Truncated Projective Plane¶
Delete one point and its full pencil of lines from a finite projective plane, retaining a partitioned incidence structure dual to an affine plane.
Core Idea¶
A truncated projective plane is made from an existing finite projective plane of order \(q\) by choosing a point, removing it and all \(q+1\) lines through it, and keeping the incidences among the other points and lines. The result is a reusable finite incidence type rather than one distinguished drawing. It has \(q^2+q\) points and \(q^2\) lines, each retained line containing \(q+1\) points. It is the point-line dual of an affine plane of order \(q\): the incidence is equivalent after exchanging the roles of points and lines, not under unchanged labels.[ref-4ca785f002b7][ref-6195293cbaca]
The local deletion has a global consequence. The removed lines, minus the chosen point, divide the retained points into \(q+1\) classes of \(q\). Every retained line meets every class once, and any two retained lines still meet. These claims follow from the projective-plane incidence axioms and from removing the entire pencil, not from a free choice of classes.[ref-4ca785f002b7][ref-6195293cbaca]
- Parent plane. Begin with a finite projective plane of order \(q\geq2\). It supplies the point-line incidence axioms and \(q^2+q+1\) points and lines. A generic hypergraph is not enough to guarantee the construction's consequences.[^ref-4ca785f002b7]
- Chosen point. A single point marks the center of deletion. A different choice may give a different realization; the definition requires a choice, not a specially named point.[^ref-4ca785f002b7]
- Complete line pencil. Remove every line through that point along with the point itself. Keeping even one such line changes the retained incidence structure and breaks the stated count and class argument.[ref-4ca785f002b7][ref-6195293cbaca]
- Inherited incidence. Do not redraw which surviving points lie on surviving lines. Their relation is the restriction of the parent plane's relation. The uniform line size, regular point degree, and pairwise line intersections are consequences of this inheritance.[ref-4ca785f002b7][ref-6195293cbaca]
As a hypergraph with points as vertices and retained lines as edges, the result is \((q+1)\)-uniform and \((q+1)\)-partite, with each point on \(q\) edges. Partiteness and cover number are diagnostics and uses of the type; they do not replace its projective-plane parent and deletion rule.[ref-4ca785f002b7][ref-6195293cbaca]
Scope of Application¶
The construction is literal in finite incidence geometry when one studies the relation between projective and affine planes: transposing point and line roles in the retained incidence recovers the affine-plane pattern. Existence is conditional on an order-\(q\) projective plane. Prime-power orders give a known family; the definition is not an existence claim for every integer \(q\).[^ref-4ca785f002b7]
In extremal hypergraph theory, use the retained points as vertices and lines as edges. Every two edges meet, while the deleted-pencil classes make the hypergraph \((q+1)\)-partite. Its matching number is one and its minimum vertex-cover size is \(q\), so for \(r=q+1\) it attains the \((r-1)\nu\) side of Ryser's proposed inequality. This is an equality witness when the parent plane exists; it does not prove the inequality for all \(r\)-partite hypergraphs.[ref-4ca785f002b7][ref-6195293cbaca]
Clarity¶
The direction of deletion fixes the object. A point is removed with the lines through it. The surviving points on each removed line form one class; each surviving line crosses all classes once. This is why the construction is partite without starting from an arbitrary partite coloring.[ref-4ca785f002b7][ref-6195293cbaca]
The cover number can be checked without treating extremality as a definition. There are \(q^2\) retained lines and each retained point lies on \(q\) of them, so at least \(q\) points are needed to hit every line. Any one induced class has exactly \(q\) points and meets every line, so \(\tau=q\). Since every pair of lines meets, \(\nu=1\). The argument proves the stated property for this construction and no converse.[^ref-4ca785f002b7]
Manages Complexity¶
One parent plane, one chosen point, and one complete pencil encode a large incidence table. Instead of listing \(q^2\) retained lines, one can derive their number, size, pairwise intersection, the \(q+1\) point classes, and the cover size from the parent axioms. The economy depends on retaining the original incidence; deleting a different collection of lines loses these deductions.[ref-4ca785f002b7][ref-6195293cbaca]
The dual-affine view is useful because it changes which side of the incidence relation one reasons about. It is a controlled role swap: claims about point counts, line counts or hitting sets must be translated, not copied verbatim from the affine presentation.[^ref-4ca785f002b7]
Abstract Reasoning¶
Treat truncation here as a rule on incidence types, not merely as a picture of a clipped figure. For each valid parent plane and selected point, the rule produces another incidence structure with provable invariants. The Fano parent gives the small Pasch realization, while a non-Desarguesian order-nine parent supplies a materially different possible input to the same rule.[ref-80fe12b9b138][ref-78c84414af94][^ref-4ca785f002b7]
This does not mean that every plane of a given order produces a nonisomorphic truncation, or that the output uniquely identifies its parent. Those reconstruction claims need their own proof. The safe inference is that the rule can be applied to each attested parent plane and that the stated parameters follow.[ref-78c84414af94][ref-4ca785f002b7]
Knowledge Transfer¶
The same retained incidence can be read geometrically as a dual affine plane and combinatorially as a uniform intersecting multipartite hypergraph.[ref-4ca785f002b7][ref-6195293cbaca] In the latter reading, lines become hyperedges and a set of points hitting every line becomes a vertex cover. These readings interpret one incidence structure; they do not create separate members of the type.
