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Fig. 10 in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 10. Hydrodynamic restoration of the Baculites compressus 3D printed model following overdamped harmonic motion. Apertural angle (θa) measured in degrees as a function of time after rotating approximately 38° from the equilibrium orientation. An angle of -90° represents a condition where the aperture is directed downwards. The function of decay in θa with time is represented by the grey dashed curve. Note that this model restores more quickly and does not oscillate about the equilibrium orientation.

Fig. 7 in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 7. Virtual and physical hydrostatic models of Nautilus pompilius with computed percentage of the phragmocone emptied for neutral buoyancy (Φ) and hydrostatic stability (St). The tip of the up-side-down pyramid = center of buoyancy. The tip of the right-side-up pyramid = total center of mass. A. External view of the virtual model. B. Medial section of the virtual model with each component of unique density (green, soft body; red, cameral gas; blue, cameral liquid; grey, shell). C. Modified virtual model with simplified internal geometry and bismuth counterweight (yellow, PLA plastic; red, air; blue, liquid; purple, bismuth counterweight). D. Neutrally-buoyant, 3D printed model. The differences in Φ and the apertural angle (θa) are a result of the mass discrepancy (Table 5) and irregular geometry of the balloon. The error in St was computed assuming that the total mass discrepancy was distributed in the positive or negative z-directions.

Fig. 9 in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 9. Virtual and physical hydrostatic models of Baculites compressus with computed percentage of the phragmocone emptied for neutral buoyancy (Φ) and hydrostatic stability (St). Green, soft body; grey, shell; red, gas; blue, liquid; yellow, PLA plastic; purple, bismuth counterweight; B, center of buoyancy; M, center of mass. A. Virtual model with an even distribution of cameral liquid and gas in the phragmocone (center of mass of cameral liquid and gas = center of volume of the phragmocone; cameral liquid and gas not shown). B. Modified virtual model with simplified internal geometry ("Modified 1" in Table 3). C. Neutrally-buoyant, 3D printed model. D. Modified virtual model with simplified internal geometry and axel hole through pivot point of rotation ("Modified 2" in Table 3). E. Neutrally-buoyant, 3D printed model fixed to an axel and silicone tubing used to supply thrust in the ventral direction. For this model, the mass discrepancy (Table 5) resulted in a slightly lower of 97.3%, but was held constant at 100%. All computed errors in St were computed assuming that the total mass discrepancy was distributed in the positive or negative z-directions.

Fig. 6 in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 6. Hydrostatic models of Baculites compressus with computed percentage of the phragmocone emptied for neutral buoyancy (Φ) and hydrostatic stability (St). All models are oriented dorsum-left. The centers of buoyancy are marked by the tip of the higher pyramid. The total centers of mass are marked by the tip of the lower pyramid. Each material of unique density is designated a color (green, soft body; red, cameral gas; blue, cameral liquid; transparent grey, shell). A. Virtual model with 40% body chamber length to total length (BCL/L). B. Virtual model with 33% BCL/L and adorally distributed cameral liquid. C. Virtual model with 33% BCL/L and adapically distributed cameral liquid. D, E. B. compressus model modified with a concave dorsum similar to B. grandis and 33% BCL/L. Adorally (D) and adapically (E) distributed cameral liquid.

Fig. 3. Full 3D in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 3. Full 3D model of Baculites compressus with model components. A. Complete, digitally-reconstructed shell rendered in X-ray view to show internal structure. B. Three-dimensional model the soft body. C. Three-dimensional model of the cameral volumes within the phragmocone.

Fig. 4 in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 4. Generation of a 3D printed model of Nautilus pompilius with theoretically equal physical properties to the virtual counterparts. A. Original virtual model from Peterman et al. (2019: fig. 2.5). B. Modified virtual model with simplified internal geometry. The center of buoyancy remains the same because external geometry does not change. The total center of mass, however, is corrected by a bismuth counterweight of known volume, density, and mass. C. 3D printed posterior half of the physical model with bismuth counterweight in the computed position. D. Anterior half of the physical model showing the one-way valve for liquid to exit upon displacement by an air-filled balloon. E. Neutrally buoyant physical model with the required volume to liquid ratio for neutral buoyancy. This computed volume of air is inserted through a one-way entrance valve into the internal balloon. Tracking points are placed parallel to the aperture in order to analyze movement in a hydrodynamic setting.

Fig. 5 in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 5. Position of the ventral tracking point (V) and umbilical tracking point (U) as a function of time measured with the physics modeling software (Tracker 4.11.0; Brown 2017). Note that the rotation of the aperture is coupled with translational motion, resulting in complex movement.

Fig. 11 in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 11. Thrust required to change the Baculites compressus model orientation (θa). A. Thrust Scenario 1: A continuous thrust supplied to the venter with a similar thrust ratio to Nautilus (Table 1). The average change from a vertical resting orientation (Δ θpeak) is 22.7°. B. Thrust Scenario 2: Periodic pulses from a pump with a simulated mantle cavity of 20% soft body volume of B. compressus (Table 1). The average change from a vertical resting orientation ( Δ θpeak) is 25.6°. C. Thrust Scenario 3: Periodic pulses from a pump with a thrust ratio between Sepia officinalis and Loligo vulgaris (Table 1). The average change from a vertical resting orientation (Δ θpeak) is 72.2°. Average peak thrust (Fpeak) error bars represent one standard deviation calibrated from 30 second intervals of pumping.

