diagonal
Enable an agent to recognise a geometric diagonal, record the figure and convention that define it, and determine which measurements, decompositions and constructions it supports.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
recalled by Codex without web access - no source was read
Researched by: Codex
Purpose and description
Enable an agent to recognise a geometric diagonal, record the figure and convention that define it, and determine which measurements, decompositions and constructions it supports.
In geometry, a diagonal is a line segment joining two nonadjacent vertices of a polygon, or two vertices of a polyhedron that do not share an edge.
It can be Validate whether a proposed endpoint pair defines a diagonal under the recorded convention.; Enumerate qualifying diagonals without counting reversed endpoint pairs twice.; Compute diagonal lengths and directions from sufficient geometric data.; Test whether a diagonal remains within the host and is usable for subdivision.; Compare diagonals for intersection, perpendicularity, equal length or bisection.; Construct or select diagonals for triangulation or geometric inference when the required assumptions hold..
Distinguishing features
A polygon diagonal joins two vertices that are not consecutive on the polygon boundary; a segment joining consecutive vertices is an edge.
A segment joining arbitrary boundary points is not a polygon diagonal unless both endpoints are qualifying vertices.
A face diagonal joins nonadjacent vertices within one face; a space diagonal requires endpoints that share no face under the convention adopted here.
Rotating a figure does not alter diagonal status, although it changes whether the segment appears slanted, horizontal or vertical.
A diagonal of a concave polygon can pass outside the polygon, so interior containment is a separate test rather than a universal defining condition.
Scope
+ Segments joining nonadjacent vertices of a polygon
+ Face diagonals and space diagonals of polyhedra under an explicit convention
+ The host figure, endpoint identities and adjacency relations that establish diagonal status
+ Length, direction and position relative to the host's interior and boundary
+ Diagonal intersections, subdivisions and permitted geometric constructions
- The complete geometry and topology of the host polygon or polyhedron
- Matrix diagonals, diagonal matrices and diagonal operators
- Diagonal relations in Cartesian products and mathematical diagonalisation arguments
- Oblique lines or strokes identified only by visual slant
- Physical braces, screen sizes and other applications beyond their underlying geometric segment
Characteristics
- Host figure
- Reference to a polygon or polyhedron with defined vertices, edges and, where applicable, faces Diagonal status is relative to a particular host and its incidence structure.
- Endpoint pair
- Unordered pair of distinct host-vertex identifiers Identifies the diagonal independently of traversal direction or drawing orientation.
- Diagonal convention
- Polygon diagonal; face diagonal; space diagonal; explicitly documented alternative Prevents incompatible uses of diagonal from being silently combined.
- Endpoint incidence
- Adjacent vertices; nonadjacent vertices sharing a face; vertices sharing no face; unresolved Provides the primary evidence for rejecting an edge or distinguishing face and space diagonals.
- Segment containment
- Relative interior inside host; on boundary; partly outside; mixed boundary contact; unresolved Controls whether the diagonal can support an interior subdivision or path.
- Length
- Nonnegative value in the host's length unit, with metric and precision stated Supports comparison, construction and dimensional calculations.
- Direction relative to reference
- Undirected angle modulo 180 degrees in a plane, or an unoriented direction in space Describes orientation without confusing visual slant with diagonal identity.
- Intersection relation
- Other segment reference plus shared endpoint, proper crossing, overlap, disjoint or skew Determines compatibility with subdivisions and distinguishes projected crossings from actual intersections.
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.diagonal
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 5 bundles · 9 layers · 14 findings · 22 questions.
Diagonal meaning Establishes which geometric sense is intended and what evidence makes a segment a diagonal.
The word can refer to a vertex relation, a visual direction or an algebraic arrangement, and these require different recognition rules.
Geometric sense
Records the adopted definition and separates geometric membership from appearance.
Definition over slant
Record the convention that establishes diagonal status; a slanted appearance alone does not establish it.
- Does diagonal mean a polygon diagonal, a face diagonal or a space diagonal in this use? definition
- Which source or task specifies this convention, and does it use diagonal merely to mean oblique? provenance
Endpoint qualification
Tests vertex membership and nonadjacency against the chosen host.
Qualifying vertex pair
Identify both endpoints as host vertices and establish the incidence conditions required by the adopted definition.
- Are both endpoints distinct vertices of the same identified host figure? boundary
- Are the endpoints connected by an edge, and which faces, if any, contain both? definition
Host incidence and inventory Relates each diagonal to the host's boundary structure and the complete set of qualifying endpoint pairs.
Coordinates alone do not identify polygon adjacency or polyhedral faces, and diagonal counts depend on these relations.
Boundary and face structure
Records the host structure needed to distinguish edges, face diagonals and space diagonals.
Host structure evidence
Use an explicit boundary order or face incidence record instead of inferring connectivity from the drawing.
- What cyclic vertex order defines the polygon, or what vertex, edge and face incidences define the polyhedron? definition
- Are self-intersections, holes, repeated vertices or degenerate faces present that require a special convention? boundary
Diagonal enumeration
Identifies qualifying pairs consistently and checks the limits of counting rules.
Unique diagonal inventory
Treat a diagonal as an unordered qualifying vertex pair and distinguish this combinatorial inventory from geometrically usable diagonals.
