Rotation
The 3-by-4 displacement establishes an oblique edge while the five-unit hypotenuse preserves exact length.
← Art + fabrication studies
3–4–5 Triangle
Digital artifact 001 · Source geometry / 345tri.dxf
3–4–5 Triangle
Progression.
An exact right-triangle relationship becomes a recursive field of orientation, scale, and proportion.
This exhibit preserves the linework of the original CAD drawing while making its construction legible. Reveal each frame, isolate the seed geometry, rotate the field, and hear the same 3:4:5 relationship translated into frequency.
72 line entities03 source layers05 recursive expansions
Complete fieldOuter span · 28.7 units
01 / The seed
One exact
right triangle.
Three integer lengths close perfectly because the square of the long side equals the sum of the other two squares.
3² + 4² = 5²9 + 16 = 25
Area6square units
Acute angles36.87° / 53.13°complementary pair
Perimeter12linear units
ClassificationPrimitive triplecoprime whole-number sides
3165 Hz4220 Hz5275 Hz
Three quiet sine tones · begin with low system volume
02 / Harmonic translation
Ratio can be
heard.
Geometry and music meet through number. Treating 3, 4, and 5 as frequencies produces a compact harmonic field—not because the drawing generates sound, but because both structures preserve the same ratios.
4:3Perfect fourth
5:4Just major third
5:3Just major sixth
03 / Recursive finding
The seed opens into
a silver progression.
The later coordinate spans in this particular DXF follow a second-order recurrence. Each new long projection is twice the previous one plus the term before it. Successive ratios oscillate around—and rapidly approach—the silver ratio.
Source spans0.7→1.4→3.5→8.4→20.3
Successive ratios2.0002.5002.4002.4167→ 1 + √2 = 2.4142…
Recurrence
The expansion is not arbitrary scaling: later coordinate terms obey an = 2an−1 + an−2.
Convergence
The alternating over-and-under ratios settle toward the silver ratio, connecting the seed to Pell-type growth.
04 / Translation
One drawing.
Many material states.
The artifact remains useful because the geometry is resolution-independent. The same source can become interface ornament, CNC toolpath, illuminated engraving, screen pattern, motion field, or spatial study.
Responsive line field
SVG preserves crisp lines from mobile scale to architectural display while allowing staged reveal and interaction.
Engraved light object
Layer separation can guide cutting, engraving depth, material changes, or independently illuminated circuits.
Screen and baffle logic
The nested field can inform perforation, relief, acoustic scattering, glazing graphics, or ornamental assemblies.
Temporal construction
Stages can rotate, accumulate, pulse, or synchronize with a 3:4:5 rhythmic and harmonic composition.
Artifact statusSource preserved · interactive interpretation active
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