A fossil skull has two shape histories. The first was made by growth: sutures closed, muscles pulled, teeth erupted and the face changed proportions as the animal matured. The second began after death. Sediment pressed from above, a slab sheared sideways, joints opened, thin walls buckled and mineral growth fixed the altered object in stone.

Both histories reach the museum together. A ruler measures their combined result.

Geometric morphometrics was built to make shape more explicit. Instead of reducing a skull to a handful of lengths and ratios, it records the coordinates of anatomically corresponding points and preserves their spatial relationships through the analysis. That makes subtle changes in an orbit, jaw joint or skull roof statistically visible. It also creates a dangerous illusion: once a fossil becomes a precise cloud of coordinates, its damage can look as quantitative as its biology. The software measures the geometry it receives; it does not know which force produced it.[1][2]

This is why the method is most revealing when its limits remain in view. It can compare shape, expose deformation and test a reconstruction. It cannot make burial disappear by calculation.

A landmark is a biological claim

Traditional measurements still answer useful questions. Skull length, jaw depth and tooth-row length are easy to understand, repeat and compare. Their weakness is that two objects can share the same selected distances while arranging the tissue between those endpoints differently. Ratios also multiply quickly, often reuse the same measurements and discard much of the original geometry.

Landmark-based morphometrics keeps that geometry. An investigator might mark the tip of a snout, the meeting point of two sutures, the rim of an eye socket and the centre of a jaw articulation on every specimen. Each landmark is supposed to represent the same anatomical location across the sample. Where a smooth curve or broad surface offers too few discrete points, semilandmarks can sample its contour; they are allowed to slide according to a stated mathematical rule so that arbitrary spacing contributes less to the comparison.[1][2]

The word “same” carries most of the scientific burden. A point at the intersection of homologous sutures has a stronger anatomical identity than “the place where this curve bends most.” A semilandmark describes geometry well but does not acquire biological homology merely because a computer assigned it coordinates. Broken edges, obscuring matrix and fused sutures can also make the intended point unavailable. Choosing a substitute, estimating a missing point or excluding the specimen changes the question the dataset can answer.[1][6]

Landmark placement is therefore not clerical work before the analysis. It is the analysis's first anatomical hypothesis.

What Procrustes alignment removes

Photograph two identical skulls at different distances and angles and their raw coordinates will disagree. The usual solution is Generalized Procrustes Analysis. Each landmark configuration is moved to a common centre, scaled to a common centroid size and rotated to minimize the summed squared distances between corresponding points. Translation, orientation and overall scale no longer dominate the comparison. What remains is variation in the relative arrangement of landmarks: operationally, shape.[1][2]

Removing scale does not prove that size is biologically irrelevant. Growth can change proportions, a relationship called allometry, so researchers commonly test shape coordinates against centroid size or divide a sample into defensible age classes. Nor does alignment erase perspective error in a two-dimensional photograph of a three-dimensional skull. A flattened object such as a leaf or footprint may tolerate a 2D treatment; a deep skull can require 3D landmarks because its visible outline changes with viewing angle.[1][3]

The aligned coordinates can then be explored with principal component analysis. PCA turns many correlated coordinate shifts into a smaller set of axes that describe the largest patterns of variation. Specimens plotted along those axes occupy a morphospace. Nearby points have similar scores on the displayed axes, although components omitted from the plot can still contain differences.

That plot is a map of variance, not a machine for naming species. Its axes do not arrive labelled “evolution,” “sex,” “age” or “crushing.” Those explanations have to be tested with specimen history, anatomy, stratigraphy, sample design and independent characters. A cluster can be biologically meaningful. It can also be a set of fossils flattened in the same direction.[2][6]

Thirty Psittacosaurus skulls became a taphonomic experiment

The Lujiatun beds of northeastern China's Yixian Formation preserve Early Cretaceous dinosaurs in three dimensions, but “three-dimensional” does not mean undistorted. In 2013, Brandon Hedrick and Peter Dodson digitized 30 psittacosaurid skulls from the same beds to examine three names then applied to the material: Psittacosaurus lujiatunensis, P. major and Hongshanosaurus houi. Keeping locality and geological interval constant reduced two obvious sources of variation before the shape analysis began.[3]

Their workflow makes the method's judgement calls visible. They initially collected 56 three-dimensional landmarks. Two skulls were too incomplete for the principal component analysis, and landmarks missing across several of the remaining specimens were removed, leaving 44 in the final configuration. They repeatedly landmarked one skull to estimate measurement error. Crucially, they did not ask coordinates to decide taxonomy alone: they also re-examined the anatomical characters that had been used to separate the named forms.[3]

In that study, the skulls did not resolve into clean biological groups. Much of the spread through morphospace followed dorsoventral crushing, lateral compression and asymmetric one-sided crushing. Specimens assigned to the same species could land far apart because one skull roof had collapsed differently from another. The authors concluded that the three named Lujiatun forms represented one species, P. lujiatunensis, expressed as different taphomorphotypes—recurring shapes made by preservation—and warned that morphometrics needed the character review, not a role as its replacement.[3]

The taxonomic conclusion belongs to that analysis and evidence set; it is not a universal rule that differently shaped fossils are synonyms. Its wider lesson is methodological. Fossils from one population need not form a tight cloud if rock has imposed structured, directional change. Conversely, a neat cluster need not be a species if a common burial process made its members resemble one another.

