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   Earth tessellation II
Self-organized breakup of Gondwana

James W. Sears

University of Montana, Missoula MT 59812

jwsears@selway.umt.edu

This webpage is a continuation of work described in:
Truncated-icosahedral breakup of Laurasia and Gondwana and anorogenic magmatism
by
James W. Sears, Gregory M. St. George & J. Chris Winne

An accompanying Powerpoint presentation is available:
Self-organized Breakup of Gondwana

by
James W. Sears

Abstract

Gondwana broke apart along a geometrically-regular, truncated-icosahedral fracture system. This tessellation minimized total crack length and therefore required the least work to nucleate and propagate fractures across the supercontinent. Fracture spacing was a function of the strength of the Gondwana lithosphere. Fracture arrangement met conditions imposed by Euler’s rule for ordering polyhedrons on a spherical shell. Plates at the vertices of an icosahedron had five-fold symmetry while intervening plates had six-fold symmetry. Large igneous provinces erupted diachronously along the fractures and triple-junctions. A family of hot spots are also congruent with the tessellation. The tensile stress field that initiated the fractures defined the geometrical dual of the fracture tessellation (an icosadeltahedron), for which each vertex corresponds to a face, and each face to a vertex. The stress field appears to have been self-organized by the pre-existing geometry of Gondwana, with tensile hoop-stress along the passive northern boundary. Regularly-spaced radial fractures abutted the northern boundary orthogonally, showing that it acted as a free surface; these defined the lateral edges of Australia, India, Arabia, Libya, and northwest Africa. The polygonal fracture network was symmetrical about the African geoid bulge when Gondwana is restored to its Triassic locus. These observations are consistent with the hypothesis of Anderson (1982) that Gondwana insulated its underlying mantle, leading to thermal expansion, uplift of the African geoid bulge, and fracturing as Gondwana stretched to accommodate an increased radius of curvature. The uplift and fracturing may have culminated during the Triassic marine lowstand. The fragments then drifted off the thermal bulge due to plate tectonic demands, with secondary rifting, large igenous province eruptions, and lingering hotspot activity. These observations favor Anderson’s top-down PLATE paradigm for the tectonic control of continental breakup and associated large igneous provinces and hot spots, and they argue against the deep-mantle-plume paradigm.

Introduction

While it is generally agreed that supercontinents break apart and re-assemble in a grand tectonic cycle, much controversy surrounds the initiation of the breakup phase of the cycle (Foulger et al., 2005). Following an original idea by J.T. Wilson (1963), the deep-mantle-plume paradigm predicts that superadiabatic plumes rising from the core-mantle boundary drive continental breakup (Morgan, 1981; Campbell, 2001). It proposes that the heads of deep mantle plumes collect at the base of the lithosphere and form broad domes that split into radiating rifts (Storey et al., 2001). The plume heads may then erupt large igneous provinces (LIPs) from the rifts (Ernst & Buchan, 2001). The rifts may propagate outward and link together to break up the continent piecemeal as fragments calve away, perhaps due to secondary plate tectonic forces. Some active volcanic hot spots may represent lingering ascents of thin plume tails (Morgan, 1981). In this paradigm, plume ascents and breakouts are episodic and depend upon deep mantle viscosity and instabilities at the core-mantle boundary (Steinberger, 2000).

Mantle tomography and receiver functions do not, however, unequivocally demonstrate that plumes cross the mantle transition zone (Foulger et al., 2000; Du et al., 2006). DeWit et al. (1988), Anderson (2001; 2002b), and Hamilton (2002), argue that continental breakup and associated large igneous outbreaks and hot spots are controlled, top-down, by lithospheric processes, rather than by rising plumes. Anderson (2006) has recently formalized this opposing view as the PLATE paradigm. Continental breakup may be initiated by thermal expansion of ordinary sub-lithospheric mantle that becomes insulated beneath a stalled or sluggish supercontinent. For example, Anderson (1982) showed that the Atlantic-African geoid anomaly coincides with the late Paleozoic locus of Pangaea and may represent a residuum of thermally-expanded sub-Pangaean mantle. The thermal expansion would lift the supercontinent and place it under uniform layer-parallel tension as it adjusted to an increased radius of curvature. The supercontinent might then rift apart, with outbreaks of LIPs along rift zones, as fragments drift off the thermal bulge toward retreating trenches. Anomalous hot spot activity that continues at fixed mantle sites within the decaying Atlantic-African geoid anomaly is consistent with this model (Anderson, 1982).