- Which unordered vertex pairs satisfy the adopted diagonal definition after edge pairs are excluded? measurement
- Does the requested count include every qualifying pair or only diagonals that satisfy an additional interior-containment condition? boundary
Embedding and measurement Records the diagonal's metric geometry and its actual position within the host.
A qualifying vertex pair does not by itself establish length, interior containment or intersection in three dimensions.
Length and direction
Determines what can be measured from coordinates, dimensions or constraints.
Metric evidence
Record sufficient geometric evidence and the metric before calculating a diagonal's length or direction.
- Which endpoint coordinates or host dimensions determine the length, in what metric and units? measurement
- If a special formula is used, what establishes its assumptions, such as a rectangle's right angle? provenance
Containment and contact
Separates diagonal identity from interior placement and boundary contacts.
Segment-host position
Test the segment apart from its endpoints for exterior portions, boundary overlap and passage through other vertices.
- Does the segment's relative interior lie inside the host, on its boundary or partly outside it? measurement
- Does it pass through another host vertex or overlap an edge, and how does the chosen operation handle that case? boundary
Diagonal relations and use Determines how diagonals interact and which geometric operations their verified properties support.
Crossing, bisection and subdivision are frequent uses of diagonals, but none follows from the label alone.
Relations between diagonals
Tests intersection and special metric relationships without generalising from familiar quadrilaterals.
Verified diagonal relations
Record actual intersections, length comparisons, angles and bisection evidence; distinguish spatial relations from their projections.
- Do the diagonals actually intersect, share an endpoint, overlap or remain disjoint or skew? measurement
- What measurements or established host properties justify claims of equal length, perpendicularity or mutual bisection? provenance
Subdivision and inference
Connects diagonal properties to triangulation, construction and conditional deductions.
Operation preconditions
Select a diagonal for an operation only after checking the host and containment assumptions that make that operation valid.
- Can this diagonal subdivide the polygon as intended without leaving its interior or creating prohibited crossings? action
- Which additional host assumptions are required before using the diagonal's measured properties to infer the figure's type or missing dimensions? action
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- No sense is recorded; this account assumes the elementary geometric meaning rather than matrix terminology or an informal slanting orientation.
- For nonconvex polyhedra, a space diagonal need not lie wholly inside the solid; check the intended convention before imposing an interior requirement.
- These are recalled mathematical facts, not findings from sources read for this entry.
- Which of these check these first hold for the sense of diagonal this model covers, and on what evidence? provenance
Kinds and varieties
Recalled without web access and unsourced; every item is a lead to verify.
- Polygon diagonal
- Face diagonal of a polyhedron
- Space diagonal of a polyhedron
- Which of these kinds and varieties hold for the sense of diagonal this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Dividing polygons into triangles for geometric computation and area calculation.
- Checking rectangular layouts in construction by comparing diagonal lengths alongside other geometric constraints.
- Specifying screen size by the diagonal length of the display area.
- Calculating distances across rectangular rooms, boxes and other geometric forms.
- Which of these real-world use hold for the sense of diagonal this model covers, and on what evidence? provenance
Typical measurements
Recalled without web access and unsourced; every item is a lead to verify.
- Number of diagonals in a simple polygon with n vertices - Exactly n(n−3)/2 for integer n ≥ 3 - count
- Diagonal length of a rectangle with side lengths a and b - Exactly √(a²+b²) in Euclidean geometry - same length unit as a and b
- Space diagonal length of a rectangular cuboid with edge lengths a, b and c - Exactly √(a²+b²+c²) in Euclidean geometry - same length unit as a, b and c
- Which of these typical measurements hold for the sense of diagonal this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Assuming every diagonal lies inside its polygon; diagonals of concave polygons can pass outside it.
- Treating a diagonal as necessarily inclined at 45 degrees; its angle depends on the geometry and reference direction.
- Using the rectangular diagonal formula for a general quadrilateral without establishing a right angle.
- Treating equal diagonals alone as proof that a quadrilateral is a rectangle.
- Confusing a polyhedron's face diagonals with its space diagonals.
- Which of these failure modes and hazards hold for the sense of diagonal this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- edge - An edge connects adjacent vertices in the boundary structure; a diagonal connects vertices that do not share an edge.
- chord - A chord joins two points on a curve, commonly a circle; a polygon diagonal is defined by nonadjacent vertices.
- diameter - A circle's diameter is a chord through its centre; a diagonal need not pass through a centre.
- matrix diagonal - A matrix diagonal is a sequence of entries selected by their row and column indices, rather than a geometric segment between vertices.
- oblique orientation - An oblique orientation describes inclination relative to a reference; a geometric diagonal is defined by its endpoints and adjacency.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of diagonal this model covers, and on what evidence? provenance
What the second pass must settle
- Does vr.tr.diagonal intend the geometric vertex-to-vertex sense, an orientation quality suggested by its domain placement, or a broader mathematical concept?
- Which reference should govern whether polyhedral diagonal includes both face and space diagonals or reserves the term for one of them?
- Should this entry cover diagonals of self-intersecting polygons, polygons with holes and higher-dimensional polytopes, and under which conventions?
- How should qualifying endpoint pairs that pass through another vertex or overlap the boundary be classified for recognition versus subdivision?
- Should algebraic and Cartesian-product senses be linked to neighbouring models or represented within this entry once its authoritative registry meaning is established?