The transfer has a limit: other intersecting hypergraphs can attain the same sharp cover bound.[^ref-b4b1f901c11c] Neither pairwise intersection nor that bound licenses calling an arbitrary hypergraph a truncated projective plane. The projective-plane parent and complete-pencil deletion, or an incidence isomorphism to such an output, must still be established.
Example¶
Fano-to-Pasch, order two. Choose a point of the Fano plane and remove it with its three incident lines. The four surviving triples on six points form the Pasch hypergraph. The three deleted lines minus their common point give three classes of two, and each retained triple meets all three classes. Here the parent, chosen point, complete pencil and inherited incidence fill the four signature roles directly. Bellmann and Reiher draw this member and identify the four-edge remainder.[^ref-80fe12b9b138]
A non-Desarguesian order-nine member, derived. Hungerbühler and Kusejko document non-Desarguesian projective planes of order nine. Choose a point in one such plane and apply the universal point-and-pencil deletion. Substituting \(q=9\) into the sourced construction gives 90 retained points, 81 retained ten-point lines and ten point classes of nine; each point has degree nine and the cover number is nine. This is an explicit mathematical deduction from two sources. The order-nine paper establishes the parent plane; it does not report this truncation as a worked object or prove that different choices yield different truncation isomorphism classes.[ref-78c84414af94][ref-4ca785f002b7]
Relationships to Other Abstractions¶
Current abstraction Truncated Projective Plane Domain-specific
Parents (1) — more general patterns this builds on
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Truncated Projective Plane is part of Relation Prime
The inherited point-line incidence relation is an identity-bearing internal constituent of every truncated projective plane.
Hierarchy path (1) — routes to 1 parentless root
- Truncated Projective Plane → Relation
Neighborhood in Abstraction Space¶
Truncated Projective Plane sits in a sparse region of the domain-specific corpus (88th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2721 abstractions)
Nearest neighbors
- Desargues's Theorem — 0.83
- Non-Euclidean geometry — 0.82
- Blocking set — 0.82
- Ordered geometry — 0.81
- Cross Section (Geometry) — 0.79
Computed from structural-signature embeddings · 2026-10-17
Not to Be Confused With¶
- Affine plane: line-and-points deletion; incidence dual to this point-and-lines deletion.[^ref-4ca785f002b7]
- Pasch hypergraph: the order-two Fano member, not every order.[^ref-80fe12b9b138]
- A merely intersecting multipartite hypergraph: one may have the class and intersection properties without this specific parent-and-pencil provenance.[^ref-b4b1f901c11c]
- An all-order existence theorem: this construction presupposes a parent finite projective plane of the stated order.[^ref-4ca785f002b7]
References¶
[^ref-4ca785f002b7]: Louis DeBiasio, Yigal Kamel, Grace McCourt, and Hannah Sheats, “Generalizations and Strengthenings of Ryser's Conjecture,” The Electronic Journal of Combinatorics 28(4), P4.37 (2021), Section 1.3, printed pp. 4–5. DOI: 10.37236/9914. https://www.combinatorics.org/ojs/index.php/eljc/article/download/v28i4p37/pdf/ [^ref-6195293cbaca]: Zoltán Király and Lilla Tóthmérész, “On Ryser's Conjecture for t-Intersecting and Degree-Bounded Hypergraphs,” The Electronic Journal of Combinatorics 24(4), P4.40 (2017), Definition 22 and following discussion, printed p. 12. https://www.combinatorics.org/ojs/index.php/eljc/article/download/v24i4p40/pdf/ [^ref-80fe12b9b138]: Louis Bellmann and Christian Reiher, “Turán's Theorem for the Fano Plane,” arXiv:1804.07673 (2018), Section 2.2 and Figure 2.2, printed pp. 4–5. https://arxiv.org/pdf/1804.07673 [^ref-78c84414af94]: Norbert Hungerbühler and Katharina Kusejko, “Poncelet's Theorem in the Four Non-Isomorphic Finite Projective Planes of Order 9,” arXiv:1406.7857v2 (2016), Introduction and Section 3.1, printed pp. 1 and 9. https://arxiv.org/pdf/1406.7857 [^ref-b4b1f901c11c]: Nevena Francetić, Sarada Herke, Brendan D. McKay, and Ian M. Wanless, “On Ryser's Conjecture for Linear Intersecting Multipartite Hypergraphs,” arXiv:1508.00951v3 (2015), abstract, introduction, and Section 3.4 / Theorem 4.1. https://arxiv.org/pdf/1508.00951