Very little is known about how morphology affects the motion, spatial stability and resulting viability of avian eggs. The limited existing research focuses on the uniquely pyriform egg shapes found in the Alcidae bird family. This unusual shell shape was originally thought to suppress displacement and prevent egg loss on the cliffside nesting habitat of the Uria genus. Unfortunately, these early studies never isolated or quantified the specific morphological features (elongation, asymmetry and conicality) of these pyriform eggs, which limits their applicability to other taxa and has hampered a robust proof of concept. We isolated each feature as an enumerated variable, produced model 3D printed eggs with incremental expressions of a single variable and then with all three features co-varying simultaneously. Recorded motion (egg-rolling) trials were conducted to test the individual and combined effects of each morphological characteristic on displacement over a range of inclines representative of the conditions found in natural habitats. Increasing elongation and asymmetry significantly increased displacement, whereas increased conicality decreased displacement in the single-variable egg models. In the multivariable egg models, only conicality consistently suppressed displacement, while lower levels of asymmetry significantly increased displacement. Our findings broadly support previous studies' assertions of the adaptive value of the pyriform eggs while also providing methodology and comparative data for future analyses of the interactions between nesting habitat, behavior and egg shape, beyond the confines of a handful of focal species.

The shape of a bird's egg and its functional significance have intrigued naturalists for hundreds of years while also presenting modern science with many questions about their immense variety across and within avian lineages (Hauber, 2014; Birkhead, 2016; Stoddard et al., 2017). A substantial body of work exists exploring the adaptive values of the shell, including how the width and length contribute to structural robustness (Bain and Solomon, 1991) as well as how the shape of an egg contributes to efficient packing underneath the incubating parent for thermal stasis (Barta and Székely, 1997). Gas and moisture exchange have been speculated to be affected by a shell's shape and relative pore density over the surface (Birkhead et al., 2017a; Ar, 1991), and even the ease of hatching for young birds has been linked to egg morphology (Cucco et al., 2012; Igic et al., 2015). Some recent and exciting work has suggested flight ability as an adaptive driver for egg shape, with strong flyers being more likely to produce asymmetrical and elongated eggs (Stoddard et al., 2017), as well as climate and microhabitat having an effect on Australian bird egg morphology (Duursma et al., 2018). In contrast, the relationship of egg shape to egg movement (e.g. rolling) is an understudied topic (Birkhead et al., 2017a; Birkhead et al., 2018), particularly considering the relevance that spatial stability might have for successful reproduction of certain colonial egg-laying taxa without distinct nest structures, breeding on rock ledges or at other, similarly precipitous sites (Goodfellow, 2011).

The largest body of work on avian egg shape and motion has been carried out on the family Alcidae, specifically on the common murre, or common guillemot (Uria aalge), and the thick-billed murre (Uria lomvia) as to the adaptive value of this genus' curiously and intensely pyriform eggs (reviewed in Birkhead, 2016; Tschanz et al., 1969; Ingold, 1980). Previous studies aimed to demonstrate whether the ‘pear’ or pyriform shape of the murre shell prevents egg loss by rolling in a consistent and tight circle, therefore reducing the distance by which an egg is displaced when the incubating bird loses hold of the egg; a common occurrence in murre colonies during social conflict or predator avoidance (Tschanz et al., 1969; Ingold, 1980; Preston, 1953). A recent study also examined the pyriform-shaped egg's ability to aid in stabilization by the incubating parent (Birkhead et al., 2018). The fitness advantage of the pear form to limit the rolling radius seems highly intuitive given that these species incubate their single egg without a nest along flat or uneven rock ledges of tall cliffs, which can be as shallow as the egg is long, ∼10 cm (Gaston and Jones, 1998; Uspenski, 1956). This rolling theory has even permeated popular culture and the ornithological field since its conception in the 1830s (Birkhead, 2016), but recent challenges to it have raised attention to alternative adaptive functions of a pyriform shape (Stoddard et al., 2017) and to the murre's egg form in particular (Birkhead et al., 2017a,b; Birkhead et al., 2018).

Literature to date asserts that the murre shell is pear shaped, distinguishing it from the presumed ancestral eggs' spheroid and soft elliptical egg forms (Uspenski, 1956; Birkhead, 2016). To deconstruct the pear shape, the U. aalge egg is deemed to have highly long and asymmetrical proportions, with the longer and volumetrically dominant half of the shell possessing a highly ‘pointed’ end (Stoddard et al., 2017; Tschanz et al., 1969; Ingold, 1980; Gaston and Jones, 1998). The pointed end typically has a highly conical quality, with a large portion of that segment of its profile ranging in shape from gently curved to tightly linear in form, which is thought to provide a stable point of contact with the resting surface (Uspenski, 1956; Birkhead et al., 2017a). The pointed end (termed the ‘conical element’ from here onward) is thought to determine both the angle at which the egg rests and its rolling radius, with the radial axis pointing to a theoretical center-locus of the rolling arch (Uspenski, 1950) (see Fig. 1). Comparatively more rounded eggs of a similar size and width-to-length proportion would produce a smaller resting angle and a larger radius and, hence, displacement (Uspenski, 1956). It has also been suggested that, as the embryo develops and retracts into the conical end of the shell, the weight shifts toward this end as opposed to the blunt end (with the air pocket), increasing the resting angle and further constricting the resulting rolling radius of the egg (Uspenski, 1950, 1956).

Fig. 1.

Model for predicting egg displacementwith resting angle and shape. The shape of an egg's profile is theorized to predict its rolling distance or maximum displacement, being twice the rolling radius. An egg at rest on a surface (A, side view) has a tilted radial axis. The tilt of the radial axis relative to the surface is defined partially by the shape of the egg and the weight distribution therein. The resting angle of the egg is thought to predict the rolling radius and affect the maximum displacement of an egg in motion (B, top view). Eggs such as the blue and orange ones above, with a highly pyriform shape and a conical element to their pointed end, are thought to reduce maximum displacement (Belopol'skii, 1961). There is no metric for conicality, but there is a measure of ‘pointedness’ (D), which is more accurately described as asymmetry as it measures the proportional distance of the widest part of the egg from the pointed end (Birkhead et al., 2017a; Deeming and Ruta, 2014). The ‘pointedness’ of the orange and the blue eggs is equal, despite their clear difference in profile, which likely affects their resting angle and the resulting maximum displacement.

Fig. 1.

Model for predicting egg displacementwith resting angle and shape. The shape of an egg's profile is theorized to predict its rolling distance or maximum displacement, being twice the rolling radius. An egg at rest on a surface (A, side view) has a tilted radial axis. The tilt of the radial axis relative to the surface is defined partially by the shape of the egg and the weight distribution therein. The resting angle of the egg is thought to predict the rolling radius and affect the maximum displacement of an egg in motion (B, top view). Eggs such as the blue and orange ones above, with a highly pyriform shape and a conical element to their pointed end, are thought to reduce maximum displacement (Belopol'skii, 1961). There is no metric for conicality, but there is a measure of ‘pointedness’ (D), which is more accurately described as asymmetry as it measures the proportional distance of the widest part of the egg from the pointed end (Birkhead et al., 2017a; Deeming and Ruta, 2014). The ‘pointedness’ of the orange and the blue eggs is equal, despite their clear difference in profile, which likely affects their resting angle and the resulting maximum displacement.

At least two researchers have conducted motion-based experiments with plaster model eggs and/or field-collected specimens that supported the notion that these species have a lower maximum displacement than more typical, ellipsoid bird eggs (Tschanz et al., 1969; Ingold, 1980). Murre egg movement dynamics are often contrasted with those of the closely related and sympatrically breeding razorbill Alca torda (Tschanz et al., 1969; Ingold, 1980). Specifically, this species will periodically roost and even nest alongside common murres, although razorbills predominantly nest in well-protected, semi-enclosed crevices, including on pebbled beds (Brooks, 1985), and have less pear-shaped eggshells (Tschanz et al., 1969). These comparative studies included analyses of cliffside egg loss patterns and some limited cross-fostering experiments with artificial and field-collected egg specimens (Tschanz et al., 1969; Ingold, 1980).

