# VectorPaint triangular-prism counts: check six minus nine plus five and an edge subdivision
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Canonical: [VectorPaint triangular-prism counts: check six minus nine plus five and an edge subdivision]()
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English: [English HTML]()
Published: 2026-11-15T00:00:00+08:00
Modified: 2026-11-15T09:50:55+08:00
Author: Bison
List vertices, edges and faces before distinguishing geometric corners from a subdivided graph vertex.
English translation of the [Chinese original](), translated on 2026-11-15. Original publication dates, attribution, code and version notes are retained.
A projected crossing need not be a vertex, and a dot on an edge need not add a geometric corner. Draw a triangular prism in VectorPaint, check 6 vertices,9 edges,5 faces, then explicitly subdivide one graph edge to obtain 7,10,5 while retaining two. Keep geometric and graph conventions distinct, without automatic topology detection.
## Use two triangles as ends of one prism
On an approximately 1800×1500 canvas draw ABC, then translate it into A′B′C′ and join corresponding vertices. Example front screen points are (400,1000),(900,1000),(500,500), translated (200,-150). This oblique projection gives no three-dimensional volume from screen lengths. Preserve endpoint order and six identities, separating end faces, connecting/hidden edges and labels. Large dots must not turn crossings into nodes.
## List six vertices as two groups of three
List A,B,C and A′,B′,C′, exactly six. A prime is part of a name, not an edge, and projected overlap does not merge spatial vertices. Add no solid node at an unconnected crossing. Adjust an ambiguous projection instead. Here vertices mean prism corners; later P requires an explicit graph-subdivision convention, not a silent change within one ledger.
## Count nine edges as three plus three plus three
Count front AB,BC,CA; rear A′B′,B′C′,C′A′; and connectors AA′,BB′,CC′: 3+3+3=9. Hidden/dashed edges remain, and shared face edges count once. Six degree-three vertices give 18 incidences, divided by two to recover 9. Eighteen confuses incidences with edges; six may omit the rear triangle. Visibility, colour and width change no model count.
## List two triangular and three lateral faces
Faces are ABC, A′B′C′ and ABB′A′,BCC′B′,CAA′C′: five. Projected lateral parallelograms are not split by decorative diagonals. Face-edge incidences total 2×3+3×4=18; each closed-prism edge belongs to two faces, giving 9. Screen background is no sixth polyhedral face. In a plane embedding one of the five serves as the exterior face, rather than adding exterior to all five.
## Use six minus nine plus five as a consistency check
Use V=6,E=9,F=5, yielding 2. This checks consistency for the closed convex prism, not complete correctness from one number. Adding one spurious vertex and edge may still give two, so retain all ledgers. Visible colour regions/segments are not face/edge definitions. Removing an end changes to a surface with boundary; do not retain the closed-model acceptance condition. Keep both ends intact here.
## Declare graph vertices when subdividing one edge
In a separate copy place interior P on AB, explicitly replacing AB by AP and PB. P is a graph vertex, not a new geometric corner. Thus V=7,E=10,F=5 still gives two. An annotation-only P leaves geometric counts 6/9/5; never count AB,AP,PB all together. P has degree two, no third branch. This declared subdivision illustrates invariant counts without automatic node creation from every dot.
## Troubleshoot projected crossings and double-counted shared edges
For seven vertices without P inspect projected crossing dots. For twelve edges inspect repeated shared outlines. Keep each named edge once, regardless of overlapping strokes. Hidden lines preserve identities. Subdivision replaces one by two, net +1, not +2. Recheck ledgers and V/E/F after each correction instead of editing the final equation. Keep original and P versions separate to avoid mixed edge names.
## Reopen both versions and restate their counting conventions
Save native work, export both PNG versions and reopen. Check six corners/nine edges, then AP,PB and degree-two P in the subdivision. Move labels rather than vertices; inspect visibility when dashes disappear, without deleting model edges. Reopen editable groups and retain the two-triangle/three-quadrilateral face ledger. Explain 6/9/5 versus 7/10/5 and no extra background face. No arbitrary holed-surface validation, automatic topology or three-dimensional precision is claimed.
## Related tutorials and sources
[VectorPaint octahedron wireframes: distinguish opposite vertices from twelve adjacent edges](); [VectorPaint wireframe boxes: distinguish visible edges, hidden edges and shared vertices]().
Sources checked on 15 November 2026: the [official VectorPaint App Store listing]() and [Apple public lookup]() show version 1.0.8, requiring iOS / iPadOS 17.0 or later. Draw manually with published shapes, paths, nodes, layers and text. Conditions, scales and layouts are original exercise arrangements; check installed controls. Keep editable native work and PNG previews. SVG/PDF may retain any bitmap content used. External calculations and static diagrams are not new automatic proof, statistics or solver features. No completed device, physical-object or universal export testing is claimed.
Counting reference: [Discrete Mathematics on planar graphs and Euler’s formula]() discusses connected plane graphs and convex-polyhedron counts. This uses a closed triangular prism and explicit edge subdivision; crossings add no automatic nodes and the formula is not generalised to every surface with holes.