Euclid30K: Geometry-Enhanced VQA Corpus

• Euclid30K is a curated multimodal dataset of 29,695 geometric problems spanning both plane and solid geometry, serving as a comprehensive curriculum for spatial reasoning.
• The dataset is rigorously filtered and standardized using duplicate removal, GPT-4o based subquestion splitting, and DeepSeek-V3.1 LaTeX conversion to ensure verifiable, reward-driven training.
• Euclid30K functions as a geometric surrogate task that enhances shape identification, relational reasoning, and deductive proof capabilities, leading to significant zero-shot gains on spatial benchmarks.

Euclid30K is a multimodal geometry VQA corpus introduced in “Euclid’s Gift: Enhancing Spatial Perception and Reasoning in Vision-LLMs via Geometric Surrogate Tasks” for training vision–LLMs on Euclidean geometry as a surrogate task for spatial intelligence (Lian et al., 29 Sep 2025). The dataset comprises approximately 30K plane and solid geometry problems and is used to fine-tune members of the Qwen2.5VL and RoboBrain2.0 families with Group Relative Policy Optimization (GRPO). In the reported formulation, Euclidean problem solving is not treated as an end in itself, but as a mechanism for inducing shape identification, counting, relational reasoning, and multi-step deductive competence that transfers zero-shot to spatial benchmarks.

1. Scope and definition

Euclid30K is defined as a curated multimodal dataset of 29,695 geometry problems spanning both plane and solid geometry. Each instance is a triple $$\langle images\ i,\ text\ problem\ p,\ answer\ a\rangle$$, where the image component contains 1–8 figures with average 1.1 figures, the text describes givens and questions with average length approximately 230 characters, and the answer is represented as either a LaTeX formula, a scalar value, or an option index (Lian et al., 29 Sep 2025).

The corpus covers 18,577 plane (2D) questions and 11,118 solid (3D) questions. Reported answer types are 16,804 mathematical-expression answers in LaTeX, 6,321 numeric answers, and 2,618 multiple-choice indices. The maximum and average question lengths are 1,598 and 229.8 characters respectively, while the maximum and average answer lengths are 501 and 8.9 characters.

Property	Value
Total problems	29,695
Plane / Solid	18,577 / 11,118
Images per instance	1–8, avg 1.1
Question length	max 1,598; avg 229.8 chars
Answer length	max 501; avg 8.9 chars

The intended coverage is broad rather than narrowly benchmark-specific. The problem categories are described as spanning the full K–12 and Olympiad syllabus. In plane geometry, examples include point-line incidence, circle-tangent problems, polygon angle sums, similarity and congruence, and distance-and-area calculations. In solid geometry, the range includes polyhedron cross-sections, volume–surface-area relations, folding and unfolding, and moving-point scenarios. This scope positions Euclid30K as a geometry curriculum for multimodal reasoning rather than merely a collection of diagram questions.

2. Data construction and curation

The dataset was assembled through a three-stage filtering pipeline that reduced approximately 32,500 raw candidates to 29,695 retained problems (Lian et al., 29 Sep 2025). The first stage applied duplicate filtering via perceptual hash on diagrams. The second stage performed subquestion splitting using GPT-4o to isolate each part. The third stage carried out LaTeX standardization of all formulas via DeepSeek-V3.1 followed by post-processing.

A distinct component of the collection process concerns solid geometry. About 3,996 solid-geometry problems were newly collected from standard junior and senior high-school and competition textbooks. The reported purpose of this addition was to broaden coverage of positional-relation, dynamic-point, and contextual word problems.

The curation protocol indicates that the dataset was designed to support reward-verifiable training rather than free-form annotation alone. During training, LaTeX-form answers are verified for exact symbolic equivalence via MathVerify. Numeric answers receive reward only when they satisfy

$\bigl|\,(p - t)/t\bigr|\;\le\;0.01,$

and multiple-choice answers use exact match. The stated motivation is to guard against trivial or grossly incorrect predictions.

Although Euclid30K is used monolithically for fine-tuning in the reported experiments, its internal composition is described as naturally spanning low-, medium-, and high-difficulty problems, from elementary angle sums to Olympiad-style multi-step proofs. Future public releases may expose explicit train/val/test splits and difficulty tags. This suggests that, in its initial form, the corpus functions primarily as a supervised or RL fine-tuning resource rather than as a standardized held-out benchmark.

3. Geometric surrogate-task formulation

The central methodological claim attached to Euclid30K is that Euclidean geometry problem solving can serve as a surrogate task for spatial perception and reasoning in Multimodal LLMs (Lian et al., 29 Sep 2025). The motivation is that spatial intelligence includes abilities such as visualizing and transforming shapes, mentally rotating objects, judging relational positions and containment, and estimating numerosity, while these capacities remain a critical unresolved challenge for MLLMs.

To operationalize this idea, the training setup encodes shape identification, counting, metric-relation inference, and multistep deductive proofs as RL tasks with verifiable zero/one reward signals. The reported interpretation is that this forces the model to internalize Euclidean axioms, including angle-sum constraints, parallel-line theorems, and similarity ratios, as surrogate supervision.