Here I argue that the breakup of Gondwana was self-organized in accordance with Euler’s theorem for convex polyhedrons to minimize crack length and therefore minimize the energy required to nucleate and propagate fractures. The fracturing appears to have been controlled by the pre-existing configuration of the Gondwana margins and the strength of the Gondwana lithosphere. Plate tectonic processes later exploited the initial fractures to widen rifts, trigger hot spots, release LIPs, and disperse rifted fragments. This model argues against the deep-mantle-plume paradigm and supports the top-down tectonic model of Anderson (1982).

Gondwanan fracture tessellation

Figure 1 presents a standard reconstruction of Gondwana, adapted from DeWit et al. (1988), Golonka et al. (1994) and Lawver et al. (1999). The argument presented in this webpage follows from recognition that Gondwana’s initial fracture architecture was closely congruent with a precise, energy-minimizing configuration, the truncated icosahedron (Sears et al., 2005; Sears, 2001). The truncated icosahedron comprises a semi-regular polyhedron made up of 12 pentagonal and 20 hexagonal faces. The pentagonal faces are centered on the vertices of an icosahedron. The tessellation is geometrically intolerant, i.e., a single pentagon fixes the entire configuration. Projected onto the Earth’s surface, each tile-edge of a truncated icosahedron subtends 23.28° of arc, or approximately 2600 km.

Figure 1. Gondwana reconstruction at 200 Ma, after Golonka et al. (1994), Lawver et al. (1999) and DeWit et al. (1988). Heavy dashed lines are major fractures that define truncated icosahedral tessellation. Black pentagons at centers of central South America, Antarctica, and Arabia occupy vertices of exact icosahedral triangle (purple) at Earth-scale. Yellow stars are major hot spot volcanoes in modern co-ordinates (note lines of latitude and longitude). Gondwana is restored so that fracture tessellation best fits hot spot tessellation. Red areas are LIPs with eruption dates shown. Note that although dates range over more than 100 million years, most LIPs erupted from fractures that restore to single tessellation. This implies that coherent tessellation dates to before oldest LIP (205 Ma), and that LIPs erupted from fractures diachronously as later plate tectonics opened fractures. Click here or on figure to enlarge.

The Gondwanan tessellation included all or parts of three pentagons and six hexagons of the exact scale and arrangement of a truncated icosahedron at the Earth’s surface (Figure 1). One measure of this congruence is that the distance between the centers of the Antarctic and central South American pentagons on the reconstruction exactly equals that between icosahedral vertices (60° or about 6600 km at Earth-scale). Gondwanan fractures with a cumulative linear distance greater than 20,000 km define segments of 16 edges of this truncated icosahedral tessellation. Their geometric coherence indicates that the fractures formed in a uniform, Gondwana-wide stress field prior to dispersal of any daughter continents. The fractures probably propagated across Gondwana in Late Permian or early Triassic time when thermal expansion culminated (Anderson, 1982). The oldest parts of some of the fractures accumulated Permian or Triassic sediments (Sengor & Natal’in, 2001). The Late Permian-Early Triassic marine lowstand (Haq, 1995) may record the culmination of Gondwana uplift.

Gondwanan stress tessellation

The Gondwanan fractures were tensional cracks. Each fracture thus propagated orthogonally to the local horizontal principal tensile stress. Construction of the principal tensile stress direction at the midpoint of each fracture generates the icosadeltahedron, the dual polyhedron of the truncated icosahedron (Figure 2). (Note: Each face of a polyhedron corresponds to a vertex of its dual. For example, the cube, with six faces and eight vertices, represents the dual of the octahedron, with six vertices and eight faces. The edges of dual tessellations bisect one another orthogonally.)

Figure 2. Gondwana tensile stress tessellation. This icosadeltahedral tessellation (thin solid lines) is dual of fracture tessellation (heavy dashed lines) such that vertices of stress tessellation occupy faces of fracture tessellation, and vice-versa. Tessellations cross one another orthogonally. Black vertices have five nearest neighbors and exactly occupy vertices of an icosahedron at Earth-scale, as required by Euler’s rule for convex polytopes (see text). Blue vertices have six nearest neighbors. Note that stress tessellation follows northern margin of Gondwana, implying it was free surface that guided tensile hoop stress. Thus, Gondwana split on radial fractures along northern rim. Note nearly perfect symmetry of stress tessellation across Gondwana. This provided shortest total fracture length and thus required least work to break up Gondwana. Click here or on figure to enlarge.

