Topological structure in human spatial representation revealed through drawing

Reviewer #1 (Public review):

The paper presents novel evidence that spatial representations prioritize coarse topological features (T‑junctions, holes, crosses) over precise Euclidean metrics like angle and length, using drawing-based memory tasks with adults and children. The study is interesting and well‑motivated, and the importance of topological relations is clear, but stronger and more nuanced evidence is needed before concluding that topological relations are more important than metric details, as task difficulty and the potentially distinct roles of metric and topological information in spatial representation have not yet been fully disentangled.

Introduction
(1) P.5: Please explain in more detail what you mean by "What is relevant is the relative prioritization of each of these features."

Results
(2) P.8: Please clarify how the "proportion of drawings with angles biased towards 90{degree sign}" was computed. Specify the criterion for counting a drawing as biased (e.g., a certain absolute deviation toward 90{degree sign} from the original angle), and explicitly state in the Results that absolute degrees of deviation were used, as described in Methods.

(3) It would help to spell out whether the findings imply that obtuse angles are typically drawn smaller (closer to 90{degree sign}) and acute angles larger (closer to 90{degree sign}). Also, would angles be more biased toward 90{degree sign} or 180{degree sign} (or 0{degree sign}) depending on the angle? (e.g., 175{degree sign} is seen more as 180{degree sign} while 95 is seen more as 90{degree sign})

(4) Figure 4B: The statement that "positive values indicate bias in the direction of 90 degrees" needs a more precise explanation. Please explain exactly how the bias metric is computed (e.g., signed difference between drawn and original angle, with the sign indicating movement toward or away from 90{degree sign}) and what the y-axis values represent. Given that the Methods refer to absolute deviations, it would be useful to reconcile where the positive/negative signs come from in this plot.

(5) Figure 4C: The description in the Results seems to use a different metric than what is plotted. Please ensure that the measure in the text matches the measure shown in the figure, and adjust labels or wording so they align clearly.

(6) P.11: Consider briefly justifying why the authors predicted that participants would also add L‑junctions, rather than only remove them.

(7) P.12: The last sentence: Weren't the overall rates of feature preservation 'higher' in the adult sample?

Methods
(8) Experiment 1: Please clarify whether the angles associated with T‑ and L‑junctions were equated or differed systematically. A short description of stimulus generation (e.g., angle ranges, line lengths, junction configurations) would be helpful.

(9) It would also be helpful to specify the statistical tests used (e.g., t‑tests, ANOVAs, mixed‑effects models), including the main factors and any random effects, so readers can clearly follow your analysis pipeline.

Discussion
(10) It may be important to note that task difficulty likely differs across feature types: junctions involve presence/absence or counting, whereas angle and length reproduction require finer metric precision. The authors' claim of "prioritization" and possible difficulty effects should be disentangled.

(11) Furthermore, would it be possible that people retain relative order/comparison of different angles/lengths rather than computing precise values?

(12) I agree that topological relations are extremely important. However, for above reasons, it seems like stronger/stricter evidence is needed to claim that topological relations are 'more' important than metric details. They also might serve different roles in spatial representations

(13) The Discussion would benefit from a short paragraph on where different junction types (T, L, crosses) typically appear in everyday scenes and objects (e.g., as cues to occlusion, surface intersections, 3D structure) and what functions they serve. This would help connect your experimental findings to the ecological importance of these features for natural vision and spatial cognition.

Reviewer #2 (Public review):

Summary:

This is an interesting study that uses drawings to evaluate the extent to which visual representations of letter- and graph-like figures (preferentially) include topological features, like junctions and holes.

The main claim is based on the observation that when participants are asked to draw presented figures from memory, they tend to (1) regularise angles towards 90deg and lengths towards the average length of the lines in the figure, while (2) preserving topological features like T-junctions more assiduously than non-topological features like L-junctions. A third experiment with 'serial reproductions' in which participants copy drawings made by other participants (like a visual version of the 'broken telephone' game) reproduce these patterns in exaggerated form. These findings were also reproduced in children (Experiment 4).

These findings are consistent with the idea that memory representations are low-bandwidth or noisy approximations to the original figure. I would suggest that when participants are asked to reproduce the figure, it is if they combine the noisy stored representation, with generic priors about angles and the average line length. The preferential preservation of T- over L-junctions indicates that they are somehow more salient or memorable. This is not inconsistent with the authors' preferred interpretation of an explicit representation of topological structure. However, it is also not inconsistent with the idea that in order to compress the visual signals for storage, high-information (complex) components of the source are given preferential treatment. This would be compatible with optimal use of limited resources when compressing the information. Additional comparisons and control conditions would help tease these alternatives apart.

Strengths:

+ Innovative use of drawing methods to probe internal visual representations
+ Experiments spanning both adults and children

Weaknesses:

- Failure to consider alternative hypotheses that are consistent with the findings

Reviewer #3 (Public review):

Kittur et al. ask whether human spatial memory is organized around topological relations (meaning coarse structural properties such as T-junctions, crosses, and holes) rather than around Euclidean properties such as angle and length. Across four experiments, adults and children studied letter-like figures and reproduced them from memory by drawing. The authors report two complementary patterns: metric features are systematically distorted, with angles pulled toward 90 degrees and line-length ratios compressed toward an average, while topologically critical features are comparatively well preserved. The central test contrasts T-junctions with L-junctions, which are visually similar but topologically distinct, since an L-junction reduces to a straight line, whereas a T-junction does not. A serial reproduction experiment amplifies both patterns across chains of participants, and a fourth experiment extends the findings to children aged five to eight.

