# VectorPaint tetrahedron nets: match three vertex copies to one folded corner
Language: en
Canonical: [VectorPaint tetrahedron nets: match three vertex copies to one folded corner]()
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Published: 2026-11-14T00:00:00+08:00
Modified: 2026-11-14T09:33:58+08:00
Author: Bison
Draw four equilateral triangles and distinguish nine planar segments, six folded edges and three seam pairs from vertex copies.
English translation of the [Chinese original](), translated on 2026-11-14. Original publication dates, attribution, code and version notes are retained.
Arrange four equal triangles into one tetrahedron net. Track multiple planar copies of one solid vertex and pair boundary edges into seams. This is a static topology and area diagram, not a production cutting pattern or an automatic folding preview.
## Draw the central equilateral triangle
Use side length 4 with central A=(0,0), B=(4,0), C=(2,2√3). Verify side lengths and altitude rather than substituting an arbitrary isosceles triangle. Choose a 1600×1400 canvas with space for negative coordinates at scale 100, recording the upward mathematical-y to screen conversion. Give central edges fold-line styling and separate the outer contour and labels. [Euclid’s tetrahedron construction]() identifies the regular solid; this planar net is not presented as the historical construction in full.
## Attach one outward triangle to each side
Add D1=(2,−2√3) outside AB, D2=(6,2√3) outside BC and D3=(−2,2√3) outside CA. Connect ABD1, BCD2 and CAD3 outward, never placing a flap over the central face and explaining the overlap with opacity. Verify all three edges of each triangle equal 4 and that it shares one complete edge with the centre, without gaps or half-edge offsets. There are six distinct planar vertex positions. Keep D1/D2/D3 separate rather than prematurely replacing them with one central dot.
## Number four faces and three fold edges
Label ABC face 0 and the flaps faces 1,2,3. AB,BC,CA are shared fold edges; six outer segments await pairing. Draw each fold edge once rather than counting duplicated object strokes as two folds. Count four faces, three internal folds and six boundary segments, totalling nine independent planar segments. Twelve triangle-edge appearances are a different quantity: the three folds occur twice each and the six outer edges once each, yielding twelve.
## Pair boundary seams by their endpoint identities
Pair AD1 with AD3, BD1 with BD2 and CD2 with CD3. Each pair shares the A, B or C identity at one end and merges D copies at the other. Equal lengths alone do not justify pairing AD1 with BD2. Use seam numbers 1,2,3 or distinct line styles for grayscale recovery. Folding merges D1,D2,D3 into one D while A,B,C remain, giving four solid vertices rather than six planar positions or twelve triangle-corner appearances.
## Recover six solid edges from face incidence
The central folds AB,BC,CA remain three solid edges; seam pairs contribute AD,BD,CD, totalling six. Twelve face-edge incidences divided by two faces per edge independently gives six. List faces ABC,ABD,BCD,CAD; each edge should occur in exactly two, such as AB in ABC and ABD. A single occurrence suggests a missing face or wrong seam. The check 4−6+4=2 applies to this convex solid; satisfying the count alone does not prove any arbitrary arrangement folds correctly.
## Preserve area while counting the net boundary separately
Each side-four equilateral face has area 4√3, giving total 16√3. The net boundary has six length-four segments, total 24. The six solid edges also total 24 here, but numerical equality does not make boundary segments and solid edges the same objects. Highlight AD1 and AD3 together: two planar segments represent the single solid edge AD after folding. Use this concrete pair to check object identity. Internal fold strokes do not subtract area. Put any added glue tabs or cutting allowances in a clearly separate copy excluded from the four-face area; thickness, folding loss and material-fit dimensions are not calculated.
## Troubleshoot misplaced flaps and mismatched seams
Hide face colours and trace the centre plus three outward regions, checking no flaps occupy the same region. From each A/B/C follow its two boundary edges to the appropriate D copies, verifying one-to-one seam numbering. If D copies were counted as three separate solid corners, repair the merger record instead of deleting a face to force four vertices. Without a folding feature, retain a paper pairing audit. Record whether a physical paper model was actually made; a static net is not evidence of a completed folding test.
## Reopen and audit planar and solid counts separately
Save, leave and reopen. Count six planar positions, nine independent segments and four faces, then merge seams into four vertices, six edges and four faces. Reconstruct all three seam pairs without answers, checking endpoint identities. Delivery should retain fold-line legend, face numbers, area 16√3 and static-diagram scope. Distinguish folds from cutting outlines. Unmade paper models and unopened formats remain unverified; every edge and face should be traceable rather than inferred manufacturable from an attractive outline.
## Related tutorials and sources
[VectorPaint six-face box nets: verify shared edges and opposite faces](), [VectorPaint wireframe boxes: distinguish visible edges, hidden edges and shared vertices]().
Checked on November 14, 2026: VectorPaint public version 1.0.8, requiring iOS / iPadOS 17.0 or later. Scope follows the [official App Store]() and [Apple public lookup](). Manual exercises use controls actually available in your installed version; external calculations, written records and diagrams imply no built-in automatic analysis. No step-by-step physical-device, private-file or all-export-environment testing is claimed.