Unfortunately, the early efforts on egg rolling lacked clear definitions and relevant metrics on egg shape, including conicality, which would be at the core of the theory. For example, the two hallmark studies conducted by Tschanz et al. (1969) and Ingold (1980) did not provide a quantifiable measure for pyriformity, elongation or asymmetry. Their models were arranged in an ordinal scale of egg sharpness or pointedness, made by visual inspection without a quantification or clear geometric description of either term. A ‘pointedness’ measurement (Birkhead et al., 2017a) is referenced in some texts but may be a misnomer because it is actually a measure of asymmetry along the length of the egg (Stoddard et al., 2017) and was not applied as a variable during the analysis of motion-based experimentation (Tschanz et al., 1969; Ingold, 1980). Pointedness, or the relationship of the distance from the apical width of the egg to the sharp end divided by the total length (Deeming and Ruta, 2014), also assumes a consistent translation from a sphere to an ellipse between all species/individuals and sidesteps the difference in curvature of the pointed end (Preston, 1953), which could affect the resting angle and displacement (see Fig. 1) but was not tested explicitly in those prior studies.

Methodologically, the contrast between razorbill and murre eggs is problematic for comparisons of egg shape per se given these species' substantial overlap in egg morphology, as well as their own intraspecific variation in form and occasional overlapping roosting and nesting habits (Ingold, 1980; Gaston and Jones, 1998). Although Ingold (1980) acknowledged the interspecific overlap of morphological features in the eggs utilized in motion experiments, he presented no quantitative method of contrasting the egg shapes of these genera.

Similarly, the actual angle that could predict the center point of a rolling radius has never before been measured, along with the exact shape of the rolling paths or the true maximum displacement during experimentation. Instead, as a proxy for egg-rolling displacement, the maximum width and depth of the roll paths were recorded (Ingold, 1980), which masks maximum displacement and confounds the results given the supposed circular rolling paths that the eggs travel. Tschanz et al. (1969) utilized a binary loss/no-loss system to compare egg rolling-related clutch loss, but without accounts of the roll path and the distance from the point of release to the cliff edge where the egg was lost, true displacement and its suppression cannot be accurately recorded (see Fig. 2). The maximum displacement of a highly semicircular roll would instead have to reflect the width measurement, while a more diagonal–linear roll of equal width (as observed during initial experimentation) would have to reflect the length of the hypotenuse (Cutnell and Johnson, 2005). Unfortunately, as mentioned above, no record of the rolling paths was made, and no empirical evidence of the highly circular rolling path was ever presented in the study, although a hand sketch produced from recollection of experimental trials attempted to illustrate a noticed difference between more pointed and less pointed eggs (Tschanz et al., 1969).

Fig. 2.

Predicting true maximum displacement. Some common roll-path configurations observed during initial experimentation are illustrated here, each with identical maximum width and depth path measurement (in gray) but with different maximum displacements (in red). Past literature utilized overall width and depth as a proxy for maximum displacement (Ingold, 1980; Tschanz et al., 1969). The obvious inaccuracy of this proxy for maximum displacement offers an argument for a pseudo-reproduction of prior experimentation with more accurate data collection.

Fig. 2.

Predicting true maximum displacement. Some common roll-path configurations observed during initial experimentation are illustrated here, each with identical maximum width and depth path measurement (in gray) but with different maximum displacements (in red). Past literature utilized overall width and depth as a proxy for maximum displacement (Ingold, 1980; Tschanz et al., 1969). The obvious inaccuracy of this proxy for maximum displacement offers an argument for a pseudo-reproduction of prior experimentation with more accurate data collection.

Contemporary research on murre eggs has yielded some alternative ideas for evolutionary drivers of their unique form, which have been presented in opposition to the displacement suppression hypothesis. Birkhead et al. (2017a,b) suggested that asymmetry (referred to as ‘pointedness’) was a method of maintaining adequate gas and moisture exchange while reducing the chances of microbial infection. More recently, in a taxon-rich comparative study, Stoddard et al. (2017) correlated asymmetry and ellipticality of egg shape to adult birds' flight ability at the species level and marked a lack of evidence for an evolutionary trend of ellipticality in shorebirds to support a general rolling depression hypothesis (Stoddard et al., 2017). Importantly, these recent challenges to the traditional view are not attributed to the lack of quantifiable morphometrics or general data collection techniques in previous studies (Stoddard et al., 2017; Birkhead et al., 2017a,b; Birkhead et al., 2018). The most recent work on murre egg shape has shown a potential for egg form to resist the commencement of motion when placed on an inclined plane, held static and in motion (Birkhead et al., 2018).

Given the dearth of quantified variables in the benchmark pyriform egg literature and the ongoing challenges to the theory based on those studies (Birkhead et al., 2017a,b; Stoddard et al., 2017), there is a strong case for verifying the rolling displacement reduction qualities found in the murre egg shape system. Therefore, the aim of this study was to quantitatively isolate the geometric properties of shape that are aggregated in Uria aalge eggs and to determine their distinct and combined value to displacement suppression, if any exists. Addressing the variables and metrics individually allows for a comparison and analysis of nesting strategies of not only closely related alcid (auks and allies) genera but also other rigid and semi-rigid egg-laying vertebrate taxa in general, some with more extreme morphological characteristics than the common murre (López-Martínez and Vicens, 2012). Although there is no complete survey of avian egg morphology, a single review (Hauber, 2014) containing only 600 extant species produced 54 examples of eggs that were of pyriform or conical typology across three separate orders. There are also a multitude of species that have particular nesting strategies such as ground-roosting birds, nestless species, and those that lay eggs on bare and flat surfaces, and in proximity to dangerous habitat elements. For example, many waders have pyriform eggs, and nest on the ground in close proximity to water, which could cause the egg to become inviable if subsumed (Lovette and Fitzpatrick, 2016; Deeming and Reynolds, 2015). Studying egg displacement and morphology could improve our understanding of the evolutionary drivers of these curious reproductive strategies.

Controlling the shape of an egg

Given that past attempts at analyzing rolling displacement and egg form lacked clearly defined and measured variables, we produced 3D-printed model egg forms to better isolate and control those variables. The pyriform shape of the murre egg deviates from a sphere in three manners, or variables: in the width to length ratio (W/L); in the distance of the apical width to the blunt end divided by the length (AWD/L); and the level of conicality on its pointed end (C) (see Fig. 3A). The AWD/L is a metric for asymmetry but differentiates from the prior ‘asymmetry’ by utilizing the measure from the apical width to the blunt end to delineate it from the poorly named ‘pointedness’ value (Birkhead et al., 2017b). For each variable, five to six iterations of egg form (n=16 total) were modeled in Trimble Sketchup™ (Sunnyvale, CA, USA), varying from completely spherical to an extreme expression of said variable, in equal increments. Conicality was manipulated by replacing a segment of the profile of the pointed end with arcs of increasingly larger radii to produce a shape that evenly approached true conicality (see Fig. 3). Fillets (Middleditch and Sears, 1985) at a 5 mm radius were used to smoothly transition the conical element with the original profile to avoid creating sharp edges in the form. Models were scaled to equal volume, as a proxy for mass, to half the mean volume/mass (107.27 g fresh mass) of the common murre (Gaston and Jones, 1998) to avoid past complications with comparing unequally sized egg models (Birkhead, 2016) as well as to increase the proportion of the observable rolling path.

