VectorPaint octahedron wireframes: distinguish opposite vertices from twelve adjacent edges

English translation of the Chinese original, translated on 2026-11-14. Original publication dates, attribution, code and version notes are retained.

An octahedron has eight faces; eight is not its vertex count. Combine a transparent wireframe with a face ledger, keeping projected crossings from creating vertices. This two-dimensional exercise supplies no solid modelling, occlusion solver or structural analysis.

Declare six vertices in a three-dimensional ledger

In external notes set N=(0,0,1), S=(0,0,−1), A=(1,0,0), B=(0,1,0), C=(−1,0,0), D=(0,−1,0). ABCD is an equatorial square with N/S above and below its centre. These six points define the object. Euclid’s octahedron construction provides the regular-octahedron context. Separate wireframe, vertex ledger and face list on the canvas. Three-dimensional coordinates are not three directly supported properties of the two-dimensional drawing interface.

Declare a projection before drawing the equator

Declare projected X=2x+y and Y=y+3z on paper. A,B,C,D project to (2,0),(1,1),(−2,0),(−1,−1), and N/S to (0,3),(0,−3). Apply one display scale and reverse screen y, then connect A→B→C→D→A. The equator appears as a parallelogram under projection although it is a square in the 3D ledger. Measuring projected slopes cannot justify changing original three-dimensional edge lengths.

Connect each pole to all four equatorial vertices

Join N to A,B,C,D and S to A,B,C,D: eight pole edges plus four equatorial edges make twelve. Declare transparent-wireframe display and retain all edges rather than erasing arbitrary crossings as though accurate solid occlusion had been calculated. Put edge numbers near their own midpoints away from intersections. Readers should recover four, not six, edges at N. Projected crossings do not add connections; only the six declared letters are vertices. Any crossing markers must be identified as nonvertex annotations.

Keep three opposite pairs unconnected

N/S, A/C and B/D are opposite pairs, without direct octahedron edges. List them to prevent adding body diagonals merely to make the wireframe look stronger. Six vertices have fifteen unordered pairs; excluding three gives twelve actual edges, independently checking the pole-based count. If NS or AC/BD are drawn for explanation, mark them explicitly as auxiliary dashed diagonals, exclude them from the twelve edges and retain opposite rather than adjacent identities.

List eight triangular faces rather than projected regions

Upper faces are NAB,NBC,NCD,NDA; lower faces SAB,SBC,SCD,SDA. Each uses one equatorial edge and one pole. NAC is not a surface face because AC is not an edge. List all eight; small projected wireframe regions do not become extra faces. Each equatorial edge belongs to one upper and one lower face, while each pole edge belongs to two neighbouring faces at that pole. Twenty-four face-edge appearances divided by two gives twelve. Audit AB and NA individually rather than trusting totals alone.

Check vertex degrees and Euler’s relation separately

Every vertex has degree four: A meets B,D,N,S but not C. Degree sum 24 divided by two yields twelve edges. Check 6−12+8=2, consistent with Euler’s relation for a convex polyhedron. In the 3D ledger every adjacent pair is distance √2 and opposite pairs distance 2, while projected lengths need not match. Keep degree, incidence and length checks separate. Correct totals alone do not turn arbitrary connections into a regular octahedron; the declared coordinates and specified edges remain essential.

Troubleshoot crossing dots and the misleading eight

If eight vertices are counted, remove solid dots mistakenly marking crossings and recover N,S,A,B,C,D. Fifteen edges usually indicate the three opposite-pair diagonals were added. Also audit A-neighbours B,D,N,S, excluding A itself; this catches a duplicate offset by a missing edge despite a correct total. Moving a point to remove a crossing requires updating the projection description or relabelling the display as a purely relational diagram. Any later dashed hidden edges require independent viewpoint and occlusion evidence; a transparent wireframe does not automatically become an accurate solid view. An exported image also contains no rotatable 3D solid.

Reopen and audit six names and eight listed faces

Save, leave and reopen. Without totals follow four neighbours from each of six names, then recover two face incidences per edge from the eight-face list. Check opposite pairs remain nonedges, axes stay auxiliary and crossings remain nonvertices. Delivery should retain transparent-wireframe scope, original coordinates, projection and separately labelled counts 6,12,8. Leave unopened formats and unverified occlusion unknown. Reconstruct the adjacency and explain six vertices despite eight faces, without claiming completion of engineering solid modelling.