arXiv subjects

Isaac Chen

Publications and source records attributed to Isaac Chen.

2 recordsLinked to original sources

arXiv2026

HistGramBench: Measuring Structural Recovery When Multimodal Models Redraw Historical Mathematical Diagrams as TikZ

Digital libraries preserve mathematical books printed from the sixteenth to the twentieth century, but the lettered diagrams in them, the figures on which Euclidean proofs depend, remain scanned images whose content cannot be searched. Redrawing such a diagram as a short drawing program would record which points lie on which lines and circles. But a printed plate offers no source code against which the redrawing could be checked. We introduce HistGramBench, a benchmark of 2,507 diagrams from 95 works printed between 1551 and 1933 in eight languages. For each diagram, two AI models from different providers agree on the facts a reader takes from the figure, such as which lines are parallel or where two lines meet, and every redrawing is checked against those facts on the model's own drawing. We evaluate six affordable AI models. The best recovers about three fifths of these facts, and most of what the models miss comes from programs that never produce a drawing. Parallels and tangencies are the hardest relations to reproduce. HistGramBench gives historians and librarians a way to measure when such tools can make old mathematical diagrams searchable.

cs.CV↗

arXiv2025

On the crossing profile of rectilinear drawings of $K_n$

We introduce the \textit{crossing profile} of a drawing of a graph. This is a sequence of integers whose $(k+1)^{\text{th}}$ entry counts the number of edges in the drawing which are involved in exactly $k$ crossings. The first and second entries of this sequence (which count uncrossed edges and edges with one crossing, respectively) have been studied by multiple authors. However, to the best of our knowledge, we are the first to consider the entire sequence. Most of our results concern crossing profiles of rectilinear drawings of the complete graph $K_n$. We show that for any $k\leq (n-2)^2/4$ there is such a drawing for which the $k^{\text{th}}$ entry of the crossing profile is of magnitude $Ω(n)$. On the other hand, we prove that for any $k \geq 1$ and any sufficiently large $n$, the $k^{\text{th}}$ entry can also be made to be $0$. As our main result, we essentially characterize the asymptotic behavior of both the maximum and minimum values that the sum of the first $k$ entries of the crossing profile might achieve. Our proofs are elementary and rely mostly on geometric constructions and classical results from discrete geometry and geometric graph theory.

math.CO↗