Fig. 3.

Experimental egg shapes and the rolling plane. The egg models were produced using these profiles. For the single-variable models (A), all three morphological variables were used: (1) elongation as width (W)/length (L), (2) asymmetry as apical width distance (AWD) divided by length and (3) conicality, or an increased radius within a finite line segment until a straight line was reached in a profile. For the multivariable models (B), the 27 egg forms incorporate three variable values simultaneously. The rolling plane (C) used for tracking the rolling distance of model eggs is displayed with a few highlighted features for quality control.

Fig. 3.

Experimental egg shapes and the rolling plane. The egg models were produced using these profiles. For the single-variable models (A), all three morphological variables were used: (1) elongation as width (W)/length (L), (2) asymmetry as apical width distance (AWD) divided by length and (3) conicality, or an increased radius within a finite line segment until a straight line was reached in a profile. For the multivariable models (B), the 27 egg forms incorporate three variable values simultaneously. The rolling plane (C) used for tracking the rolling distance of model eggs is displayed with a few highlighted features for quality control.

The egg forms were 3D printed as per Igic et al. (2015) with one MakerBot™ Replicator 2 but with a consistent 2 mm ABS shell thickness for durability. A high polygon count (2590) was used to provide a smooth surface when constructing the models, and all protruding blemishes and hard edges were removed with 200 grit wet-grade sanding paper. A low elasticity Smoothon™ Mold Star® 15 Slow Set silicon with a density 97.2% comparable to combined yolk/albumen (https://www.smooth-on.com/tb/files/MOLD_STAR_15_16_30_TB.pdf; Burley, 1989) was used as fill in lieu of any appropriately dense 3D print medium and was piped into a 5 mm hole at the bluntest point of the shell to ±0.5% of the desired mass. To emulate the most favorable conditions for possible rolling radius reduction, the eggs were filled to 87% volume to mimic contents that include a well-developed air sac (Deeming and Reynolds, 2015). The silicone was cured with the radial axis of the egg form perpendicular to the grade, allowing the air sac to form evenly at the blunt end of the shell.

To explore the interaction of the three variables simultaneously, additional series of eggs with multiple traits were produced with identical methods. There were three iterations of each variable tested, which culminated in 27 (or 3³) unique models (see Fig. 3B). Variable traits were selected based roughly on a sampling of 380 alcid eggs from 11 species, including common murres, thick-billed murres and razorbills (I.R.H. and M.E.H., unpublished data).

True maximum displacement and motion tracking

We used similar protocols to Ingold (1980) for egg rolling/displacement trials. The egg forms were placed on a rolling platform (1×1.5 m) (see Fig. 3C). The rolling platform was fabricated from a solid-core, unfinished wooden door. Four 2.5 cm-diameter threaded steel dowels acted as adjustable legs and were affixed to the door with threaded inserts, allowing for a varying incline on the rolling plane surface. A large protractor with an integrated 60 cm contractor's level at the 1 m long edge of the door allowed accurate measurement of the incline of the rolling plane. Two other contractor's levels were affixed down the length of the door on the underside to monitor lateral levelness. Three white markers spaced 90 cm apart in an ‘L’ configuration provided a calibration method for scale during motion tracking. To increase the fidelity of the motion-tracking software, a dark non-sealing, pore-penetrating, water-based wood stain (Minwax Onyx®) was used to create high black–white contrast between the egg models and the rolling plane surface. The friction coefficient for planed oak, such as that provided by our rolling plane surface, ranges from 0.19 to 0.48 μd (kentric) depending on the orientation of the object to the wood grain (Aira et al., 2014), and the grain orientation was tangent to the slope of the incline.

Each egg model was placed on the plane with its radial axis perpendicular to the direction of the slope and held with a squared metal stop. Upon release, the model's rolling path was recorded with a Sony FDR-AX33 Handycam (at 60 frames s−1), with the lens of the camera oriented true to the flat surface of the rolling plane to minimize visual distortion. This exercise was duplicated five times for each egg form, with the rolling plane set at inclines of 2, 4, 6, 8, 10 and 15 deg. Kinovea™ (https://www.kinovea.org/) motion-tracking software was utilized to extract the course of the roll and maximum displacement.

Statistical analysis of resting angle, egg shape and displacement

In the basic model diagrammed by Uspenski (1950), as an angle of the egg at rest relative to the surface it sits on decreases, the rolling radius should increase. To assess the foundational hypothesis that the egg's pitch can in any capacity predict displacement, each egg model's radial axis angle was measured relative to a leveled plane. Angular measurements were taken from legacy markings from the ABS printing process and reproduced four times at equal rotational increments around the diameter of the egg to account for any inconsistency caused by an uneven distribution of the air sac. These angles were reincorporated into the original computer models used for printing to create a predicted radius, by mean, and then compared with a Pearson's (R) correlation (with OriginPro™; https://www.originlab.com/) to the actual rolling radius, by proxy of the maximum displacement of the experimentally generated rolling paths (as 2×radius=diameter=maximum displacement) (Cutnell and Johnson, 2005). Correlation analysis was performed at each rolling plane incline (2–15 deg).

To verify the principle that there is an inverse relationship between resting angle and displacement, a comparative linear, exponential and polynomial model fit was utilized to determine any interaction between these two variables for all (N=41) of the recorded rolls released on a 2 deg rolling plane incline.

To clarify how egg form affects egg pitch, the resting angle of the single-variable model forms was compared with the gradient of variable expression in each model iteration. As a spherical form elongates, its broadest and most stable point of contact (and fulcrum) should move toward the apical width of the egg, with the air sac creating an uneven balance, and a diminished resting angle. A similar reduced resting angle effect should occur with the increasing asymmetry, given that one pole of the half is growing by proportion. Conicality, however, may decrease the resting angle, given that a 45 deg profile is being imposed on a spherical form that otherwise may have the lightest portion (the air sac) of the egg drawn upward, creating a perpendicular resting angle. A comparative linear, exponential and polynomial model fit was utilized to determine any relationship of the resting angle to both the W/L and AWD/L of the single variable models. Friedman's test was used for models of increasing conicality, given the lack of a viable metric (Stoddard et al., 2017).

Finally, maximum displacement and shape were analyzed to test the angle–shape displacement model by each morphological variable. Pearson's test with elongation and asymmetry was used to determine whether the increased intensity of these variables produces a positive correlation with displacement on single-variable model forms. Friedman's test was also utilized to confirm any paired incremental increase of displacement with conicality, which was repeated for all variables on the multivariable egg forms. Single-variable egg models were grouped and analyzed as a set by rolling plane angle for consistent comparison and to expose the effects of increased incline (to match the varied inclines of nesting surfaces in the field; Uspenski, 1950), as well as utilizing the increasing influence of gravity as a proxy for projectile force (Cutnell and Johnson, 2005). This also allowed for a comparative measure for the effect of different inclines on the conical versus spherical models, irrespective of resting radius, because an uneven form of a conical egg could provide a greater resistance to an incline as a function of its weight distribution and a fixed fulcrum point (Uspenski, 1950; Cutnell and Johnson, 2005), despite its comparative resting angle. Multi-variable eggs were tested in kind but with each incline angle utilized as another subject (N) within Friedman's test to produce a single result for each geometric variable.

Egg forms that did not produce results for maximum displacement at a specific angle as a result of loss from the surface of the rolling plane were excluded from the statistical analysis given their inability to accurately contribute to a maximum displacement. Multi-variable eggs that experienced 100% loss by rolling off the platform led to the exclusion of that class of variables from Friedman's test, because of the requirement for equal data ‘participation’ at each treatment/variable level (Frieman et al., 2017).