Fig. 8 in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 8. Hydrodynamic restoration of the Nautilus pompilius 3D printed model following underdamped harmonic oscillation. Apertural angle (θa) measured in degrees as a function of time after rotating approximately 38° from the equilibrium orientation. An angle of zero represents a condition where the aperture is horizontally oriented. Open dots represent the peaks used to calculate decay in amplitude with time (grey dashed curves).

Fig. 2 in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 2. Shell and septum thickness measured from three specimens of Baculites compressus (WSU-1400, WSU-1401, and WSU-1405). Exponential curves were fit to these points to define thickness for the full 3D model as a function of whorl height.

Fig. 1 in Mode of life and hydrostatic stability of orthoconic ectocochleate cephalopods: Hydrodynamic analyses of restoring moments from 3D printed, neutrally buoyant models

Fig. 1. Three-dimensional reconstruction of a fragmentary baculite Baculites compressus Say, 1820 (WSU-1400) from the late Campanian Pierre Shale of Meade County, South Dakota. A. Model of a fragmentary specimen generated by photogrammetry with the software (3DF Zephyr). B. Broken septum isolated from the photogrammetry model. C. Suture pattern. D. Complete septum created by reconstructing the higher-order frilling with the suture pattern as a template.

Fig. 7 in Large onychites (cephalopod hooks) from the Upper Jurassic of the Boreal Realm

Fig. 7. Fitting of inner and outer edges of hooks to logarithmic spiral segments (thick lines). A. PMO 223.380. B. PMO 223.381. C. PMO 223.382 D. PMO 223.379. E. PMO 223.383. F. PMO 210.229.

Fig. 5 in Large onychites (cephalopod hooks) from the Upper Jurassic of the Boreal Realm

Fig. 5. Mega− (Me) and micro−onychites (retouched). From Statoil well 33/9−C27, depth 3187.7 m (Upper Jurassic). PMO 223.405.

Fig. 8 in Large onychites (cephalopod hooks) from the Upper Jurassic of the Boreal Realm

Fig. 8. Whorl expansion coefficients (b) for inner and outer spirals. Onychites quenstedti as figured by Engeser (1987: pl. 2: 5).

Fig. 9 in Large onychites (cephalopod hooks) from the Upper Jurassic of the Boreal Realm

Fig. 9. Functional morphology of a hook or claw in a logarithmic spiral shape, penetrating a surface. A, B. Thrusting action, pole proximal. The angle φ at the contact point and the direction of the force vector (arrow) are constant during penetration. C, D. Pulling action, pole distal.

Fig. 2 in Large onychites (cephalopod hooks) from the Upper Jurassic of the Boreal Realm

Fig. 2. Simplified stratigraphy of the upper part of the Agardhfjellet Formation in the Sassenfjorden area, Spitsbergen, with positions of mega−onychites. The Upper Volgian is highly condensed. Modified from Hammer et al. (2011).

Fig. 4 in Large onychites (cephalopod hooks) from the Upper Jurassic of the Boreal Realm

Fig. 4. Belemnoid mega− (Onychites quenstedti Engeser, 1987) and microonychites from the Agardhfjellet Formation (probably Oppdalsåta Member), Upper Jurassic, Spitsbergen. PMO 223.381. A. Photograph. B. Explanatory drawing. Micro−onychites marked as hooks (complete specimens), lines and dots (incomplete specimens). Density of micro−onychites in gray scale, computed using the kernel density method.

Fig. 3 in Large onychites (cephalopod hooks) from the Upper Jurassic of the Boreal Realm

Fig. 3. Belemnoid mega−onychites Onychites quenstedti Engeser, 1987 from Boreal localities. A. PMO 223.380, seep carbonate, earliest Late Volgian, Knorringfjellet, Spitsbergen; entire specimen. B. Close−up of shaft on PMO 223.380, showing rows of circular tubercles. C. PMO 223.378, latest Early Volgian, Janusfjellet, Spitsbergen. D. PMO 223.379, latest Early Volgian, Janusfjellet, Spitsbergen. E. PMO 223.382, Middle Volgian, Janusfjellet, Spitsbergen. F. PMO 210.229, latest Early or early Middle Volgian, well 7/4−1, North Sea. G. PMO 223.383, Late Jurassic, well 34/7−23A, North Sea. H. PMO 74149, Late Jurassic, Andøya, northern Norway.

Fig. 1 in Large onychites (cephalopod hooks) from the Upper Jurassic of the Boreal Realm

Fig. 1. Geological map of the Sassenfjorden area, central Spitsbergen. Sampling locality: 1, Janusfjellet; 2, Knorringfjellet. Redrawn from Dallmann et al. (2001).