The optimization method is Group Relative Policy Optimization, described as a variant of PPO tailored for VQA with verifiable rewards. For each question $q$ and policy $\pi_{\theta_{\rm old}}$, the method generates $G$ rollouts and optimizes the token-level clipped surrogate objective

$$\mathcal{J}(\theta) = \frac{1}{\gamma}\sum_{i=1}^G\sum_{t=1}^{|o_i|} \min\!\Bigl(r_{i,t}(\theta)\,\hat A_i,\;\mathrm{clip}(r_{i,t}(\theta),\,1\pm\epsilon)\,\hat A_i\Bigr),$$

with $\gamma=\sum_i |o_i|$, followed by the update

$$\mathcal{L}_{\rm GRPO}(\theta) =\mathbb{E}\bigl[\mathcal{J}(\theta)\bigr]\;-\;\beta\;\mathrm{KL}\bigl[\pi_{\theta}\,\|\,\pi_{\rm ref}\bigr]$$

using $\beta=10^{-2}$.

In this formulation, Euclid30K is not merely a dataset of answers; it is a source of reward-verifiable geometric structure. A plausible implication is that the value of the corpus depends as much on formal answer verifiability and deductive depth as on raw problem count.

4. Training integration and model families

The models fine-tuned with Euclid30K are Qwen2.5VL-3B, Qwen2.5VL-7B, Qwen2.5VL-72B, RoboBrain2.0-7B, and RoboBrain2.0-32B (Lian et al., 29 Sep 2025). The reported training configuration uses 10 epochs on 64 $\times$ H100 GPUs with Adam, learning rate $\bigl|\,(p - t)/t\bigr|\;\le\;0.01,$0, weight decay $\bigl|\,(p - t)/t\bigr|\;\le\;0.01,$1, global batch size 128, 8 rollouts per prompt, clipping parameter $\bigl|\,(p - t)/t\bigr|\;\le\;0.01,$2, context window 2048 tokens, and rollout maximum length 512.

The integration objective is described as imbuing vision–LLMs with Euclidean priors. Within that framing, Euclid30K supplies training instances whose rewards can be checked symbolically, numerically, or by exact match depending on answer type. The dataset therefore supports RL fine-tuning that is more constrained than generic preference optimization and more structured than unconstrained multimodal instruction tuning.

The paper’s terminology emphasizes “geometry-centric fine-tuning” and “geometry-as-surrogate.” These phrases distinguish the approach from direct adaptation to downstream tasks such as egocentric video reasoning or 3D localization. The stated design choice is that no task-specific adaptations are used for the downstream benchmarks, so any gains are intended to reflect transferred spatial competence rather than benchmark-specialized engineering.

5. Empirical transfer to spatial reasoning benchmarks

Zero-shot evaluation is reported on four spatial benchmarks: Super-CLEVR for 2D compositional reasoning, Omni3DBench for 3D localization and size/distance, VSI-Bench for egocentric-video spatial tasks, and MindCube for partial-viewpoint spatial reasoning (Lian et al., 29 Sep 2025). The headline result is that Euclid30K fine-tuning yields substantial gains across all four benchmarks without task-specific adaptation.

The reported mean VSI-Bench accuracy across all evaluated models rises from 34.5% to 40.5%. Among individual systems, RoboBrain2.0-Euclid-7B reaches 49.6% accuracy, exceeding Spatial-MLLM-4B at 48.4%. On Super-CLEVR, Qwen2.5VL-7B increases from 76.1% to 86.2%. On MindCube, Qwen2.5VL-Euclid-3B reaches 38.9% compared with 20.4% for the base model.

These results are presented as evidence that Euclid30K provides broadly transferable spatial priors rather than overfitting to a single task domain. The broader claim in the paper is that this is the first systematic study showing that geometry-centric fine-tuning can confer vision-LLMs with broadly transferable spatial skills. A cautious reading is that the reported evidence supports transfer across heterogeneous benchmark formats—2D synthetic reasoning, 3D scene understanding, egocentric video, and partial-viewpoint tasks—while still leaving open the question of how far this transfer extends beyond the specific benchmarks tested.

6. Significance, applications, and interpretive boundaries

The broader impact claim attached to Euclid30K is that embedding Euclidean theory into vision–LLMs teaches shape recognition, metric estimation, positional inference such as parallelism and intersection, and multi-step logical deduction (Lian et al., 29 Sep 2025). The paper further states that this “geometry-as-surrogate” paradigm reduces the distributional gap to downstream spatial tasks, invoking domain adaptation theory as motivation.

Potential applications listed for the resulting models include advanced visual question answering in robotics and AR/VR, autonomous navigation and path planning in indoor and outdoor environments, 3D reconstruction and CAD blueprint interpretation, geospatial analysis for mapping and volumetric estimation, and integration with temporal datasets for full spatio-temporal reasoning. These are presented as applications of the induced spatial priors rather than as directly benchmarked deployment outcomes.

Several interpretive boundaries are explicit or implicit in the reported setup. First, Euclid30K is used monolithically for fine-tuning rather than as a benchmark with explicit public train/validation/test splits. Second, the transfer claims are zero-shot and benchmark-based, not yet accompanied in the provided material by ablations on curriculum structure, explicit difficulty tags, or comparative data-scaling studies. Third, the central premise is surrogate-task transfer: the corpus is designed to teach Euclidean principles that generalize outward, not to mirror every downstream spatial distribution directly. This suggests that the significance of Euclid30K lies less in coverage of all possible spatial tasks than in demonstrating that a compact, well-curated geometry curriculum can serve as a training substrate for generalizable spatial reasoning in vision–LLMs.