Resting angle and displacement

At the lowest incline of the rolling plane (2 deg), the predicted maximum displacement derived from the resting angle of the egg models was a strong predictor of actual maximum displacement (t39=12.198, P<0.001) (see Fig. 4A) and displacement variance (R2=0.787, F1,39=148.780, P<0.001). This near one-to-one ratio of predicted and actual displacement deteriorated at a 4 deg incline (t27=3.225, P<0.003, R2=0.251, F1,27=10.405, P<0.003) and this was even more apparent at 6 deg (t23=2.878, P<0.009, R2=0.233, F1,23=8.281, P<0.009). The results for rolling plane inclines of 8, 10 and 15 deg were excluded from the analysis because of the low sample size (N<10), resulting from models leaving the plane's trackable surface.

Fig. 4.

Predicted displacement and resting angles of the single-variable models. The relationship between resting angle and displacement was strong on near-to-level surfaces. (A) The maximum displacement of all egg models (N=47, mean of five rolling trials) at a rolling plane incline of 2 deg was strongly predicted by the projected maximum displacement (y=0.840x−0.036). Projected displacement was obtained from the resting angle of the egg form's radial axis. The effect of shape on the resting angles of the models was also examined; as each egg shape variable (B, elongation; C, asymmetry; and D, conicality) increased in severity, the resting angle decreased. The conicality chart has no line of best fit given its ordinal scale. Each point represents one of four resting angle measurements taken from each egg for plotting against the strength of each morphological variable.

Fig. 4.

Predicted displacement and resting angles of the single-variable models. The relationship between resting angle and displacement was strong on near-to-level surfaces. (A) The maximum displacement of all egg models (N=47, mean of five rolling trials) at a rolling plane incline of 2 deg was strongly predicted by the projected maximum displacement (y=0.840x−0.036). Projected displacement was obtained from the resting angle of the egg form's radial axis. The effect of shape on the resting angles of the models was also examined; as each egg shape variable (B, elongation; C, asymmetry; and D, conicality) increased in severity, the resting angle decreased. The conicality chart has no line of best fit given its ordinal scale. Each point represents one of four resting angle measurements taken from each egg for plotting against the strength of each morphological variable.

The regression analysis showed that, as the resting angle decreased, the displacement increased significantly (R2=0.780, χ²=0.00133, P<0.001) with an exponential decay model showing the best fit for the rolling plane incline of 2 deg (see Fig. 5).

Fig. 5.

Displacement and resting angle. As the resting angle of an egg increases, the theoretical displacement should increase (Belopol'skii, 1961; Ingold, 1980; Tschanz et al., 1969). The displacement increased substantially between lower resting angles (A), which was borne out statistically (P<0.05) when observing the relationship between resting angle and displacement (B) with all the tracked models (N=41) at a rolling plane incline of 2 deg.

Fig. 5.

Displacement and resting angle. As the resting angle of an egg increases, the theoretical displacement should increase (Belopol'skii, 1961; Ingold, 1980; Tschanz et al., 1969). The displacement increased substantially between lower resting angles (A), which was borne out statistically (P<0.05) when observing the relationship between resting angle and displacement (B) with all the tracked models (N=41) at a rolling plane incline of 2 deg.

Resting angle and shape

Resting angle increased as the values of W/L and AWD/L increased (as the extent of the variable expression decreased) in the single-variable models (see Fig. 4B), with a quadratic model showing the strongest fit (R²=0.992, χ²=4.595, P<0.001) for W/L and similarly for AWD/L (R²=0.999, χ²=3.719, P<0.001) (see Fig. 4C,D). Increasing increments of conicality also caused decreasing resting angles (χ²4=15.25, P<0.004) (see Table 1), and Dunn's post hoc analysis of Friedman's test (Friedman, 1937) showed that the two most conical egg forms, 3 and 4, had significantly different resting angles (P<0.012 and P<0.025, respectively) from the spherical base model (form 0).

Table 1.

Single-variable conical models: maximum displacement comparison

Single-variable conical models: maximum displacement comparison
Single-variable conical models: maximum displacement comparison

Shape and displacement

In the assessment of the relationship between shape and displacement, W/L positively correlated with maximum displacement, which was strong and significant at 2 (Pearson's R=0.967) and 4 (R=0.892) degrees at P<0.001 on both variables, whereas all other angles resulted in 100% loss from the platform. For asymmetry, AWD/L also showed a significant but negative correlation to displacement at 2 deg (R=−0.880, P<0.001), 4 deg (R=−0.823, P<0.001) and 8 deg (R=−0.726, P<0.002), although the set released at 6 deg lacked strong correlation and significance (R=−0.122, P<0.630). From angles of 4–8 deg, each increase in the rolling plane incline resulted in the exclusion of the most spherical model from the previous incline as a result of 100% of the models leaving the rolling plane. For conicality, Friedman's test produced significant differences (P<0.05) between models at each plane incline, except 2 deg (χ²4=6.880, P<0.142; see Table 1 for full results). Only the most conical model (4) remained on the rolling plain at a 15 deg incline, excluding it from intra-incline comparison. At inclines of 4 deg (χ²4=10.720, P<0.030), 6 deg (χ²2=10.000, P<0.007), 8 deg (χ²2=10.000, P<0.007) and 10 deg (χ²4=5.771, P<0.016), there was a significant consistent rank order at angles of 6–10 deg (see Table 1). Dunn's post hoc analysis verified a significant difference between the most conical model form (4) and the least conical model at inclines of 4 deg (P<0.027), 6 deg (P<0.005) and 8 deg (P<0.005), with no significant sum difference between any pairs at 2 and 10 deg. From the angle of 4 deg, and greater, each increase of the rolling plane's incline resulted in the next most spherical model being excluded from analysis as a result of 100% roll loss (see Fig. 6).

Fig. 6.

Displacement of the single-variable models. For the single-variable egg models, shape appears to affect displacement. Maximum displacement of all rolls from every single-variable egg model (five replicate rolls) was plotted against each model's morphological variable strength and this was repeated with the rolling plane inclined at 2 deg increments. The single-variable egg model results showed that (A) increased asymmetry produces larger rolling radii for all rolling inclines (except at 6 deg), (B) elongation negatively correlates with displacement and (C) increased conicality produces incrementally lower displacement (except for the 2 deg incline angle trials). One conicality egg form, no. 4, persisted on the rolling plane up to 15 deg but was omitted for lack of comparison and graphic clarity. Egg forms with increased conicality and asymmetry persisted at higher inclines. The above data exclude trials that terminated in ‘egg loss’ from eggs falling off the edge of the rolling plane before terminating motion.

Fig. 6.

Displacement of the single-variable models. For the single-variable egg models, shape appears to affect displacement. Maximum displacement of all rolls from every single-variable egg model (five replicate rolls) was plotted against each model's morphological variable strength and this was repeated with the rolling plane inclined at 2 deg increments. The single-variable egg model results showed that (A) increased asymmetry produces larger rolling radii for all rolling inclines (except at 6 deg), (B) elongation negatively correlates with displacement and (C) increased conicality produces incrementally lower displacement (except for the 2 deg incline angle trials). One conicality egg form, no. 4, persisted on the rolling plane up to 15 deg but was omitted for lack of comparison and graphic clarity. Egg forms with increased conicality and asymmetry persisted at higher inclines. The above data exclude trials that terminated in ‘egg loss’ from eggs falling off the edge of the rolling plane before terminating motion.

