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Euclid's Gift: Enhancing Spatial Perception and Reasoning in Vision-Language Models via Geometric Surrogate Tasks

Shijie Lian, Changti Wu, Laurence Tianruo Yang, Hang Yuan, Bin Yu, Lei Zhang, Kai Chen

arXiv:2509.24473v3cs.CVcs.AIcs.CLcs.LG

TL;DR

MLLMs still fall short of spatial intelligence, and task-specific spatial datasets may not generalize across tasks. This paper trains models on Euclidean geometry problems as a surrogate task; the resulting models report gains across four unseen spatial benchmarks.

  • Problem

    State-of-the-art MLLMs still fall short of spatial intelligence, including reasoning about spatial relationships and geometric configurations.

    Source: §1 · p. 1

  • Method

    The paper treats Euclidean geometry problem-solving as a surrogate task and fine-tunes seven model variants from three model families using GRPO and Euclid30K.

    Source: §1 · p. 2

  • Results

    Euclid30K fine-tuning yields consistent gains across four unseen spatial benchmarks for seven models; VSI-Bench average accuracy improves by 5.2% across evaluated variants.

    Source: §4.1 · p. 7, §6 · p. 8

  • Takeaways & Limitations

    The authors conclude that basic Euclidean priors learned from planar and solid geometry can transfer to multiple spatial tasks without extra fine-tuning.

    Source: §6 · p. 8

    Euclid30K lacks solution-process annotations, making it less suited to SFT routines that rely on explicit reasoning traces.

    Source: §5 · p. 8

Abstract

from arXiv · show

Spatial intelligence spans a rich suite of abilities, including visualising and transforming shapes, mentally rotating objects, judging relational positions and containment, and estimating numerosity. However, it still remains a critical unresolved challenge for Multimodal Large Language Models (MLLMs). To fill this gap, we propose to treat Euclidean geometry problem-solving as a surrogate task. Specifically, we meticulously constructed a curated multimodal dataset, called Euclid30K, comprising approximately 30K plane and solid geometry problems. Furthermore, to enable the model to learn and apply Euclidean principles from these geometry problems, we fine-tuned seven model variants (spanning 3--72B parameters) from the Qwen2.5VL, Qwen3VL, and RoboBrain2.0 families using Group Relative Policy Optimization (GRPO), inspiring the models to identify shapes, count, and relate entities, and perform multi-step deductive reasoning using Euclidean principles. Our experiments demonstrate that the resulting models achieve substantial zero-shot gains across four spatial reasoning benchmarks (Super-CLEVR, Omni3DBench, VSI-Bench, and MindCube) without any task-specific adaptations. Notably, after training on the Euclid30K, the mean VSI-Bench accuracy rose from 36.6\% to 41.8\% (+5.2\%), and the mean MindCube accuracy rose from 31.4\% to 38.1\% (+6.7\%). To our knowledge, this is the first systematic study showing that geometry-centric fine-tuning can confer vision-language models with broadly transferable spatial skills. Code and Euclid30K dataset can be found in \href{https://zgca-ai4edu.github.io/Euclids_Gift}{this}.

1. Introduction

State-of-the-art MLLMs still fall short of spatial intelligence, while spatially focused training may not generalize beyond its covered tasks. This work uses Euclidean geometry problems as a surrogate task and reports gains across four spatial benchmarks.

1. Introduction

  • Task-specific spatial datasets may yield strong in-domain results yet fail to cultivate generalizable spatial intelligence, as Spatial-MLLM accuracy drops on out-of-domain MindCube.This motivates learning from a broader range of spatial phenomena rather than a single dataset.Source: §1 · p. 2
  • Euclidean geometry is proposed as a source of universal spatial principles that can support generalization across downstream tasks.The paper argues that planar and solid geometry problems encourage models to internalize Euclidean axioms and constraints.Source: §1 · p. 2
  • Geometry problem-solving exercises shape recognition, spatial relations, geometric measurement, and multistep reasoning that are also needed for static spatial perception and reasoning.The authors frame geometry as a surrogate task for instilling foundational Euclidean priors.Source: §1 · p. 2
  • Euclid30K contains 29,695 geometry problems, including 3,996 solid-geometry problems added to address the existing corpus’s plane-geometry bias.The solid problems expose models to concepts including perspective, occlusion, polyhedral structures, and spatial rotation.Source: §1 · p. 2
  • Geometry-trained models show consistent gains on Super-CLEVR, Omni3D-Bench, VSI-Bench, and MindCube across Qwen2.5VL, Qwen3VL, and Robo-Brain2.0.The authors use GRPO with Euclid30K to fine-tune seven model variants.Source: §1 · pp. 2–3
  • The authors present geometry problems as a principle-driven alternative to single-skill fine-tuning, intended to support zero-shot transfer without task-specific training.They describe this as the first work using geometry problems as surrogate tasks to cultivate spatial intelligence in general-purpose VLMs.Source: §1 · pp. 2–3

2. Related Work

Related work covers spatial-intelligence benchmarks and spatially aware VLMs, alongside multimodal mathematics and geometry datasets. The paper positions geometry problem-solving as a surrogate task using readily available Euclidean problems.

