Modular Space: M = 32

Prime Factorization: 2⁵

ConfigurationModulus M = 32
Prime Factorization: 2⁵
Display Mode: Projection Lines
Color Scheme: Coprime vs Non-coprime
Statisticsφ(M)
16
Reducible Residues
16
Reducibility Ratio
50.0%
Farey Channels
6
Color KeyCyan = Irreducible (gcd=1)
Red = Reducible (gcd>1)
Gold = Farey Channels & Lines
White = Outer ring (M circle)
Farey Lines:
Curved gold lines connect each reducible residue r/M on the outer ring to its reduced form r'/M' on the corresponding inner channel ring.
AnalysisModulus M = 32 = 2⁵
Coprime: 16 residues (φ(M))
Reducible: 16 residues
Ratio: 50.0% reducible
Farey Channels:
                                
                            
Key ResultEvery composite M has reducible residues projecting onto simpler Farey channels.
The number projecting to channel M' is exactly d = M/M' (multiplicity).
Note:

                                    r=0 (with gcd(0,M)=M) is always placed at 0° = 3 o'clock position. 
                                    All points are plotted at angle θ = 2πr/M measured counterclockwise from the right.
                                

Research Controls

Animation Controls

Speed: 500 ms

Loop Animation

Record Frames

Modulus Configuration

M = 32

Multi-Modulus View

Enable Multi-Modulus Mode

Display Mode

From: 1

To: 10

Color Scheme

Layer Opacity: 0.7

Stack Layers (Offset)

Visualize multiple moduli simultaneously

View Controls

Zoom: 1.00×

Statistical Summary

Modulus

φ(M)

Coprime

Non-coprime

Channels

Density

50.0%

Classification

Composite ω(M)=3

Prime Constellations

Twin (2)

Cousin (4)

Sexy (6)

Gap 8

Visual Appearance

Global Thickness Scale: 1.00×

Scales all points and lines uniformly. Increase for small M, decrease for large M.

Color Scheme

Point Size: 3.0

Line Opacity: 100%

Geometric Parameters

Ring Spacing: 1.00

Minimum Ring (M'): 1

Rotation Angle: 0°

Point Labels

Show Point Labels

Label Mode

Inner Ring Labels

Label Size: 12px

Label Distance: 1.15×

Distance from point center (1.0 = at point, >1.0 = outside)

Label Background

Label Coprimes Only

Label Every Nth Point: 1

For large M, label every 5th or 10th point to reduce clutter

Display Options

Projection Lines

Channel Rings

Inner Ring Points

Color Inner Points

Show Symmetry Axes

Highlight Prime Residues

Constellation Highlight

Filter Residues

Advanced Analysis

Number Generators

Generate special numbers with rich mathematical structure

Product of first n primes (e.g., 2×3×5 = 30)

n! - divisible by all numbers up to n

Numbers with maximum divisors for their size

Data Export

Export Resolution (px)

3D Farey Divisor Lattice

Hierarchical Structure of Modular Reduction in ℤ/Mℤ

Modulus

M = 30

3D Rotation

X-Tilt: 45°

Y-Rotation: 0°

Z-Spin: 0°

Spacing

Vertical: 1.0

Radial: 1.0

💡 Tip: Drag on canvas to rotate! Press Space for auto-rotate.

Lattice Structure: The divisors of M form a partially ordered set (lattice) under divisibility. This visualization embeds the lattice vertically, with quotient rings ℤ/M'ℤ represented as horizontal layers. Farey chains connect each residue to its canonical representative under the reduction homomorphism.

M = 30

Lattice Levels: 8

Reduction Maps: 22

Ring Dynamics

Inner→Outer: 0°/s

Outer→Inner: 0°/s

Individual: 0°/s

Animate Rings

Display

Show Farey Chains

Show Labels

Auto-Rotate

Color Legend

Coprime (gcd(r,M)=1)

Reducible (gcd(r,M)>1)

Identity (r≡0)

⌨️ Shortcuts:
R Reset rotation
Space Auto-rotate
C Toggle Farey chains
L Toggle labels

Analysis & Patterns

Deep Mathematical Analysis of Modular Structures

Primary Modulus (M₁)

Compare with (M₂) - Optional

Range Analysis

Key Statistics

Farey Channel Breakdown

Gap Distribution

Coprime Density

Euler Totient φ(M)

Divisor Structure

Pattern Detection

Export Analysis

Custom Sequences Explorer

Visualize Any Sequence Modulo M

Sequence Selection

Preset Sequences:

Parameters

Modulus M = 30

Terms to Show: 100

Display Options

Show Connections

Show Value Labels

Highlight Period

Animate Sequence

💡 Examples:
• Fibonacci mod 10 (Pisano period)
• Primes mod 6 (always 1 or 5)
• Squares mod 8 (only 0,1,4)
• Custom: n²+n+41 (prime-rich)

Period Analysis

Distribution

Sequence Values

Sieve Animation

Watch Prime & Modular Sieves in Action

Sieve Algorithm

Parameters

Upper Limit N = 100

Speed: Medium

Slow Fast

Display

Show Numbers

Show Explanation

Show Statistics

Current Status:

Step Explanation

Numbers Processed

Kept (Primes/Coprime)

Eliminated

Progress

Comparative Visualizer

Side-by-Side Comparison of Multiple Moduli

Select Moduli to Compare (up to 4)

M₁

M₂

M₃ (Optional)

M₄ (Optional)

Sync Zoom/Pan

Comparative Statistics

Relationship Analysis

Export Comparison

❦

Modular Reduction Projection

A Geometric Visualization of Residue Class Structures in $\mathbb{Z}/M\mathbb{Z}$

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I. Mathematical Foundations

§1. The Ring of Integers Modulo M

Let $M \in \mathbb{N}$, $M \geq 2$. The ring of integers modulo $M$ is $\mathbb{Z}/M\mathbb{Z} = \{0, 1, 2, \ldots, M-1\}$ with addition and multiplication modulo $M$.