Within the multivariable egg forms, increasing conicality was the most consistent variable linked to displacement reduction (see Fig. 7). The three levels of conicality in model form were significantly different (Friedman's statistic χ²2=39.08, P<0.001), and a Dunn post hoc test of Friedman's test (Friedman, 1937) confirmed a significant difference (P<0.05) between each pair of levels (see Table 2), with the rank demonstrating a decrease in maximum displacement as the form increased in conicality. Displacement was significantly different between classes of asymmetry (χ²2=9.091, P<0.011), but a Dunn post hoc test of Friedman's test (Friedman, 1937) revealed that only the two lower levels of asymmetry (AWD/L=0.375 to 0.45) showed statistical difference (P<0.008) with 0.375 having significantly greater displacement. However, different intensities of elongation yielded no significant results (P<0.05) (see Table 2).

Fig. 7.

Displacement of the multivariable models. The maximum displacement of all the multivariable egg models is portrayed by rolling in a matrix, by each rolling plane incline increment. Each node represents a three-variable egg form, whereas each circle on the node represents the displacement value of that model at a specific rolling plane incline, designated by its color. The size of the circle represents the displacement at that incline (mean maximum displacement of five rolls). As the conicality increases, the circles grow darker, showing persistence at higher inclines, and the circles of the same color decrease in size, showing a decrease in displacement, which was verified as significant by Friedman's test. No significant trends were found across all three iterations of the other two variables, although the highly asymmetrical models also persisted at higher levels of rolling plane incline.

Fig. 7.

Displacement of the multivariable models. The maximum displacement of all the multivariable egg models is portrayed by rolling in a matrix, by each rolling plane incline increment. Each node represents a three-variable egg form, whereas each circle on the node represents the displacement value of that model at a specific rolling plane incline, designated by its color. The size of the circle represents the displacement at that incline (mean maximum displacement of five rolls). As the conicality increases, the circles grow darker, showing persistence at higher inclines, and the circles of the same color decrease in size, showing a decrease in displacement, which was verified as significant by Friedman's test. No significant trends were found across all three iterations of the other two variables, although the highly asymmetrical models also persisted at higher levels of rolling plane incline.

Table 2.

Three variable models: maximum displacement comparison

Three variable models: maximum displacement comparison
Three variable models: maximum displacement comparison

Resting angle and predicted displacement

The foundation of the displacement suppression model for pyriform eggs is the role of the resting angle in determining the diameter (and maximum displacement) of a roll path. The ability to accurately predict the actual displacement from the resting angle at low inclines (2 deg, see Fig. 4A) and the increase in displacement with the decrease in resting angle (see Fig. 5A,B) provide novel numerically based data-driven evidence to support older claims (Ingold, 1980; Tschanz et al., 1969). Prior work presenting profile images of eggs at rest on a level plane in an ordinal scale of ‘pointedness’ remarked only on the resting angle while lacking any measurements of that resting angle or attempts to extrapolate a displacement value. The decrease in correlation at inclines of 4 deg (R²=0.251) and 6 deg (R²=0.233), and the loss of ∼75% of the rolling data at 8 deg provide evidence that this relationship erodes as the incline (or level of applied force) increases.

This decline may be attributed to the radial axis rocking, as observed on multiple occasions in the video recordings from the displacement trials of our current study. As the egg forms moved across the arch of their rolling path, their radial axis orientation changed from tangent to parallel with the direction of the incline, possibly causing conflict between the projectile force in the direction of the incline and the effect of gravity on the distribution of the weight (Cutnell and Johnson, 2005) along that egg's axis. This weight redistribution would affect the angle and therefore the displacement (Belopol'skii, 1961).

Resting angle and shape

A strong positive relationship between shape and resting angle was demonstrated by the model fit of the single-variable eggs, given that as the eggs were made both increasingly asymmetrical and increasingly elongated, their resting angles consistently decreased (see Fig. 6). Increasing conicality also reduced the resting angle incrementally, although this was only significant at the highest levels of conicality. The non-linear relationship demonstrated between the resting angle and both W/L and AWD/L demonstrated that a small variation in geometry, from W/L=1 to 0.875, produced a substantial reduction in the mean resting angle (from 61.8 to 31.9 deg) (see Fig. 4B). The predicted displacement at such high resting angles varied by only ∼5 cm, which is only slightly less than the length of many of the egg models themselves.

Shape and displacement

To affirm the shape–angle displacement relationship of the model, the findings consistently supported predictions for two of the three single variables. Elongation affected the single-variable eggs significantly (P<0.05) by increasing the displacement on every rolling-plane incline (see Fig. 6B). The multivariable models produced no significant difference in elongation. Greater asymmetry produced an increase in displacement on single-variable eggs (see Fig. 6A), which was also supported by the multivariable model of the lowest level of asymmetry (AWD/L=0.45, mean rank=1.546) having significantly lower displacement (P<0.008) than the intermediate (AWD/L=0.375, mean rank=2.455) level (see Fig. 7 and Table 2).

The conical model-set results varied substantially by the level of incline. As the single-variable models increased in conicality, the displacement increased significantly only at the lowest incline level (2 deg), while every other incline produced decreasing displacement with increasing conicality (4–8 deg significantly so; see Fig. 6C and Table 1). This outcome deviated from the predicted outcome that could be derived from the resting angle (see Fig. 4D), that higher resting angles predict lower displacement (Uspenski, 1950). The multivariable models reinforced this deviation from the model, with a significant decrease in displacement with the increase in conicality between each ordinal level (see Table 2 and Fig. 6).

Conicality and stochastic rolling suppression

Although there is broad support for the shape–angle displacement hypothesis in the present study, the lack of fidelity to the predicted results within the conical model highlights the significance of shape for displacement at higher inclines. Whereas it was implied in some published work that the cone shape of murre eggs causes a circular rolling pattern (Ingold, 1980; Tschanz et al., 1969), it had yet to be incorporated into a full mechanical scenario based on data or accounting for the effects of an incline. At an even incline, with minimal applied force, a spherical egg with an air sac may have a smaller displacement than a conical egg, according to the results, but at higher inclines (or with the application of greater force), rounder eggs are more susceptible to rolling off the edge in a multi-axis stochastics roll or ‘tumble’. In the single-variable conicality model set, at every increase of incline, the most rounded model from the previous incline was incrementally excluded from displacement analysis because of 100% loss over the side of the rolling plane (see Fig. 6C). Our video recordings showed that the radial axis of rounded eggs both rocked heavily back and forth along their rolling path and engaged in ‘head-over-heel’ stochastic multi-axial rolling before exiting the rolling plane. Similarly, a discord between rolling suppression and loss in the AWD/L models occurred (see Fig. 6A); a highly asymmetrical form traveled farther but resisted loss as the rolling plane increased in incline. A possible explanation for why highly asymmetrical and conical models resist stochastic rolling even at high angles may be their uneven shape and profile, which redistributes weight and the fulcrum point of an egg, reducing the likelihood of tipping or rocking (Uspenski, 1950), which can be seen at the initiation of ‘head-over-heel’ free rolling. An unevenly balanced spheroid in motion, intuitively, is at risk of shifting weight along that curve. A likely impact of that highly conical form is that it prevented such shifting, fixing the axial angle relative to the surface. This was supported with the maximum displacement of the most conical single-variable model ranging from ∼3 to 8 cm up to a 10 deg incline, whereas highly elongated and asymmetrical models broke the 40 and 10 cm marks at a 2 deg incline, respectively (Fig. 7).

Limitations of protocols, alternative hypotheses and further study

Although the results strongly support the link between conicality and displacement depression, many factors detailed here require further exploration. The extent of embryo/air sac development and how its variation affects the resting angle of heavily conical, asymmetrical and elongated egg shapes remains unverified. Previous studies were conducted with solid plaster models that were shown to take slightly more time and distance to stop moving than biologically produced eggs. Uria egg shapes, whether real or modeled, still consistently rolled shorter distances than the rounder models (Ingold, 1980). The insides of an egg, with the suspended yolk, chalza and various membranes and fluid matrix (Hauber, 2014), present a complex set of structures to mimic accurately (Burley, 1989). The yolk-to-hatchling gradient would present an increasingly difficult set of conditions to mimic, particularly to account for every developmental stage. The existence of these internal structures is fairly consistent throughout the vast majority of bird species (Hauber, 2014). Our assumption was that the forces exerted on displacement may be consistent between eggs of a similar developmental stage (as a constant variable) and that a constant (not a differential) effect on displacement across various species would occur.

