Euclidean geometry is among the earliest forms of mathematical thinking. While the geometric primitives underlying its constructions, such as perfect lines and circles, do not often occur in the natural world, humans rarely struggle to perceive and reason with them. Will computer vision models trained on natural images show the same sensitivity to Euclidean geometry? Here we explore these questions by studying few-shot generalization in the universe of Euclidean geometry constructions. We introduce Geoclidean, a domain-specific language for Euclidean geometry, and use it to generate two datasets of geometric concept learning tasks for benchmarking generalization judgements of humans and machines. We find that humans are indeed sensitive to Euclidean geometry and generalize strongly from a few visual examples of a geometric concept. In contrast, low-level and high-level visual features from standard computer vision models pretrained on natural images do not support correct generalization. Thus Geoclidean represents a novel few-shot generalization benchmark for geometric concept learning, where the performance of humans and of AI models diverge. The Geoclidean framework and dataset are publicly available for download.

Euclidean geometry has historically played a central role in cultivating logical reasoning and abstract thinking within mathematics education, but has experienced waning emphasis in recent curricula. The resurgence of interest, driven by advances in artificial intelligence and educational technology, has highlighted geometry's potential to develop essential cognitive skills and inspired new approaches to automated problem solving and proof verification. This article presents an ontology-based framework for annotating and optimizing geometry problem sets, originally developed in the 1990s. The ontology systematically classifies geometric problems, solutions, and associated skills into interlinked facts, objects, and methods, supporting granular tracking of student abilities and facilitating curriculum design. The core concept of 'solution graphs'--directed acyclic graphs encoding multiple solution pathways and skill dependencies--enables alignment of problem selection with instructional objectives. We hypothesize that this framework also points toward automated solution validation via semantic parsing. We contend that our approach addresses longstanding challenges in representing dynamic, procedurally complex mathematical knowledge, paving the way for adaptive, feedback-rich educational tools. Our methodology offers a scalable, adaptable foundation for future advances in intelligent geometry education and automated reasoning.

Euclidean geometry is among the earliest forms of mathematical thinking. While the geometric primitives underlying its constructions, such as perfect lines and circles, do not often occur in the natural world, humans rarely struggle to perceive and reason with them. Will computer vision models trained on natural images show the same sensitivity to Euclidean geometry? Here we explore these questions by studying few-shot generalization in the universe of Euclidean geometry constructions. We introduce Geoclidean, a domain-specific language for Euclidean geometry, and use it to generate two datasets of geometric concept learning tasks for benchmarking generalization judgements of humans and machines.

Artificial neural networks (ANNs) were inspired by the architecture and functions of the human brain and have revolutionised the field of artificial intelligence (AI). Inspired by studies on the latent geometry of the brain, in this perspective paper we posit that an increase in the research and application of hyperbolic geometry in ANNs and machine learning will lead to increased accuracy, improved feature space representations and more efficient models across a range of tasks. We examine the structure and functions of the human brain, emphasising the correspondence between its scale-free hierarchical organization and hyperbolic geometry, and reflecting on the central role hyperbolic geometry plays in facilitating human intelligence. Empirical evidence indicates that hyperbolic neural networks outperform Euclidean models for tasks including natural language processing, computer vision and complex network analysis, requiring fewer parameters and exhibiting better generalisation. Despite its nascent adoption, hyperbolic geometry holds promise for improving machine learning models through brain-inspired geometric representations.

Since the publication of CLIP, the approach of using InfoNCE loss for contrastive pre-training has become widely popular for bridging two or more modalities. Despite its wide adoption, CLIP's original design choices of L2 normalization and cosine similarity logit have rarely been revisited. We have systematically experimented with alternative geometries and softmax logits for language-image pre-training and identified that variants with intuitive Euclidean geometry, Euclidean CLIP (EuCLIP), match or exceed the performance of CLIP and support hierarchical relationships at least as well as more complicated hyperbolic alternative.

A recent numerical study observed that neural network classifiers enjoy a large degree of symmetry in the penultimate layer. Namely, if $h(x) = Af(x) +b$ where $A$ is a linear map and $f$ is the output of the penultimate layer of the network (after activation), then all data points $x_{i, 1}, \dots, x_{i, N_i}$ in a class $C_i$ are mapped to a single point $y_i$ by $f$ and the points $y_i$ are located at the vertices of a regular $k-1$-dimensional tetrahedron in a high-dimensional Euclidean space. We explain this observation analytically in toy models for highly expressive deep neural networks. In complementary examples, we demonstrate rigorously that even the final output of the classifier $h$ is not uniform over data samples from a class $C_i$ if $h$ is a shallow network (or if the deeper layers do not bring the data samples into a convenient geometric configuration).

Randomized dimensionality reduction techniques, such as random projection (RP) [7, 15] and Ho's random subspace method (RS) [12] are popular approaches for data compression, with many empirical studies showing the utility of both for machine learning and data mining tasks in practice [26, 11, 21, 19, 18, 27]. For RP a key theoretical motivation behind their use is the Johnson-Lindenstrauss lemma (JLL), the usual constructive proof of which also implies an algorithm with high-probability geometry preservation guarantees for projected data. However RP is costly to apply to large or high-dimensional datasets since it requires a matrix-matrix multiplication to implement the projection, and furthermore the projected features may be hard to interpret. On the other hand RS is a particularly appealing approach for dimensionality reduction because it involves simply selecting a subset of data feature indices randomly without replacement, and so does not require a matrix-matrix multiplication to implement the projection and it retains (a subset of) the original features. RS is therefore computationally far more efficient in practice, and more interpretable than RP, but there is little theory to explain its effectiveness. Focusing on this latter problem, here we prove data-dependent norm-preservation guarantees for data projected onto a random subset of the data features.

Methods of Approach At the outset it might be well to distinguish sharply between two general approaches to the problem of machine learning.

We determine the implicit assumptions and the structure of the Single and Double Cross Calculi within Euclidean geometry, and use these results to guide the construction of analogous calculi in mereogeometry.  The systems thus obtained have strong semantic and deductive similarities with the Euclidean-based Cross Calculi although they rely on a different geometry. This fact suggests that putting too much emphasis on usual classification of qualitative spaces may hide important commonalities among spaces living in different classes.