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| <title>Mathematics Batch 04 - Geometry & Topology - Programming Framework Analysis</title> | |
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| <h1>Mathematics Batch 04 - Geometry & Topology - Programming Framework Analysis</h1> | |
| <p>This document presents geometry and topology processes analyzed using the Programming Framework methodology. Each process is represented as a computational flowchart with standardized color coding: Red for triggers/inputs, Yellow for structures/objects, Green for processing/operations, Blue for intermediates/states, and Violet for products/outputs. Yellow nodes use black text for optimal readability, while all other colors use white text.</p> | |
| <h2>1. Euclidean Geometry Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| graph TD | |
| A1[Geometric Objects] --> B1[Euclidean Axioms] | |
| C1[Point Line Plane] --> D1[Distance Measurement] | |
| E1[Angle Measurement] --> F1[Geometric Constructions] | |
| B1 --> G1[Parallel Postulate] | |
| D1 --> H1[Pythagorean Theorem] | |
| F1 --> I1[Circle Constructions] | |
| G1 --> J1[Triangle Properties] | |
| H1 --> K1[Area Calculations] | |
| I1 --> L1[Geometric Proofs] | |
| J1 --> M1[Congruence Theorems] | |
| K1 --> L1 | |
| L1 --> N1[Similarity Theorems] | |
| M1 --> O1[Geometric Transformations] | |
| N1 --> P1[Coordinate Geometry] | |
| O1 --> Q1[Euclidean Geometry Process] | |
| P1 --> R1[Euclidean Geometry Validation] | |
| Q1 --> S1[Euclidean Geometry Verification] | |
| R1 --> T1[Euclidean Geometry Result] | |
| S1 --> U1[Euclidean Geometry Analysis] | |
| T1 --> V1[Euclidean Geometry Parameters] | |
| U1 --> W1[Euclidean Geometry Output] | |
| V1 --> X1[Euclidean Geometry Analysis] | |
| W1 --> Y1[Euclidean Geometry Final Result] | |
| X1 --> Z1[Euclidean Geometry Analysis Complete] | |
| style A1 fill:#ff6b6b,color:#fff | |
| style C1 fill:#ff6b6b,color:#fff | |
| style E1 fill:#ff6b6b,color:#fff | |
| style B1 fill:#ffd43b,color:#000 | |
| style D1 fill:#ffd43b,color:#000 | |
| style F1 fill:#ffd43b,color:#000 | |
| style G1 fill:#ffd43b,color:#000 | |
| style H1 fill:#ffd43b,color:#000 | |
| style I1 fill:#ffd43b,color:#000 | |
| style J1 fill:#ffd43b,color:#000 | |
| style K1 fill:#ffd43b,color:#000 | |
| style L1 fill:#ffd43b,color:#000 | |
| style M1 fill:#ffd43b,color:#000 | |
| style N1 fill:#ffd43b,color:#000 | |
| style O1 fill:#ffd43b,color:#000 | |
| style P1 fill:#ffd43b,color:#000 | |
| style Q1 fill:#ffd43b,color:#000 | |
| style R1 fill:#ffd43b,color:#000 | |
| style S1 fill:#ffd43b,color:#000 | |
| style T1 fill:#ffd43b,color:#000 | |
| style U1 fill:#ffd43b,color:#000 | |
| style V1 fill:#ffd43b,color:#000 | |
| style W1 fill:#ffd43b,color:#000 | |
| style X1 fill:#ffd43b,color:#000 | |
| style Y1 fill:#ffd43b,color:#000 | |
| style Z1 fill:#ffd43b,color:#000 | |
| style M1 fill:#51cf66,color:#fff | |
| style N1 fill:#51cf66,color:#fff | |
| style O1 fill:#51cf66,color:#fff | |
| style P1 fill:#51cf66,color:#fff | |
| style Q1 fill:#51cf66,color:#fff | |
| style R1 fill:#51cf66,color:#fff | |
| style S1 fill:#51cf66,color:#fff | |
| style T1 fill:#51cf66,color:#fff | |
| style U1 fill:#51cf66,color:#fff | |
| style V1 fill:#51cf66,color:#fff | |
| style W1 fill:#51cf66,color:#fff | |
| style X1 fill:#51cf66,color:#fff | |
| style Y1 fill:#51cf66,color:#fff | |
| style Z1 fill:#51cf66,color:#fff | |
| style Z1 fill:#b197fc,color:#fff | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Geometric Methods | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Geometric Operations | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 1.</strong> Euclidean Geometry Process. This geometry process visualization demonstrates Euclidean geometric constructions and proofs. The flowchart shows geometric object inputs and axioms, geometric methods and theorems, geometric operations and constructions, intermediate results, and final Euclidean geometry outputs. | |