Considering the development of the air sac, it is likely that, without a weight differential, the elongated ellipsoid form would produce a resting angle parallel to an even grade given the general principles of a fulcrum (Uspenski, 1956; Cutnell and Johnson, 2005). The present study utilized gravity as a proxy for an affecting force such as collision, but the real-world impact from birds or wind (Uspenski, 1956) would transfer the directional component of that force to the form, possibly interrupting the resting angle and its effect. In this light, a limiting factor in our study is the relative value of each variable of egg shape to suppress displacement without such a favorable weight distribution.

Similarly, the current study did not measure how eggs behaved when spun, impacted or otherwise subject to stochastic applied forces that may occur in a natural habitat. Relatively smooth and planar surfaces are common throughout murre and razorbill nesting sites (Ingold, 1980). The greater contact a more ‘pointed egg’ makes with the surface it rests on (Birkhead et al., 2017a) should increase its probability of encountering smaller debris that could alter or stop the rolling path (Tschanz et al., 1969). Prior studies have introduced varied rolling surfaces that presented elements to impact and disrupt a highly stereotyped rolling pattern to add a level of stochasticity, and they had a noted effect on the path of the object. These older laboratory and field experiments with live birds still produced results that were consistent with the current study (Ingold, 1980; Tschanz et al., 1969), which presented an argument against expanding the current study's methodology in the minutia of impact mechanics and varied stone substrates and other field conditions. The data collection for our analysis (1548 motion-tracked rolling trials) for the four independent variables presented in a gradient would have substantially increased when considering the impact force typology, angle of impact relative to the egg shape and intensity of applied force. Observations of adult common murres placing small pebbles and debris next to their incubating eggs (Uspenski, 1950) may warrant some further study of egg shape within the context of motion and substrate character.

One source of possible error of measurement in the current protocols may exist in the tolerance for displacement at the smallest distances. Several of the single-variable conical models experienced displacement of only 2–4 cm in some of the lower rolling plane inclines. The motion-tracking software's locus point, from our observations, could vary from the center of the model egg during movement by 1.5 cm, which could have a significant effect on the 2 deg and possibly 4 deg incline rolls. Any greater deviation from this triggered an intervention and manual adjustment of the tracking locus to be set back at the center of the egg, and no tracking locus point was permitted to leave the boundary of the egg.

A primary flaw in all past studies, including the current one, is the lack of a standard metric for conicality. A survey of the literature yielded no discrete measure, which has been echoed by the lack of a pyriformity metric and perhaps explains its exclusion as a comparative metric from the study by Stoddard et al. (2017). The 3D modeling and printing allowed for the generation of model shapes that incrementally approached conicality without requiring a formal measure, leaving no rubric for studying real-world eggs. Proxies such as the aforementioned ‘pointedness’ (which is actually the asymmetry of the egg) and ‘ellipticality’ are incapable of determining the presence or magnitude of a straight line within the profile of a shell, given that both measures rely on the assumption of a continuous curve throughout the shell's profile (Preston, 1953; Alexander and Koeberlein, 2014). This ignores the obvious spectrum of convex, straight to concave profiles (see Fig. 8) in the pointed or conical element of Uria aalge shells (Birkhead, 2016). Knowing the magnitude of this variable could provide useful data for the study of motion/displacement and be fruitful in various avenues of inquiry. Our efforts to address variable independence by single-variable transforms could, in future experimentation, be aided by a solid and verified metric for conicality (Stoddard et al., 2017). The use of ‘pointedness’ (Birkhead et al., 2017b) as a proxy for conicality may hint that, by manipulating the length of a pole of an egg, a certain amount of conicality is produced as a by-product of that elongation. Further investigation to this end is needed.

Fig. 8.

Egg shape and nesting habitat. Little work has been done to correlate egg shape and net typology. (A) Many species of the alcidae family, including Uria aalge, Uria lomvia and Pinguinus impennis, that roost on rigid planar cliffs and shores without a nest have very pyriform eggs with highly conical elements. Razorbills (Alca torda) and black guillemots (Cepphus grylle) have more-rounded eggs and most often nest in crevices with pebble beds, as compared with exposed microhabitats (Gaston and Jones, 1998). (B) Some non-avian theropods have highly conical eggs, but there is limited information on their nesting habits (Zelenitsky and Therrien, 2008), which could be illuminated by egg shape and displacement dynamics. (C) Species of avian theropods such as Gallus gallus and Pernis apivorous have rather spheroid eggs in contrast to the conical forms. Instances of birds that incubate eggs without a nest on rigid and planar surfaces are fairly rare. One exception is the emperor penguin (Aptenodytes forsteri), which not only has a strong conical element to the pointed end of its egg but also must limit the egg's contact with the packed snow and ice of their maternity colonies to prevent thermal shock (Williams, 1995).

Fig. 8.

Egg shape and nesting habitat. Little work has been done to correlate egg shape and net typology. (A) Many species of the alcidae family, including Uria aalge, Uria lomvia and Pinguinus impennis, that roost on rigid planar cliffs and shores without a nest have very pyriform eggs with highly conical elements. Razorbills (Alca torda) and black guillemots (Cepphus grylle) have more-rounded eggs and most often nest in crevices with pebble beds, as compared with exposed microhabitats (Gaston and Jones, 1998). (B) Some non-avian theropods have highly conical eggs, but there is limited information on their nesting habits (Zelenitsky and Therrien, 2008), which could be illuminated by egg shape and displacement dynamics. (C) Species of avian theropods such as Gallus gallus and Pernis apivorous have rather spheroid eggs in contrast to the conical forms. Instances of birds that incubate eggs without a nest on rigid and planar surfaces are fairly rare. One exception is the emperor penguin (Aptenodytes forsteri), which not only has a strong conical element to the pointed end of its egg but also must limit the egg's contact with the packed snow and ice of their maternity colonies to prevent thermal shock (Williams, 1995).

One argument against the rolling hypothesis originates from thick-billed murres' notably ‘less-pointed’ eggs by rough description and by the geometric measure of asymmetry while nesting on narrower ledges than common murres (Ingold, 1980), although the eggs of both Uria species appear to be highly conical relative to those of other taxa analyzed (Birkhead et al., 2017b). According to the current findings, less-symmetrical egg forms increase the axial resting angle to reduce displacement, negating the seeming incongruence of a more ‘pointed’ egg-laying species incubating on deeper ledges (Ingold, 1980; Birkhead et al., 2017b). This again highlights the importance of defining morphology with geometric principles as compared with rough analogy.

A further challenge to the adaptive value of displacement suppression arose from a study that showed the ‘pointed’ asymmetrical form of common murre eggs likely provides microbial infection defense and promotes proper gas exchange; this is supported by a higher pore density around the blunt end of the shell, which is elevated from the ground by the resting angle, while the pore density is lower on the conical portion, which often accumulates a thick layer of microbial-rich guano (Birkhead et al., 2017a). The rolling and infection theories generate an excellent complementary set of hypotheses, though, and need not compete for exclusivity in the realm of the adaptive value of murre egg shape. Nesting ledges vary in depth greatly, ranging from 10 cm to over 3 m, and it is well documented that guano accumulates in less-exposed areas (sheltered from rain) such as caves, deep but narrow cleaves in the cliff side and indentations with little natural drainage (Johnson, 1941; Birkhead, 2016; Uspenski, 1956). In deeper cliff formations, falling eggs are less likely (Harris et al., 1997), whereas at a shallower, less-populated ledges, the guano accumulation is less prevalent (Uspenski, 1956), leaving room for the egg shape to function adaptively by rolling at a narrow radius.