Strengths:

The question is a good one and sits at a productive intersection of topics. It bears on debates about the representational format of cognitive maps, on proposals about the primitives of visual perception, and on a classic developmental claim from Piaget and Inhelder that has rarely been tested directly.

The drawing paradigm is well chosen and offers something that the group's earlier forced-choice work could not. Because participants produce an open-ended response, distortion of metric detail and preservation of structure can be observed within a single response, and the relationship between them can be examined directly. The serial reproduction experiment is a particularly effective use of this affordance. The choice of the T-junction versus L-junction contrast as the primary test is well-motivated, since it holds the number of junctions constant and varies only topological relevance. It is also worth noting for readers that the central claim of a representational privilege for topologically distinct features was previously established by this group using forced-choice paradigms in both adults and children. That a similar conclusion emerges from free generation is a genuine strength, since the two methods have very different sources of error.

The work is carefully executed. Sample sizes, dependent variables, and analyses were preregistered; stimuli were purpose-built for each question, including the deliberate exclusion of 90-degree angles so that no reference angle was available; drawings were double-coded; and the full set of raw drawings is being released publicly.

Weaknesses:

Drawing is treated as a transparent window onto representation, and motor limitations are not considered. Drawing is a motor act, drawing skill varies widely across individuals, and the manuscript does not discuss motor limitations at any point. As the study is designed, representational imprecision cannot be separated from difficulty of precise reproduction. The clearest way to resolve it might be asking adults to copy the figures exactly while the stimulus remains visible. If the biases persist under direct copying, then some portion of the effect is production rather than memory. Because the topological findings have already been demonstrated in keypress-only paradigms, this concern affects the metric distortion results most heavily, which are the novel contribution of the present paper.

Three distinct claims are treated as one, and the data speak mainly to the weakest of them. The paper moves between a claim about mnemonic robustness (topological features survive degradation better than metric features), one about representational architecture (topology is a base layer with metric detail superimposed on top), and a claim about priority (topology is encoded prior to metric detail). The experiments show evidence for robustness, which is a claim about what is lost first. Robustness does not entail architecture: an encoder with a single layer, whose loss happens to spare structure, produces the same pattern with no layered format and no claim about encoding order. Earlier work does address format, because false "same" judgments to topologically matched but metrically different items show that topology plays a role in what the system treats as equivalent. Preservation counting measures robustness, rather than equivalence.

Some alternative hypotheses to consider/address: (a) A capacity-limited memory that reconstructs from a prior produces the metric distortions with no commitment to topology. A literal absence of angle encoding, which the authors invoke, predicts noisy and unconstrained recall rather than recall pulled toward a particular value. The observed pattern reflects a structured prior. (b) The result that does discriminate might be confounded with local salience. A memory that adds uniform noise to all parts of a figure does not predict that T-junctions are preserved better than L-junctions; however, a three-way branch point is plausibly more locally distinctive than a corner, so a salience-weighted-but-topology-free account predicts the same ordering. (c) Motor simplification also predicts the same ordering, since omitting an L-junction converts a bend into a straight line, which is easier to draw, whereas omitting a T-junction requires dropping a stroke.

The better a feature works as a topological marker, the less variance it produces and the harder it is to test, so the method is best powered where the theoretical signal is weakest. Holes are the textbook case of a topological invariant and are reported to disappear from drawings less than one percent of the time, but they are excluded from formal analysis because they are at ceiling and have no matched comparison feature. It would be useful to see bidirectional rates for holes (both how often a hole disappears and how often participants spuriously close an open figure into a loop).

In Experiment 4, the conclusion of developmental stability rests on a nonsignificant effect of age, which is failure to detect a change rather than evidence of stability. An equivalence test or an estimate of the precision of the null is better support for developmental stability. Motor skill is confounded with age throughout. So this is an experiment that shows that the effect generalizes to childhood, but cannot adjudicate a developmental question.

In the serial reproduction experiment, chains were intermixed so that each participant contributed one drawing to each of ten chains. The final drawings are therefore linked through shared intermediate participants, and an individual with an idiosyncratic drawing style influences ten chains at once, so the reported degrees of freedom are somewhat generous. The analysis also focuses on the final drawings and sets aside the nine hundred intermediate ones, which are the data that would show where in a chain metric detail collapses and whether structure ever breaks.

To formalize the topology is to strengthen the argument: each figure is a one-dimensional complex (its underlying graph), treated intrinsically and up to homeomorphism. The homeomorphism type is what remains after suppressing all degree-2 vertices. Under this definition, every feature in this paper's taxonomy becomes one kind of object, namely a homeomorphism invariant of the graph: number of components, first Betti number, and the degree sequence of three or greater with its adjacency structure. Relatedly, the term "metric" needs to be unpacked. The 90-degree bias concerns angle, whereas the 4:2:1 result concerns length ratios, which are affine rather than strictly metric. The stronger statement available is that distortion appears at every level above topology in the transformation hierarchy while preservation occurs at the topological level, which connects directly to Chen's (2005) invariance hierarchy that is already cited.

Appraisal and impact:

The authors aimed to show that topological structure is preferentially retained in memory while metric detail is lost, and in the sense of relative preservation they succeed. The dissociation is real, replicates across two stimulus sets, amplifies under serial reproduction, and appears in young children. What the data do not establish is the stronger architectural claim that topology is a base representational layer, nor that the metric distortions specifically implicate topology rather than general properties of reconstructive memory. Separating these claims would communicate the well-supported result better. Conceptually, the work strengthens a growing case that coarse relational structure deserves a place alongside Euclidean properties in accounts of spatial representation. Practically, the public release of the full set of adult and child drawings, including excluded ones, is a resource that will support analyses well beyond those reported in this paper, and the serial reproduction design is a method that others will want to borrow.