| </div> | |
| </div> | |
| <h2>2. Topology Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| graph TD | |
| A2[Topological Space] --> B2[Open Sets] | |
| C2[Topology Definition] --> D2[Neighborhood Analysis] | |
| E2[Continuity Analysis] --> F2[Homeomorphism] | |
| B2 --> G2[Topology Verification] | |
| D2 --> H2[Connectedness] | |
| F2 --> I2[Compactness] | |
| G2 --> J2[Separation Axioms] | |
| H2 --> K2[Homotopy Theory] | |
| I2 --> L2[Fundamental Group] | |
| J2 --> M2[Topological Invariants] | |
| K2 --> L2 | |
| L2 --> N2[Covering Spaces] | |
| M2 --> O2[Topology Analysis] | |
| N2 --> P2[Topology Validation] | |
| O2 --> Q2[Topology Process] | |
| P2 --> R2[Topology Verification] | |
| Q2 --> S2[Topology Result] | |
| R2 --> T2[Topology Output] | |
| S2 --> U2[Topology Analysis] | |
| T2 --> V2[Topology Parameters] | |
| U2 --> W2[Topology Final Result] | |
| V2 --> X2[Topology Analysis] | |
| W2 --> Y2[Topology Analysis Complete] | |
| X2 --> Z2[Topology Analysis Complete] | |
| style A2 fill:#ff6b6b,color:#fff | |
| style C2 fill:#ff6b6b,color:#fff | |
| style E2 fill:#ff6b6b,color:#fff | |
| style B2 fill:#ffd43b,color:#000 | |
| style D2 fill:#ffd43b,color:#000 | |
| style F2 fill:#ffd43b,color:#000 | |
| style G2 fill:#ffd43b,color:#000 | |
| style H2 fill:#ffd43b,color:#000 | |
| style I2 fill:#ffd43b,color:#000 | |
| style J2 fill:#ffd43b,color:#000 | |
| style K2 fill:#ffd43b,color:#000 | |
| style L2 fill:#ffd43b,color:#000 | |
| style M2 fill:#ffd43b,color:#000 | |
| style N2 fill:#ffd43b,color:#000 | |
| style O2 fill:#ffd43b,color:#000 | |
| style P2 fill:#ffd43b,color:#000 | |
| style Q2 fill:#ffd43b,color:#000 | |
| style R2 fill:#ffd43b,color:#000 | |
| style S2 fill:#ffd43b,color:#000 | |
| style T2 fill:#ffd43b,color:#000 | |
| style U2 fill:#ffd43b,color:#000 | |
| style V2 fill:#ffd43b,color:#000 | |
| style W2 fill:#ffd43b,color:#000 | |
| style X2 fill:#ffd43b,color:#000 | |
| style Y2 fill:#ffd43b,color:#000 | |
| style Z2 fill:#ffd43b,color:#000 | |
| style M2 fill:#51cf66,color:#fff | |
| style N2 fill:#51cf66,color:#fff | |
| style O2 fill:#51cf66,color:#fff | |
| style P2 fill:#51cf66,color:#fff | |
| style Q2 fill:#51cf66,color:#fff | |
| style R2 fill:#51cf66,color:#fff | |
| style S2 fill:#51cf66,color:#fff | |
| style T2 fill:#51cf66,color:#fff | |
| style U2 fill:#51cf66,color:#fff | |
| style V2 fill:#51cf66,color:#fff | |
| style W2 fill:#51cf66,color:#fff | |
| style X2 fill:#51cf66,color:#fff | |
| style Y2 fill:#51cf66,color:#fff | |
| style Z2 fill:#51cf66,color:#fff | |
| style Z2 fill:#b197fc,color:#fff | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Topology Methods | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Topology Operations | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 2.</strong> Topology Process. This topology process visualization demonstrates topological space analysis and invariants. The flowchart shows topological space inputs and definitions, topology methods and properties, topology operations and analysis, intermediate results, and final topology outputs. | |
| </div> | |
| </div> | |
| <h2>3. Differential Geometry Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| graph TD | |
| A3[Manifold] --> B3[Tangent Space] | |
| C3[Metric Tensor] --> D3[Curvature Analysis] | |
| E3[Geodesic Equations] --> F3[Parallel Transport] | |
| B3 --> G3[Vector Fields] | |
| D3 --> H3[Riemann Curvature] | |
| F3 --> I3[Levi Civita Connection] | |
| G3 --> J3[Lie Derivatives] | |
| H3 --> K3[Ricci Curvature] | |
| I3 --> L3[Scalar Curvature] | |
| J3 --> M3[Differential Forms] | |
| K3 --> L3 | |
| L3 --> N3[Exterior Derivatives] | |