Displacement suppression was also questioned recently in a morphology-based comparative egg survey, suggesting that there was no strong trend of ellipticality in shorebirds; instead, the authors proposed that asymmetry and ellipticality are primarily correlated to flight ability (Stoddard et al., 2017). Although the sample was taxonomically broad, and the assertions made about common forces that affect the evolution of egg shape were very well supported, the methods and analysis did not consider the scarcity of species that incubate eggs on the flat, planar, highly rigid surfaces that these alcids utilize (Johnson, 1941; Uspenski, 1956; Stoddard et al., 2017). It is possible that only three extant and one extinct auk roost(ed) in this exact manner (Gaston and Jones, 1998), as do emperor penguins, Aptenodytes forsteri (Williams, 1995), which from their appearance also have a visible conical element to one side of their shells (see Fig. 8). The lack of strong evolutionary trends across an entire taxonomic order should not discount unique adaptive strategies and highly specific niches of individual taxa (Hall, 2011). Furthermore, common murre eggs were an extreme outlier even in the previous survey, as the third most asymmetrical and fifth most elliptical of all 1400 species sampled, and with no accounting for conicality (Stoddard et al., 2017), which justifies an analysis focusing on its extreme properties (Barnett and Lewis, 1994).

One value of the present study is the ability to look beyond auks and use the individual variables to assess eggs of different proportions and typologies for motion and displacement. Perhaps the measurement of conicality can be used to discern why Charadriiformes are noted for such pyriform eggs (Lovette and Fitzpatrick, 2016; Hauber, 2016), although there is a lack of a significant trend of ellipticality and asymmetry in shorebirds (Stoddard et al., 2017). Many waders and non-alcid shorebirds, such as plovers and sandpipers, produce eggs with highly pyriform or just conical shells and nest on the ground in close proximity to water and other potentially dangerous conditions for an egg (Lovette and Fitzpatrick, 2016; Hauber, 2016) where displacement suppression may be valuable. Emperor penguin eggs can also now be comparatively studied in the context of displacement and egg morphology to closely related sister taxa with less-extreme nesting conditions (Williams, 1995) and more-elliptical egg shapes (Hauber, 2016).

Similarly, highly conical, asymmetrical and elongated shapes are common in rigid/semi-rigid flightless egg-producing organisms, such as many species of dinosaur (López-Martínez and Vicens, 2012), and could be used to elucidate nesting behavior within those taxa. Data on non-avian theropod nesting systems are sparse at best (Zelenitsky and Therrien, 2008), with a large proportion of species having intensely elongated and asymmetrical eggs. The distended and incredibly conical eggs of the Citipati and Troodon dinosaur genera (each containing numerous species) from the late Cretaceous rival that of the Uria (Fig. 8) (López-Martínez and Vicens, 2012), and their well-established terrestrial (non-flying) existence (Gatesy, 1995) excludes them from the strong flight hypothesis (Stoddard et al., 2017). Additionally, egg-poaching specialists existed alongside dinosaurs (Wellnhofer, 1971; Kirkland et al., 1994), presenting any distance between a guarded nest and a parent as a potential hazard, let alone exposure to possibly egg-damaging habitat around nests that are common for contemporary fauna (Lovette and Fitzpatrick, 2016). Given the dearth of information on dinosaur nesting behavior (Zelenitsky and Therrien, 2008), we feel that this morphology study has relevance beyond our initial model species and class of organism.

Conclusion

Breaking down multivariable systems into their fundamental components is a core pursuit of scientific investigation (Carey, 2011). The results of the present study support unquantified proposals of past studies that a resting angle of an egg's radial axis relative to the grade determines its displacement at low inclines (and applied force). The angle of rest is determined by the egg's shape; increasing elongation and asymmetry decrease the resting angle while increasing displacement. Although greater conicality mildly increased displacement at low inclines, it suppressed displacement at inclines higher than 2 deg. Both conicality and asymmetry reduce multi-axis ‘tumbling’, which provides a form of displacement suppression. This supports assertions that the distinct pyriform shape of Uria genus eggs may have aspects of form that possess an adaptive value in the context of their nestless and often shallow, cliff-edge roosting microhabitat (Ingold, 1980; Tschanz et al., 1969). The present study supported this by actually measuring the individual morphological values of elongation, asymmetry, conicality and resting angle, which were not analyzed or measured in past experiments (Ingold, 1980; Tschanz et al., 1969). Importantly, the analysis of these individual and combined morphological characteristics poses no conflict with alternative theories for the adaptive value of ‘pointed’ or pyriform eggs (Birkhead et al., 2017a,b; Stoddard et al., 2017; Birkhead et al. 2018). A true measure of conicality could further elucidate the ‘point’, as put by Birkhead et al. (2017a), of this morphological feature within Uria species (Hauber, 2014) and provide a better understanding of other taxa, extant and not, with similarly shaped eggs.

For assistance and/or discussions, we thank Miri Dainson, Phill Cassey, Cassie Stoddard, Jim Dale, Paul Ingold, Paul Hays, Peter Moller, Gabriela Cruz-Garcia, Erpur Hansen, The Baltic Seabird Project and many others.

Author contributions

Conceptualization: I.R.H., M.E.H.; Methodology: I.R.H.; Software: I.R.H.; Validation: I.R.H., M.E.H.; Formal analysis: I.R.H.; Investigation: I.R.H.; Resources: I.R.H., M.E.H.; Data curation: I.R.H.; Writing - original draft: I.R.H.; Writing - review & editing: I.R.H., M.E.H.; Visualization: I.R.H.; Supervision: I.R.H., M.E.H.; Project administration: M.E.H.; Funding acquisition: I.R.H., M.E.H.

Funding

For funding, we thank the Human Frontier Science Program [RGY0083 to M.E.H.], the HJ Van Cleave Professorship at the University of Illinois, Urbana-Champaign [to M.E.H.], and the Hunter College Animal Behavior and Conservation Program [to I.R.H.].