Each element $r \in \mathbb{Z}/M\mathbb{Z}$ represents an equivalence class of integers congruent to $r$ modulo $M$. The structure of this ring is fundamental to algebraic number theory, cryptography, and Diophantine equations.

§2. Euler's Totient Function

Euler's totient function $\varphi(M)$ counts integers $r \in \{1, 2, \ldots, M\}$ where $\gcd(r, M) = 1$. These form the group of units $(\mathbb{Z}/M\mathbb{Z})^*$.

For $M = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$: $$\varphi(M) = M \prod_{i=1}^{k} \left(1 - \frac{1}{p_i}\right)$$

§3. Geometric Representation

We map each residue $r \in \{0, 1, \ldots, M-1\}$ to the unit circle via:

$$z_r = e^{2\pi i r / M}$$

Important: This places $r=0$ at angle $0$ (the positive real axis, or 3 o'clock position), and subsequent residues proceed counterclockwise. This matches the standard unit circle convention in mathematics.

For residue $r$ with $\gcd(r, M) = d$, define $r' = r/d$ and $M' = M/d$. We call $M'$ the reduction channel of $r$.

II. Key Properties

The number of reduction channels equals $\tau(M)$, the divisor function.

Coprime residues ($\gcd(r,M)=1$) appear on the outer circle. Non-coprime residues project to inner circles corresponding to their channels $M' \mid M$.

Prime Constellations

The visualization reveals patterns in coprime pairs $(r_1, r_2)$ with fixed gaps $k = r_2 - r_1$:

Twin Primes ($k=2$): Adjacent coprime residues
Cousin Primes ($k=4$): Gap-4 coprime pairs
Sexy Primes ($k=6$): Gap-6 coprime pairs

III. The Chinese Remainder Theorem

For coprime $m_1, m_2$ with $M = m_1 m_2$, the isomorphism:

$$\mathbb{Z}/M\mathbb{Z} \cong \mathbb{Z}/m_1\mathbb{Z} \times \mathbb{Z}/m_2\mathbb{Z}$$

manifests geometrically as factorization of the circular structure.

IV. Applications

This visualization serves multiple purposes in mathematical research and education:

Visual intuition for abstract algebraic concepts
Pattern discovery before formal proof
Connection between number theory and geometry
Computational experimentation platform

"The purpose of computation is insight, not numbers." — Richard Hamming

V. Special Cases

For prime $M=p$: All $p-1$ non-zero residues are coprime, yielding a uniform circle.

Primorial numbers $M = 2 \cdot 3 \cdot 5 \cdot 7 \cdots$ exhibit maximal channel richness, while highly composite numbers provide intricate symmetry patterns with numerous divisors.

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For interactive exploration, return to the Portal tab above.

About This Research Tool

⁂

Author

Wessen Getachew
@7dview

This Modular Reduction Projection tool represents an expansion of the standalone composite visualization originally featured in the Farey Triangle Explorer. All tools in this series employ the fundamental $2\pi r/M$ unit circle method combined with Farey fraction analysis of $r/M$ to reveal deep patterns in number theory.

Related Research Tools

Explore the complete suite of interactive number theory visualizations:

Farey Triangle & Cayley Transform
Composite visualization suite featuring Farey fractions on the unit circle

GCD Farey Circle
Farey sequences mapped to unit circle with GCD analysis

Composite v2
Composite v2

Prime Constellation Explorer
Advanced twin/cousin/sexy prime detection up to mod 3000

Euler Product Calculator
Computes π and ζ(2) using prime-only relative error methods

Fractional Slice Primality
Unit circle fraction decomposition for primality testing

Overview

The Modular Reduction Projection Research Portal is a comprehensive web-based platform for exploring number-theoretic structures through geometric visualization. It combines rigorous mathematical foundations with interactive exploration capabilities.

Features

Visualization

Real-time SVG rendering with smooth animations
Touch-optimized controls (pinch zoom, pan)
6 different color schemes
Configurable rotation and symmetry display
Prime residue highlighting

Analysis Tools

Automatic pattern detection and analysis
Statistical summaries and comparisons
Prime constellation identification
Mathematical theorem reference

Export Capabilities

High-resolution PNG images (up to 8000×8000px)
Animated GIF sequences
CSV and JSON data formats
LaTeX document generation
Frame-by-frame export for video creation

Mathematical Content

This tool implements and visualizes:

Euler's totient function φ(M)
Greatest common divisor calculations
Prime factorization
Reduction channel decomposition
Chinese Remainder Theorem structure
Prime constellation patterns

Educational Applications

Suitable for:

Undergraduate number theory courses
Abstract algebra visualization
Cryptography foundations
Mathematical research and exploration
Computational mathematics demonstrations

Technical Details

Browser Requirements: Modern browser with JavaScript and SVG support

Performance: Optimized for M up to 10,000

External Libraries: GIF.js (animation export), MathJax (equation rendering)

Usage Tips

Start with small moduli (M < 100) to understand patterns
Try primorials (30, 210, 2310) for rich structure
Use prime moduli (17, 101) to see uniform distributions
Enable animation recording before playback
Experiment with different color schemes
Use filters to isolate specific residue properties

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Exploring the intersection of number theory, geometry, and computation
By Wessen Getachew • @7dview