y_i} \right)$, but the global $(\sigma, \phi, \theta)$ provides a system-wide perspective.

15. **Scale-Adjusted Time Transformation for Dynamics**
For dynamic systems, introduce a time variable $\tau$ such that $\frac{dt}{d\tau} = e^{\alpha \sigma}$, where $\alpha > 0$ is chosen based on the system (e.g., $\alpha = 1$ for inverse-square forces). This adjusts the time parametrization to simplify kinetic energy and dynamics.

16. **Kinetic Energy Simplification**
The time transformation ensures that the kinetic energy $T$, in $\tau$-time, is expressed as $T = \frac{1}{2} M \left( \frac{d\sigma}{d\tau} \right)^2 + T_{\text{shape}}\left( \theta, \frac{d\theta}{d\tau}, \phi, \frac{d\phi}{d\tau} \right)$, free of exponential factors in $\sigma$, making large-scale behavior tractable.

## Derived Properties
The axioms yield a rich set of properties that enhance the system’s utility for individual points, configurations, and dynamic systems.

1. **Compactification of Space**
The unbounded x-plane is compactified in the τ-plane, with infinite distances collapsing to $\tau = (0,0)$ and infinitesimal distances expanding to the boundary.

2. **Metric Inversion**
Euclidean distance $d_\tau = \sqrt{\tau_x^2 + \tau_y^2}$ inverts in the x-plane: small $d_\tau$ maps to large $d_x = \sqrt{x^2 + y^2}$, and large $d_\tau$ maps to small $d_x$.

3. **Asymptotic Duality**
Behavior as $x \to \infty$ aligns with $\tau \to 0$, and as $x \to 0$, $|\tau| \to \infty$, enabling dual limit analysis.

4. **Series Transformation**
A Laurent series around $x = \infty$ becomes a Taylor series around $\tau = 0$.

5. **Scale Compactification in Configurations**
Infinite separations ($s \to \infty$) map to $\sigma \to \infty$, with the time transformation ensuring dynamics remain finite.

6. **Shape Invariance**
Shape coordinates $\theta$ isolate relative geometry from scale, remaining finite across all $\sigma$.

7. **Energy Decomposition**
Potentials like gravitational energy can be expressed as $V = -e^{-\sigma} U(\theta)$, separating scale and shape.

8. **Simplified Dynamics**
The kinetic energy in $\tau$-time is free of exponential growth, ensuring manageable analysis at large scales.

## Practical Applications
The framework provides powerful tools for analysis and computation.

1. **Asymptotic Analysis**
- **Example**: Analyze $f(x) = x^2$ as $x \to \infty$ via $f\left(\frac{1}{\tau}\right) = \frac{1}{\tau^2}$ near $\tau = 0$.

2. **Numerical Computation**
- **Example**: Transform unbounded integrals like $$ \int_1^\infty \int_1^\infty e^{-(x^2 + y^2)} \, dx \, dy $$ to bounded domains in the τ-plane.

3. **Differential Equations**
- **Example**: Solve singular equations like $x^2 u'' + u = 0$ by mapping to regular problems in τ.

4. **Multi-Body Dynamics**
- **Example**: Study three-body interactions with $(\sigma, \phi, \theta)$, using $\tau$-time to simplify large-separation dynamics.

5. **Gravitational Systems**
- **Example**: Express potentials as $V = -e^{-\sigma} U(\theta)$, facilitating perturbation methods.

6. **Numerical Simulation**
- **Example**: Simulate multi-body motion with $\sigma \in (-\infty, \infty)$, leveraging the time transformation for efficient computation.

## Conclusion
This geometric system, with infinity at the origin and zero excluded, offers a robust framework for mathematics and physics. Its unified axioms for points, configurations, and dynamics provide tools for asymptotic analysis, computation, and multi-body problems, surpassing traditional geometry. Future work may refine coordinates or integrate computational methods, solidifying its role as a versatile instrument in science.