Aira
,
J. R.
,
Arriaga
,
F.
,
Íñiguez-González
,
G.
and
Crespo
,
J.
(
2014
).
Static and kinetic friction coefficients of Scots pine (Pinus sylvestris L.), parallel and perpendicular to grain direction
.
Materiales de Construcción
64
,
030
.
Alexander
,
D. C.
and
Koeberlein
,
G. M.
(ed.) (
2014
).
Elementary Geometry for College Students
.
Boston
,
USA
:
Cengage Learning
.
Ar
,
A. M. O. S.
(
1991
).
Roles of water in avian eggs
. In
Egg Incubation: Its Effects on Embryonic Development in Birds and Reptiles
(ed.
C.
Deeming
and
M. W. J.
Ferguson
), pp.
229
-
243
.
Cambridge
,
UK
:
Cambridge University Press
.
Bain
,
M.
and
Solomon
,
S.
(
1991
).
Cracking the secret of eggshells
.
New Sci.
129
,
27
-
29
.
Barnett
,
V.
and
Lewis
,
T.
(
1994
).
Outliers in Statistical Data
, Vol.
3
.
New York
,
USA
:
Wiley
.
Barta
,
Z.
and
Székely
,
T.
(
1997
).
The optimal shape of avian eggs
.
Funct. Ecol.
11
,
656
-
662
.
Belopol'skii
,
L. O.
(
1961
).
Ecology of Sea Colony Birds of the Barents Sea
.
Israel
:
Israel Program for Scientific Translations
.
Birkhead
,
T. R.
(
2016
).
The Most Perfect Thing: The Inside (and Outside) of a Bird's Egg
.
New York
,
USA
:
Bloomsbury
.
Birkhead
,
T. R.
,
Thompson
,
J. E.
,
Jackson
,
D.
and
Biggins
,
J. D.
(
2017a
).
The point of a Guillemot's egg
.
Ibis
159
,
255
-
265
.
Birkhead
,
T. R.
,
Thompson
,
J. E.
and
Biggins
,
J. D.
(
2017b
).
Egg shape in the Common Guillemot Uria aalge and Brunnich's Guillemot U. lomvia: not a rolling matter?
J. Ornithol.
158
,
679
-
685
.
Birkhead
,
T. R.
,
Thompson
,
J. E.
and
Montgomerie
,
R.
(
2018
).
The pyriform egg of the common murre (Uria aalge) is more stable on sloping surfaces
.
Auk
135
,
1020
-
1032
.
Brooks
,
D. J.
(
1985
).
Handbook of the birds of Europe, the Middle East and North Africa: the birds of the Western Palearctic. 4. Terns to Woodpeckers
.
Oxford
,
UK
:
Oxford University Press
.
Burley
,
R. W.
(
1989
).
The Avian Egg: Chemistry and Biology
.
New York
,
USA
:
Wiley
.
Carey
,
S. S.
(
2011
).
A Beginner's Guide to Scientific Method
.
Boston
,
USA
:
Cengage Learning
.
Cucco
,
M.
,
Grenna
,
M.
and
Malacarne
,
G.
(
2012
).
Female condition, egg shape and hatchability: a study on the grey partridge
.
J. Zool.
287
,
186
-
194
.
Cutnell
,
J. D.
and
Johnson
,
K. W.
(
2005
).
Essentials of Physics
.
Hoboken
,
USA
:
Wiley-VCH
.
Deeming
,
D. C.
and
Reynolds
,
S. J.
(ed.). (
2015
).
Nests, Eggs, and Incubation: New Ideas about Avian Reproduction
.
Oxford
,
UK
:
Oxford University Press
.
Deeming
,
D. C.
and
Ruta
,
M.
(
2014
).
Egg shape changes at the theropod–bird transition, and a morphometric study of amniote eggs
.
R. Soc. Open Sci.
1
,
140311
.
Duursma
,
D. E.
,
Gallagher
,
R. V.
,
Price
,
J. J.
and
Griffith
,
S. C.
(
2018
).
Variation in avian egg shape and nest structure is explained by climatic conditions
.
Sci. Rep.
8
,
4141
.
Friedman
,
M.
(
1937
).
The use of ranks to avoid the assumption of normality implicit in the analysis of variance
.
J. Am. Stat. Assoc.
32
,
675
-
701
.
Frieman
,
J.
,
Saucier
,
D. A.
, and
Miller
,
S. S.
(ed.) (
2017
).
Principles & Methods of Statistical Analysis
.
Thousand Oaks, CA
:
SAGE Publications
.
Gaston
,
A. J.
and
Jones
,
I. L.
(
1998
).
The Auks: Alcidae
.
New York
,
USA
:
Oxford University Press
.
Gatesy
,
S. M.
(
1995
).
Functional evolution of the hind limb and tail from basal theropods to birds
. In
Functional Morphology in Vertebrate Paleontology
(ed.
J.
Thomason
), pp.
219
-
234
.
New York
,
USA
:
Cambridge University Press
.
Goodfellow
,
P.
(
2011
).
Avian Architecture: How Birds Design, Engineer, and Build
.
Princeton
,
USA
:
Princeton University Press
.
Hall
,
B. K.
(
2011
).
Evolution: Principles and Processes
.
Burlington
,
USA
:
Jones & Bartlett Publishers
.
Harris
,
M. P.
,
Wanless
,
S.
,
Barton
,
T. R.
and
Elston
,
D. A.
(
1997
).
Nest site characteristics, duration of use and breeding success in the guillemot Uria aalge
.
Ibis
139
,
468
-
476
.
Hauber
,
M. E.
(
2014
).
The Book of Eggs: A Life-Size Guide to the Eggs of Six Hundred of the World's Bird Species
.
Chicago
,
USA
:
University of Chicago Press
.
Igic
,
B.
,
Nunez
,
V.
,
Voss
,
H. U.
,
Croston
,
R.
,
Aidala
,
Z.
,
López
,
A. V.
and
Hauber
,
M. E.
(
2015
).
Using 3D printed eggs to examine the egg-rejection behaviour of wild birds
.
PeerJ
3
,
e965
.
Ingold
,
P.
(
1980
).
Anpassungen der Eier und des Brutverhaltens von Trottelummen (Uria aalge aalgePont.) an das Brüten auf felssimen*
.
Z. Tierpyschol.
53
,
341
-
388
.
Johnson
,
R. A.
(
1941
).
Nesting behavior of the Atlantic Murre
.
Auk
58
,
153
-
163
.
Kirkland
,
J. I.
,
Carpenter
,
K.
,
Hirsch
,
K. F.
and
Horner
,
J. R.
(
1994
).
Predation of dinosaur nests by terrestrial crocodilians
. In
Dinosaur Eggs and Babies
(
K.
Carpenter
,
K. F.
Hirsch
and
J. R.
Horner
), pp.
124
-
133
.
New York
:
Cambridge University Press
.
López-Martínez
,
N.
and
Vicens
,
E.
(
2012
).
A new peculiar dinosaur egg, Sankofa pyrenaica oogen. nov. oosp. nov. from the Upper Cretaceous coastal deposits of the Aren Formation, south-central Pyrenees, Lleida, Catalonia, Spain
.
Palaeontology
55
,
325
-
339
.
Lovette
,
I. J.
and
Fitzpatrick
,
J. W.
(
2016
).
Handbook of Bird Biology
.
New York
,
USA
:
Wiley
.
Middleditch
,
A. E.
and
Sears
,
K. H.
(
1985
).
Blend surfaces for set theoretic volume modelling systems
.
ACM SIGGRAPH Computer Graphics
19
,
161
-
170
.
Preston
,
F. W.
(
1953
).
The shapes of birds’ eggs
.
Auk
70
,
160
-
182
.
Stoddard
,
M. C.
,
Yong
,
E. H.
,
Akkaynak
,
D.
,
Sheard
,
C.
,
Tobias
,
J. A.
and
Mahadevan
,
L.
(
2017
).
Avian egg shape: Form, function, and evolution
.
Science
356
,
1249
-
1254
.
Tschanz
,
B.
,
Ingold
,
P.
and
Lengacher
,
H.
(
1969
).
Eiform und Bruterfolg bei Trottellummen (Uria aalge)
.
Orn. Beob.
66
,
25
-
42
.
Uspenski
,
S. M.
(
1950
).
Adaptive Züge des Eis die Dickschnabelleumme
.
Sammelband Naturschutz
11
,
95
-
100
.
Uspenski
,
S. M.
(
1956
).
The bird bazaars of Novaya Zemlya (English translation)
.
Russian Game Reports
4
,
1
-
159
.
Wellnhofer
,
P.
(
1971
).
Die Atoposauridae (Crocodylia, Mesosuchia) der Oberjura-Plattenkalke Bayerns
.
Palaeontogr. Abt. A
133
-
165
.
Williams
,
T. D.
(
1995
).
The Penguins: Spheniscidae
, Vol.
2
.
New York
,
USA
:
Oxford University Press
.
Zelenitsky
,
D. K.
and
Therrien
,
F.
(
2008
).
Unique maniraptoran egg clutch from the Upper Cretaceous Two Medicine Formation of Montana reveals theropod nesting behaviour
.
Palaeontology
51
,
1253
-
1259
.

Competing interests

The authors declare no competing or financial interests.