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

arXiv: https://arxiv.org/abs/2509.24473 · HTML: https://mlsanity.com/paper/2509.24473/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 59 of its 96 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)
- Multimodal LLMs / vision-language models / [Embodied and 3D vision-language models](https://mlsanity.com/mllm/embodied-and-3d-vision-language-models/)

## Builds on (most cited first, 30 of 59)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Visual Instruction Tuning](https://mlsanity.com/paper/2304.08485/) (2023)
- [BLIP-2: Bootstrapping Language-Image Pre-training with Frozen Image Encoders and Large Language Models](https://mlsanity.com/paper/2301.12597/) (2023)
- [Qwen2.5-VL Technical Report](https://mlsanity.com/paper/2502.13923/) (2025)
- [GPT-4o System Card](https://mlsanity.com/paper/2410.21276/) (2024)
- [Flamingo: a Visual Language Model for Few-Shot Learning](https://mlsanity.com/paper/2204.14198/) (2022)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context](https://mlsanity.com/paper/2403.05530/) (2024)
- [ScanNet: Richly-annotated 3D Reconstructions of Indoor Scenes](https://mlsanity.com/paper/1702.04405/) (2017)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [LLaVA-OneVision: Easy Visual Task Transfer](https://mlsanity.com/paper/2408.03326/) (2024)
- [InternVL: Scaling up Vision Foundation Models and Aligning for Generic Visual-Linguistic Tasks](https://mlsanity.com/paper/2312.14238/) (2023)
- [MMMU: A Massive Multi-discipline Multimodal Understanding and Reasoning Benchmark for Expert AGI](https://mlsanity.com/paper/2311.16502/) (2023)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Gemma 3 Technical Report](https://mlsanity.com/paper/2503.19786/) (2025)
- [MathVista: Evaluating Mathematical Reasoning of Foundation Models in Visual Contexts](https://mlsanity.com/paper/2310.02255/) (2023)
- [Expanding Performance Boundaries of Open-Source Multimodal Models with Model, Data, and Test-Time Scaling](https://mlsanity.com/paper/2412.05271/) (2024)
- [VGGT: Visual Geometry Grounded Transformer](https://mlsanity.com/paper/2503.11651/) (2025)
- [LlamaFactory: Unified Efficient Fine-Tuning of 100+ Language Models](https://mlsanity.com/paper/2403.13372/) (2024)
- [CLEVR: A Diagnostic Dataset for Compositional Language and Elementary Visual Reasoning](https://mlsanity.com/paper/1612.06890/) (2016)
- [ViLT: Vision-and-Language Transformer Without Convolution or Region Supervision](https://mlsanity.com/paper/2102.03334/) (2021)
- [PaLI: A Jointly-Scaled Multilingual Language-Image Model](https://mlsanity.com/paper/2209.06794/) (2022)
- [ScanNet++: A High-Fidelity Dataset of 3D Indoor Scenes](https://mlsanity.com/paper/2308.11417/) (2023)
- [DeepSeek-VL: Towards Real-World Vision-Language Understanding](https://mlsanity.com/paper/2403.05525/) (2024)
- [SpatialVLM: Endowing Vision-Language Models with Spatial Reasoning Capabilities](https://mlsanity.com/paper/2401.12168/) (2024)
- [VILA: On Pre-training for Visual Language Models](https://mlsanity.com/paper/2312.07533/) (2023)
- [Thinking in Space: How Multimodal Large Language Models See, Remember, and Recall Spaces](https://mlsanity.com/paper/2412.14171/) (2024)
- [LLaVA-Video: Video Instruction Tuning With Synthetic Data](https://mlsanity.com/paper/2410.02713/) (2024)
- [Long Context Transfer from Language to Vision](https://mlsanity.com/paper/2406.16852/) (2024)

## Built on by (most cited first, 13 of 13)

- [MMFineReason: Closing the Multimodal Reasoning Gap via Open Data-Centric Methods](https://mlsanity.com/paper/2601.21821/) (2026)
- [Embodied-R1.5: Evolving Physical Intelligence via Embodied Foundation Models](https://mlsanity.com/paper/2606.11324/) (2026)
- [ScalSelect: Scalable Training-Free Multimodal Data Selection for Efficient Visual Instruction Tuning](https://mlsanity.com/paper/2602.11636/) (2026)
- [Persistent Visual Memory: Sustaining Perception for Deep Generation in LVLMs](https://mlsanity.com/paper/2605.00814/) (2026)
- [PhysBrain 1.0 Technical Report](https://mlsanity.com/paper/2605.15298/) (2026)
- [DynaSolidGeo: A Dynamic Benchmark for Genuine Spatial Mathematical Reasoning of VLMs in Solid Geometry](https://mlsanity.com/paper/2510.22340/) (2025)
- [MMVistaReason: Toward Open-Data and Post-Training Recipes for Multimodal Reasoning](https://mlsanity.com/paper/2610.01352/) (2026)
- [Why MLLMs Struggle to Determine Object Orientations](https://mlsanity.com/paper/2604.13321/) (2026)
- [RoboPIN: Grounded Embodied Reasoning via Pinned Chain-of-Thought](https://mlsanity.com/paper/2606.15753/) (2026)
- [MentalThink: Shaping Thoughts in Mental SVG World](https://mlsanity.com/paper/2607.03530/) (2026)
- [When Prompts Become Pixels: Prompt-Region Grounding for Multimodal Reasoning](https://mlsanity.com/paper/2608.04726/) (2026)
- [EvoSpatial: Experience-Driven Skill and Tool Evolution for Spatial Reasoning Agents](https://mlsanity.com/paper/2610.16679/) (2026)
- [ABot-ER: Unified Embodied Reasoning through Agentic Data-Model Co-Evolution](https://mlsanity.com/paper/2610.18355/) (2026)

## Abstract (arXiv)

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 hrefhttps://zgca-ai4edu.github.io/EuclidsGiftthis.