2.1. Spatial Intelligence

  • Spatial intelligence spans mental rotation, viewpoint switching, and object counting, while existing spatial datasets cover only part of real-world scenarios and can be costly to build.The paper contrasts this approach with training on readily available Euclidean geometry problems, aiming to transfer their principles to visual-spatial perception and reasoning without architectural changes.Source: §2.1 · p. 3

2.2. Multimodal Math & Geometry Datasets

  • Multimodal math datasets progressed from image-text plane-geometry QA to broader math tasks, while SolidGeo3K focuses on 3D geometry with skill tags and difficulty annotations.Examples include GeoQA, Geometry3K, and UniGeo for plane geometry, followed by datasets such as MathVista, We-Math2, and GeoSense.Source: §2.2 · p. 3

3. Method

The method uses geometric problem-solving as a surrogate curriculum for spatial intelligence, motivating this choice through domain adaptation and implementing it with Euclid30K and GRPO fine-tuning.

3.1. Motivation: A Domain-Adaptation Perspective

  • Euclidean geometry’s broad spatial constraints may reduce its distribution gap to downstream tasks, and gains across four benchmarks align with this hypothesis.The target distribution is not directly observable, so the H∆H distance cannot be measured in practice.Source: §3.1 · pp. 3–4

3.3. RL Training in Euclid30K

  • GRPO fine-tunes Qwen2.5VL, Qwen3VL, and RoboBrain2.0 variants on sampled outputs, using token-level policy gradients and filtering prompts with uniformly correct or wrong answers.Training also uses over-sampling to ensure batches have valid gradients.Source: §3.3 · p. 5
  • Answer-dependent rewards use symbolic equivalence for LaTeX expressions, a ±1% tolerance for numeric answers, and exact matching for multiple-choice answers.The numeric threshold filters rough answers while allowing rounding or floating-point truncation; unit prompts are appended when ground-truth answers include units.Source: §3.3 · pp. 5–6

4. Experiment

The experiments test whether geometry-trained models transfer spatial skills zero-shot across four benchmarks and whether the gains exceed those from equal-sized spatial-data training. Euclid30K improves accuracy across the reported benchmarks, with gains over the spatial-data comparison in the ablation.

4. Experiment

  • The evaluation measures zero-shot generalization on Super-CLEVR, Omni3D-Bench, VSI-Bench, and MindCube, with compared models using a shared reasoning prompt and 1024-token response budget.The VSI-Bench prompt setup is intended to expose intermediate reasoning while assessing transferable spatial priors.Source: §4 · p. 6, §4.1 · pp. 6–7

4.1. Comparisons on VSI-Bench

  • 5.2% average accuracy improvement across evaluated Qwen2.5VL, Qwen3VL, and Robo-Brain2.0 variants on VSI-Bench follows Euclid30K fine-tuning.The reported Robo-Brain2.0 7B and 32B variants reach 49.6% overall, above the cited open-source and proprietary baselines.Source: §4.1 · p. 7

4.2. Comparisons on SuperClevr & Omni3DBench

  • Euclid30K improves both Super-CLEVR and Omni3D-Bench accuracy, including gains of 5.2 points and 1.8 points for Qwen2.5VL-3B, respectively.The reported Qwen2.5VL-72B gains are 10.5 points on SuperCLEVR and about 2.5 on Omni3D-Bench; the Qwen3VL variants also improve.Source: §4.2 · p. 7, Table 3 · p. 7

4.3. Comparisons on MindCube

  • 6.7 percentage points is the average MindCube accuracy gain across variants in the three model families after Euclid30K fine-tuning.Qwen2.5VL-Euclid-3B scores 38.9% on MindCube versus Spatial-MLLM's 32.1%, despite the latter's larger spatial corpus.Source: §4.3 · pp. 7–8, Table 4 · p. 7

4.4. Ablation Study

  • 44.3% average accuracy with Euclid30K exceeds 41.4% with an equal-sized Clevr-CoGenT split under the same GRPO setup.The respective gains from the 38.5% baseline are +5.8 and +2.9 points, supporting an advantage beyond additional data or GRPO alone.Source: §4.4 · p. 8, Table 5 · p. 8

5. Discussion

The discussion notes that Euclid30K lacks process annotations, limiting its fit for some supervised fine-tuning routines, and that gains vary by model. The authors associate larger gains with models that already have rudimentary spatial concepts.

