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What Grassmann Knew: Incidence Theorems on Cubics

Will Traves

TL;DR

The paper revisits the problem of testing whether 10 points lie on a plane cubic using Hermann Grassmann’s incidence-geometry framework, presenting a self-contained, straightedge-based approach that mirrors and extends Traves and Wehlau’s results. A central construction encodes a cubic as $Q=igl\{ x\in \mathbb{P}^2: (xaAa_1.xbBkCb_1.xc)=0 \bigr\}$, yielding explicit, projectively invariant procedures to intersect a line with $Q$, to construct the tangent to $Q$ at a point, and to locate the third point of intersection of that tangent with $Q$. The authors also solve a six-point conic–cubic intersection problem through auxiliary cubics and conics, and they demonstrate how Grassmann’s parameterization can be made to pass through nine prescribed points, providing a direct, geometric route to solve the open problems posed by Traves and Wehlau. Beyond solving these problems, the paper discusses the historical context, links to the broader literature on Grassmann-Cayley algebra, and several open questions, including potential higher-degree generalizations and the extent of plane-only constructions for more complex incidence configurations. The results highlight the enduring relevance of classical projective methods for modern synthetic geometry and geometric constraint systems.

Abstract

Traves and Wehlau recently gave a straightedge construction that checks whether 10 points lie on a plane cubic curve. They also highlighted several open problems in the synthetic geometry of cubics. Hermann Grassmann investigated incidence relations among points on cubic curves in three papers appearing in Crelle's Journal from 1846 to 1856. Grassmann's methods give an alternative way to check whether 10 points lie on a cubic. Using Grassmann's techniques, we solve the synthetic geometry problems introduced by Traves and Wehlau. In particular, we give straightedge constructions that find the intersection of a line with a cubic, find the tangent line to a cubic at a given point, and find the third point of intersection of this tangent line with the cubic. As well, given 5 points on a conic and a cubic and 4 additional points on the cubic, a straightedge construction is given that finds the sixth intersection point of the conic and the cubic. The paper ends with two open problems.

What Grassmann Knew: Incidence Theorems on Cubics

TL;DR

The paper revisits the problem of testing whether 10 points lie on a plane cubic using Hermann Grassmann’s incidence-geometry framework, presenting a self-contained, straightedge-based approach that mirrors and extends Traves and Wehlau’s results. A central construction encodes a cubic as , yielding explicit, projectively invariant procedures to intersect a line with , to construct the tangent to at a point, and to locate the third point of intersection of that tangent with . The authors also solve a six-point conic–cubic intersection problem through auxiliary cubics and conics, and they demonstrate how Grassmann’s parameterization can be made to pass through nine prescribed points, providing a direct, geometric route to solve the open problems posed by Traves and Wehlau. Beyond solving these problems, the paper discusses the historical context, links to the broader literature on Grassmann-Cayley algebra, and several open questions, including potential higher-degree generalizations and the extent of plane-only constructions for more complex incidence configurations. The results highlight the enduring relevance of classical projective methods for modern synthetic geometry and geometric constraint systems.

Abstract

Traves and Wehlau recently gave a straightedge construction that checks whether 10 points lie on a plane cubic curve. They also highlighted several open problems in the synthetic geometry of cubics. Hermann Grassmann investigated incidence relations among points on cubic curves in three papers appearing in Crelle's Journal from 1846 to 1856. Grassmann's methods give an alternative way to check whether 10 points lie on a cubic. Using Grassmann's techniques, we solve the synthetic geometry problems introduced by Traves and Wehlau. In particular, we give straightedge constructions that find the intersection of a line with a cubic, find the tangent line to a cubic at a given point, and find the third point of intersection of this tangent line with the cubic. As well, given 5 points on a conic and a cubic and 4 additional points on the cubic, a straightedge construction is given that finds the sixth intersection point of the conic and the cubic. The paper ends with two open problems.
Paper Structure (6 sections, 5 theorems, 14 equations, 1 figure)

This paper contains 6 sections, 5 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Projective Geometry from Incidence Structures
  3. Grassmann on Cubics
  4. Synthetic Geometry
  5. Extensions and Open Problems.
  6. Acknowledgments

Key Result

Theorem 3.1

When the nine points $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$ and $i$ lie in linearly general position (no three on a line), the construction described above produces a cubic $(xaAa_1.xbBkCb_1.xc)=0$ that passes through the nine given points.

Figures (1)

  • Figure 1: The point $x$ lies on the cubic $(xaAa_1.xbBkCb_1.xc)=0$.

Theorems & Definitions (17)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Theorem 4.3
  • proof
  • Remark 4.4
  • Theorem 4.5
  • proof
  • ...and 7 more