The cover photograph is useful precisely because it is not a diagram. It shows the kind of object that enters the measurement chain: a museum skull identified on Commons as P. lujiatunensis, with boundaries that appear obvious at page scale but become choices when every suture, opening and displaced surface must receive a coordinate. The image is illustrative; the Commons record gives no specimen number, does not say whether the display is original fossil material or a cast, and does not identify it as one of the 30 skulls in the 2013 dataset.[3][8]

Diictodon shows that the answer depends on scale

A later study tested the same problem with Diictodon feliceps, a small dicynodont therapsid abundant in the Permian rocks of South Africa's Karoo Basin. The sample was valuable because numerous skulls preserve many versions of damage rather than one exceptional, tidy specimen. Researchers recognized seven recurrent deformation styles, including side-to-side and top-to-bottom compression, shortening from front to back, localized “saddle” collapse and several kinds of shear. Only about six percent of the examined skulls were classed as undistorted.[4]

In the empirical morphometric datasets, deformation was the dominant source of shape variation. Signals expected from growth and sexual dimorphism were weak or obscured. Skulls tended to occupy regions of morphospace associated with how they had been crushed, much as the Lujiatun Psittacosaurus skulls had. Rock pressure was not random visual noise around a pristine form; skull architecture and burial orientation channelled it into repeatable outcomes.[4]

Yet the study also found an important boundary. Within Diictodon, deformation could overwhelm the relatively small differences at issue. In a broader analysis across anomodont therapsids, the larger anatomical differences among groups still carried a strong phylogenetic signal, while the total effect of deformation was comparatively minor. For some high-level disparity questions, species means could approximate the undistorted position well enough to remain useful.[4]

So there is no honest label reading “too deformed for morphometrics.” Fitness depends on scale and claim. Damage fatal to a proposed difference between sexes might be tolerable when comparing body plans separated by much larger evolutionary distances. The uncertainty belongs beside the conclusion, not merely in a methods appendix.

Retrodeformation is an explicit model of the missing skull

Digital retrodeformation tries to estimate shape before geological distortion. Bilaterally symmetrical anatomy offers one route: mirror corresponding landmarks or semilandmarks from the two sides, estimate a plane of symmetry and warp the model toward a more symmetrical form. A 2018 implementation in the R package Morpho spread semilandmarks across curves and surfaces, tested the procedure on a deliberately deformed gorilla cranium and then applied it to a distorted Neanderthal skull.[5]

That is more reproducible than silently pushing fragments until a skull “looks right,” but it is not time reversal. The result depends on which points were paired, how much surface was preserved, the interpolation used between control points and the assumption that observed asymmetry is damage. Living organisms are not perfectly symmetrical. A disease, injury or normal developmental difference can be real biology; an algorithm instructed to restore symmetry may erase it. Symmetrical flattening can also escape a method that relies on left-right discrepancy because both sides were altered together.[2][5][7]

A responsible reconstruction should therefore remain linked to the unaltered scan, its landmark set and alternative parameter choices. If a downstream bite-force model, taxonomic diagnosis or evolutionary tree changes when a plausible second restoration is used, the restoration uncertainty is part of the result.

Read the coordinate chain, not just the coloured plot

The strongest geometric-morphometric fossil studies make five boundaries inspectable. They define why each landmark corresponds anatomically; report which fossils and points were excluded or estimated; separate size, shape and allometry; test observer error and taphonomic pattern; and keep exploratory separation in morphospace distinct from a confirmed biological group. When retrodeformation is involved, they also preserve a route back to the fossil as found.[1][2][5][6][7]

None of this diminishes the method. It explains its real power. Geometric morphometrics can retain spatial information that rulers lose, compare whole configurations reproducibly and reveal that a supposed evolutionary signal follows a burial direction instead. In the best cases, the method does not simply rescue biological shape from damaged fossils. It turns damage itself into evidence about compaction, orientation and preservation.

The fossil skull still has two histories. Coordinates make both measurable. Paleontology begins when the analysis refuses to confuse them.

Sources

1. Mark Webster and H. David Sheets, “A Practical Introduction to Landmark-Based Geometric Morphometrics,” The Paleontological Society Papers 16 (2010)—landmarks, semilandmarks, Procrustes superimposition and common paleobiological analyses.
2. Philipp Mitteroecker and Katrin Schaefer, “Thirty years of geometric morphometrics: Achievements, challenges, and the ongoing quest for biological meaningfulness,” American Journal of Biological Anthropology 178 (2022)—shape variables, alignment, visualization and interpretation limits.
3. Brandon P. Hedrick and Peter Dodson, “Lujiatun Psittacosaurids: Understanding Individual and Taphonomic Variation Using 3D Geometric Morphometrics,” PLOS ONE 8 (2013)—30 skulls, landmark workflow, taphomorphotypes and the Lujiatun taxonomic result.
4. Christian F. Kammerer et al., “Effects of taphonomic deformation on geometric morphometric analysis of fossils: a study using the dicynodont Diictodon feliceps,” PeerJ 8 (2020)—seven deformation styles and the scale-dependent effect of distortion on biological signals.
5. Stefan Schlager et al., “Retrodeformation of fossil specimens based on 3D bilateral semi-landmarks: Implementation in the R package ‘Morpho’,” PLOS ONE 13 (2018)—a reproducible symmetry-based restoration method, simulations and a fossil application.
6. Alessandro Palci and Michael S. Y. Lee, “Geometric morphometrics, homology and cladistics: review and recommendations,” Cladistics 35 (2019)—landmark homology and why shape coordinates complement rather than replace anatomical characters.
7. Stephan Lautenschlager, “Reconstructing the past: methods and techniques for the digital restoration of fossils,” Royal Society Open Science 3 (2016)—a review of restoration workflows, symmetry assumptions and how retrodeformation can alter biological information.
8. Ghedoghedo, “Psittacosaurus lujiatunensis skull,” Wikimedia Commons (CC BY-SA 4.0; photographed at the Royal Belgian Institute of Natural Sciences, 2015)—source page for the article's museum photograph, resized here to 1,600 × 1,427 pixels; the page provides no specimen number or original-versus-cast identification.