Limitations

  • Euclid30K lacks solution-process annotations, making it less suited to SFT routines that require explicit reasoning traces.The authors cite the higher cost of collecting step-by-step annotations than answer-only labels and plan to extend the dataset with process annotations.Source: §5 · p. 8

Model-Specific Performance Variations

  • Spatial gains vary by model; the authors suggest Euclid30K refines existing spatial concepts more readily than it builds first principles from scratch.RoboBrain2.0 is given as an example of a model with larger improvements.Source: §5 · p. 8

6. Conclusion

The conclusion presents geometry problem-solving as a surrogate task for spatial perception and reasoning. It reports that Euclid30K-trained models transfer Euclidean priors to multiple spatial tasks without additional fine-tuning.

6. Conclusion

  • Geometry serves as a surrogate task for developing transferable spatial perception and reasoning in MLLMs.The conclusion frames this as an alternative approach to transferable spatial skills.Source: §6 · p. 8
  • Models trained on planar and solid geometry transfer basic Euclidean priors to multiple spatial tasks without extra fine-tuning.The conclusion reports consistent, significant gains across four unseen benchmarks for seven models.Source: §6 · p. 8

A. Proof of Standard Domain-Adaptation Bound

The appendix presents a conditional domain-adaptation argument for geometry as a surrogate, then reviews educational-psychology evidence relevant to the geometry–spatial link. It also notes that a later appendix section summarizes experimental settings.

A. Proof of Standard Domain-Adaptation Bound

  • When source-target joint errors are negligible and the HΔH distance is sufficiently small, the bound supports a small target-source population gap.Under those conditions, the source distribution can serve as a reliable surrogate for the target distribution [49] [50].Source: A. Proof of Standard Domain-Adaptation… · p. 13

B. Evidence from Educational Psychology

  • Educational-psychology studies link geometry performance to spatial ability through correlations and its use as a proxy for spatial proficiency.The cited evidence includes a moderate positive correlation and geometry skills and error patterns that vary with spatial-intelligence levels.Source: B. Evidence from Educational Psychology · pp. 13–14
  • Intervention studies report spatial-skill gains after structured geometric activities, while animal neuroscience findings provide convergent evidence about geometric experience and spatial coding.The neuroscience evidence is from animal models and is presented as a principle that may extend to learning systems.Source: B. Evidence from Educational Psychology · p. 14

C. Detailed Experimental Setup

C.1. Training setup

The paper details GRPO training and evaluation setups, then reports consistent gains across four spatial benchmarks and qualitative changes in model reasoning.

C.1. Training setup

  • Training follows VeRL and EasyR1 defaults for 10 epochs on 64 NVIDIA H100 GPUs, with Adam, a 1 × 10−6 learning rate, and eight GRPO rollouts per question.The KL coefficient is 1 × 10−2; other specified settings include a 512-sample rollout batch and global batch size of 128.Source: §C.1 · p. 14

C.2. Test setup

  • Evaluation uses lmms-eval, temperature 0 for VSI-Bench and MindCube, and a 1024-token response limit; models are tested with native and unified reasoning prompts, reporting the better result.VSI-Bench prompts are adjusted to match its original settings, including a video-frames prefix.Source: §C.2 · p. 14, §C.3 · p. 15

C.3. Prompt templates

  • The unified Euclid prompts request step-by-step reasoning and boxed answers, while RoboBrain2.0 uses answer tags and Qwen prompts retain an instruction-style cue.The prompt cue is retained for Qwen models to align evaluation inputs with their instruction-tuning style.Source: §C.3 · pp. 14–15

C.4. Dataset Setup

  • VSI-Bench covers video-based numerical and multiple-choice tasks, while Super-CLEVR, Omni3D-Bench, and MindCube test complementary 2D, 3D, and partial-observability spatial reasoning.MindCube evaluates exact-match accuracy across Rotation, Around, and Among camera trajectories.Source: §C.4 · p. 15

D.1. More experiments about main results

  • Euclid30K-tuned models improve across Super-CLEVR, Omni3DBench, VSI-Bench, and MindCube for Qwen2.5VL, Qwen3VL, and RoboBrain2.0 variants.Figure 4 also includes Qwen2.5VL-7B and Qwen2.5VL-32B, which show improvements across all four benchmarks.Source: §D.1 · p. 15
  • After geometry tuning, qualitative traces show more coherent multiview reasoning, clearer distance and size estimation, more reliable counting, and fewer unsupported shortcuts.Reported examples include similarity relations, cardinal-direction cues, and perspective-based size scaling.Source: §D.1 · p. 15

D.2. More experiments about ablation study

The ablation contrasts Euclid30K with equal-sized non-geometric spatial data, finding broader overall transfer from geometry alongside task-specific strengths for spatial data.

