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The 3–4–5 Triangle and 10368 (2⁷·3⁴): A Practical Engineering Note

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Τίτλος: The 3–4–5 Triangle and 10368 (2⁷·3⁴): A Practical Engineering Note
Συγγραφείς: Delph, Andrew
Στοιχεία εκδότη: Zenodo
Έτος έκδοσης: 2025
Συλλογή: Zenodo
Θεματικοί όροι: Mathematics — Geometry — Euclidean geometry (right triangles, circle radii/areas) Mathematics — Number Theory — Diophantine equations / Pythagorean triples Mathematics — Foundations / Notation — Alternative radix systems (duodecimal/base-12)
Περιγραφή: Technical note (corrected version).This short note derives and verifies exact relations between the inradius r and circumradius R of right triangles, using the canonical 3:4:5 triangle as a worked example. We prove scale-invariance of the ratio (R/r)^2 and obtain the closed form (R/r)^2 = 25/4 for 3:4:5. We then present a duodecimal (base-12) normalization based on the constant 10,368, offering a dimensionless way to compare circle/triangle ratios across Pythagorean triples without loss of generality. The note includes compact algebraic derivations, numeric checks, and a concise template to replicate the calculation for other triples (e.g., 5:12:13, 8:15:17). Highlights Closed-form expressions for R and r in right triangles; explicit evaluation for 3:4:5. Proof of scaling invariance; dimensionless comparison via (R/r)^2. A 10,368 (base-12) normalization for cross-triple comparability. Worked examples and a quick-check procedure for additional triples. Intended useGeometric pedagogy, quick verification of circle/triangle relations, normalized comparison of Pythagorean triples, and groundwork for radix-aware geometric indices in number-theoretic studies. Version notesThis “fixed” version corrects arithmetic slips and clarifies the duodecimal normalization step; notation standardized and examples extended. Mini “recipe” box (for methods section) Place the 3 ⁣− ⁣4 ⁣− ⁣53\!-\!4\!-\!53−4−5 triangle; scale all lengths by t=123t=12^3t=123. Then r=t, s=6t=10,368, P=124r=t,\ s=6t=10{,}368,\ P=12^4r=t, s=6t=10,368, P=124; areas scale to 12612^6126. Use modulus N=10,368=27 ⁣⋅ ⁣34N=10{,}368=2^7\!\cdot\!3^4N=10,368=27⋅34 for encoders (128×81), FFT frames, and CRT phase estimation; compare shapes via I10368=N (R/r)2I_{10368}=N\,(R/r)^2I10368=N(R/r)2. Copyright 2025 Andrew Delph
Τύπος εγγράφου: text
Γλώσσα: English
Relation: https://zenodo.org/records/17009996; oai:zenodo.org:17009996; https://doi.org/10.5281/zenodo.17009996
DOI: 10.5281/zenodo.17009996
Διαθεσιμότητα: https://doi.org/10.5281/zenodo.17009996
https://zenodo.org/records/17009996
Rights: Creative Commons Attribution 4.0 International ; cc-by-4.0 ; https://creativecommons.org/licenses/by/4.0/legalcode ; Copyright (C) 2025 Andrew Delph
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  Data: The 3–4–5 Triangle and 10368 (2⁷·3⁴): A Practical Engineering Note
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  Data: <searchLink fieldCode="DE" term="%22Mathematics+—+Geometry+—+Euclidean+geometry+%28right+triangles%22">Mathematics — Geometry — Euclidean geometry (right triangles</searchLink><br /><searchLink fieldCode="DE" term="%22circle+radii%2Fareas%29+Mathematics+—+Number+Theory+—+Diophantine+equations+%2F+Pythagorean+triples+Mathematics+—+Foundations+%2F+Notation+—+Alternative+radix+systems+%28duodecimal%2Fbase-12%29%22">circle radii/areas) Mathematics — Number Theory — Diophantine equations / Pythagorean triples Mathematics — Foundations / Notation — Alternative radix systems (duodecimal/base-12)</searchLink>
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  Data: Technical note (corrected version).This short note derives and verifies exact relations between the inradius r and circumradius R of right triangles, using the canonical 3:4:5 triangle as a worked example. We prove scale-invariance of the ratio (R/r)^2 and obtain the closed form (R/r)^2 = 25/4 for 3:4:5. We then present a duodecimal (base-12) normalization based on the constant 10,368, offering a dimensionless way to compare circle/triangle ratios across Pythagorean triples without loss of generality. The note includes compact algebraic derivations, numeric checks, and a concise template to replicate the calculation for other triples (e.g., 5:12:13, 8:15:17). Highlights Closed-form expressions for R and r in right triangles; explicit evaluation for 3:4:5. Proof of scaling invariance; dimensionless comparison via (R/r)^2. A 10,368 (base-12) normalization for cross-triple comparability. Worked examples and a quick-check procedure for additional triples. Intended useGeometric pedagogy, quick verification of circle/triangle relations, normalized comparison of Pythagorean triples, and groundwork for radix-aware geometric indices in number-theoretic studies. Version notesThis “fixed” version corrects arithmetic slips and clarifies the duodecimal normalization step; notation standardized and examples extended. Mini “recipe” box (for methods section) Place the 3 ⁣− ⁣4 ⁣− ⁣53\!-\!4\!-\!53−4−5 triangle; scale all lengths by t=123t=12^3t=123. Then r=t, s=6t=10,368, P=124r=t,\ s=6t=10{,}368,\ P=12^4r=t, s=6t=10,368, P=124; areas scale to 12612^6126. Use modulus N=10,368=27 ⁣⋅ ⁣34N=10{,}368=2^7\!\cdot\!3^4N=10,368=27⋅34 for encoders (128×81), FFT frames, and CRT phase estimation; compare shapes via I10368=N (R/r)2I_{10368}=N\,(R/r)^2I10368=N(R/r)2. Copyright 2025 Andrew Delph
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        Type: general
      – SubjectFull: circle radii/areas) Mathematics — Number Theory — Diophantine equations / Pythagorean triples Mathematics — Foundations / Notation — Alternative radix systems (duodecimal/base-12)
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