| M3 --> O3[Differential Geometry Analysis] | |
| N3 --> P3[Differential Geometry Validation] | |
| O3 --> Q3[Differential Geometry Process] | |
| P3 --> R3[Differential Geometry Verification] | |
| Q3 --> S3[Differential Geometry Result] | |
| R3 --> T3[Differential Geometry Output] | |
| S3 --> U3[Differential Geometry Analysis] | |
| T3 --> V3[Differential Geometry Parameters] | |
| U3 --> W3[Differential Geometry Final Result] | |
| V3 --> X3[Differential Geometry Analysis] | |
| W3 --> Y3[Differential Geometry Analysis Complete] | |
| X3 --> Z3[Differential Geometry Analysis Complete] | |
| style A3 fill:#ff6b6b,color:#fff | |
| style C3 fill:#ff6b6b,color:#fff | |
| style E3 fill:#ff6b6b,color:#fff | |
| style B3 fill:#ffd43b,color:#000 | |
| style D3 fill:#ffd43b,color:#000 | |
| style F3 fill:#ffd43b,color:#000 | |
| style G3 fill:#ffd43b,color:#000 | |
| style H3 fill:#ffd43b,color:#000 | |
| style I3 fill:#ffd43b,color:#000 | |
| style J3 fill:#ffd43b,color:#000 | |
| style K3 fill:#ffd43b,color:#000 | |
| style L3 fill:#ffd43b,color:#000 | |
| style M3 fill:#ffd43b,color:#000 | |
| style N3 fill:#ffd43b,color:#000 | |
| style O3 fill:#ffd43b,color:#000 | |
| style P3 fill:#ffd43b,color:#000 | |
| style Q3 fill:#ffd43b,color:#000 | |
| style R3 fill:#ffd43b,color:#000 | |
| style S3 fill:#ffd43b,color:#000 | |
| style T3 fill:#ffd43b,color:#000 | |
| style U3 fill:#ffd43b,color:#000 | |
| style V3 fill:#ffd43b,color:#000 | |
| style W3 fill:#ffd43b,color:#000 | |
| style X3 fill:#ffd43b,color:#000 | |
| style Y3 fill:#ffd43b,color:#000 | |
| style Z3 fill:#ffd43b,color:#000 | |
| style M3 fill:#51cf66,color:#fff | |
| style N3 fill:#51cf66,color:#fff | |
| style O3 fill:#51cf66,color:#fff | |
| style P3 fill:#51cf66,color:#fff | |
| style Q3 fill:#51cf66,color:#fff | |
| style R3 fill:#51cf66,color:#fff | |
| style S3 fill:#51cf66,color:#fff | |
| style T3 fill:#51cf66,color:#fff | |
| style U3 fill:#51cf66,color:#fff | |
| style V3 fill:#51cf66,color:#fff | |
| style W3 fill:#51cf66,color:#fff | |
| style X3 fill:#51cf66,color:#fff | |
| style Y3 fill:#51cf66,color:#fff | |
| style Z3 fill:#51cf66,color:#fff | |
| style Z3 fill:#b197fc,color:#fff | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Differential Methods | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Differential Operations | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 3.</strong> Differential Geometry Process. This differential geometry process visualization demonstrates manifold analysis and curvature calculations. The flowchart shows manifold inputs and metric tensors, differential methods and connections, differential operations and curvature analysis, intermediate results, and final differential geometry outputs. | |
| </div> | |
| </div> | |
| <div class="navigation"> | |
| <h3>Navigation</h3> | |
| <div class="nav-links"> | |
| <a href="mathematics_index.html" class="nav-link">← Back to Mathematics Index</a> | |
| <a href="mathematics_batch_03.html" class="nav-link">← Previous: Abstract Algebra</a> | |
| <a href="mathematics_batch_05.html" class="nav-link">Next: Applied Mathematics →</a> | |
| <a href="index.html" class="nav-link">Programming Framework Home</a> | |
| </div> | |
| </div> | |
| <div class="footer"> | |
| <p><strong>Generated using the Programming Framework methodology</strong></p> | |
| <p>Each flowchart preserves maximum detail through optimized Mermaid configuration</p> | |
| <div class="contact-info"> | |
| <p><strong>Gary Welz</strong></p> | |
| <p>Retired Faculty Member</p> | |
| <p>John Jay College, CUNY (Department of Mathematics and Computer Science)</p> | |
| <p>Borough of Manhattan Community College, CUNY</p> | |
| <p>CUNY Graduate Center (New Media Lab)</p> | |
| <p>Email: gwelz@jjay.cuny.edu</p> | |
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