D.2. More experiments about ablation study

  • Qwen2.5VL-32B reaches 55.8% on object size with Euclid30K versus 44.3% with Clevr-CoGenT, and 46.0% on room size versus 40.8%.The table compares Euclid30K with a 30K-sample Clevr-CoGenT subset under matched training settings.Source: §D.2 · p. 16, Table 6 · p. 17
  • Euclid30K provides a more general-purpose foundation across task types, while task-specific spatial data can improve categories aligned with its design.Clevr-CoGenT training gives Qwen2.5VL-3B 40.5% on object counting, slightly above the 38.3% with Euclid30K.Source: §D.2 · p. 16
  • The geometry advantage holds across Qwen2.5VL, Qwen3VL, and RoboBrain2.0, across 3B–72B scales, and across dense and MoE variants.The authors interpret the consistency as evidence that the advantage is not confined to one architecture or capacity regime.Source: §D.2 · p. 16
  • The ablation concludes that Euclid30K delivers superior overall performance and broader generalization than an equal-sized non-geometric spatial dataset.Table 6 reports the comparison on VSI-Bench.Source: §D.2 · p. 16, Table 6 · p. 17

D.3. Comparison with Other Surrogate Tasks

The paper compares geometry with another surrogate-task approach and evaluates geometry-based supervised fine-tuning, reporting benchmark gains and a dataset limitation.

D.3. Comparison with Other Surrogate Tasks

  • On VSI-Bench, Euclid30K scores 31.4 versus Spatial-SSRL-7B’s 29.2 using 30K rather than 81K training samples.Euclid30K gains concentrate on size and distance estimation, while Spatial-SSRL performs better on route planning and appearance order.Source: §D.3 · p. 18
  • Euclid30K improves Super-CLEVR and remains competitive on Omni3D-Bench and MindCube under the same inference protocol.The cross-benchmark comparisons are reported in Table 8.Source: §D.3 · p. 18

E.1. Supervised Fine-tuning (SFT).

  • Geo170K supplies explicit reasoning trajectories for supervised fine-tuning, unlike Euclid30K, which lacks process annotations because such annotations are costly to collect.The authors use Geo170K to test geometry as a surrogate task under SFT as well as reinforcement learning.Source: §E.1 · pp. 18–19
  • Qwen3VL-30B-A3B rises from 40.0 to 43.7 on VSI-Bench after Geo170K SFT, while cross-benchmark scores rise on SuperCLEVR and Omni3D-Bench but fall on MindCube.MindCube changes from 39.8 to 38.3; the authors suggest plane-geometry-heavy Geo170K offers little direct supervision for viewpoint changes or 3D mental imagery.Source: Table 9 · p. 18, Table 11 · p. 18, §E.1 · p. 19

E.2. Model-Specific Performance Variations

The appendix reports model-specific transfer after Euclid30K training, discusses prior-knowledge boundaries, and provides further dataset and reasoning examples.

E.2. Model-Specific Performance Variations

  • VST-7B-RL’s VSI-Bench overall score moves from 44.3 to 55.5 after Euclid30K GRPO, with small gains on SuperCLEVR and Omni3DBench and a small MindCube decrease from 35.5 to 34.8.The authors relate the variation to the model’s inherited spatial capabilities and data coverage.Source: §E.2 · p. 19
  • Transfer may be limited when pretraining weakly represents key concepts such as viewpoint transformation, and geometric fine-tuning may offer little benefit or slightly interfere with existing heuristics.The authors note that MindCube relies heavily on viewpoint-change and spatiotemporal-consistency patterns, which VST data covers only weakly.Source: §E.2 · p. 19

F. More Visualization about Euclid30K

  • Additional examples span geometry problem types, diagram styles, and reasoning complexity, alongside spatial questions about object size, counting, distance, and relative direction.The appendix describes these examples as illustrating the range of configurations and formulations in Euclid30K.Source: F. More Visualization about Euclid30K · pp. 19, 20, 21, 25, 26, 27

A. True proposition

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