The stress tessellation of Figure 2 obeys Euler’s rule for convex polytopes. A sphere with n ≥ 12 vertices requires 12 vertices with five nearest neighbors (pentamers) and n-12 vertices with six nearest neighbors (hexamers). The pentamers occupy the vertices of an icosahedron. The vertices in central South America, Antarctica, and Arabia have five nearest neighbors (not all shown). As predicted by Euler’s rule, these precisely occupy the vertices of an icosahedron at the scale of the Earth. The remaining vertices in Figure 2 have six nearest neighbors. This precise geometric relationship indicates that the spherical Gondwanan shell failed as an integrated whole in a uniform, non-random fashion.

The edges of Gondwana appear to have acted as free surfaces that organized the stress tessellation. Six edges of the stress tessellation coincided with the northern rim of Gondwana and parts of the Gondwanides trench margin.

Tensile hoop-stress swept around the northern rim of Gondwana from Central America to New Zealand. Six major radial Gondwanan fractures abutted this rim because the fractures formed orthogonally to the tensile hoop-stress. These fractures separated New Zealand, Australia, India, Arabia, Libya, northwest Africa, and Central America. Most of these fractures were equally spaced, as a function of the strength of the Gondwanan lithosphere.

Tensile hoop-stress forms in response to expansion of an enclosed region. The stress tessellation within the Gondwanan interior reflected the expansion of the enclosed Gondwana surface area. It was nearly perfectly symmetrical across Gondwana, providing the most balanced stress distribution that was consistent with the boundary conditions.

The stress field derived from the fracture tessellation is consistent with a uniformly and simultaneously expanding Gondwana, as predicted by the hypothesis of Anderson (1982). The plume paradigm predicts, conversely, that the fractures were caused by random and unrelated plumes that impinged on the base of the moving lithosphere.

Scale of the fracture polygons

The scale of the fracture tessellation was a function of the strength of the Gondwanan lithosphere and the geometry of a spherical shell. The truncated icosahedral fractures are analogous to the hexagonal crack patterns seen in columnar-jointed basalt. For ideally hexagonal basalt columns such as parts of the Devils Postpile, Long Valley Caldera, California, cracks began to form when tension caused by shrinkage exceeded the strength of the crystallized and cooling basalt. A planar crack would propagate to a critical length of about 25 cm, and would then spontaneously split into two cracks at an angle of approximately 120° to the original crack direction. Each new crack would branch when it reached the critical length and, together with similar cracks, formed a network of polygonal columns of surprisingly uniform sizes. Where a propagating crack encountered the edge of a flow, it bent into an orthogonal intersection, because the edge and the propagating crack coincided with orthogonal principal planes of the local stress ellipsoid.

Hexagonal fracture systems develop in an homogenous material undergoing uniform layer-parallel tension, such as a cooling and contracting basalt flow, because they provide the greatest stress relief for the least work to nucleate and propagate cracks (Jagla & Rojo, 2002). This is because a regular hexagonal pattern requires the shortest total crack length to pave a given area, and provides the most stable triple-junctions. The energy used for the work of propagating cracks is stored as elastic strain within the volume of the contracting layer. Stronger layers crack into larger hexagons because fewer tiles are required to pave the area and these require a shorter total crack length. More strain energy is required to initiate the fractures, but because the layer is stronger, it stores more energy before failing.

Each crack in an array of hexagonal tension cracks corresponds to a principal plane of a local stress ellipsoid. The corresponding local tensile stress is normal to the crack. Tensile stresses for the entire hexagonal array of cracks define a triangular array, with the apices of the triangles at the centers of the hexagons. The apices form null points of no strain in the shrinking medium; strain increases radially outward from them to the distance at which the material cracks. Hexagonal close-packing of fractures best relieves strain between neighboring domains.

The edge lengths of the polygons increase with strength of the shell to minimize the total crack length. Once the polygons become large enough to reflect the curvature of a spherical shell, Euler’s rule becomes evident; pentagonal polygons will occupy the 12 vertices of an icosahedron, and intervening polygons will be hexagonal. With increasing shell strength, the 12 pentagonal faces will increase in size, and the intervening hexagons will also increase in size because they share edges with the pentagons. All edge lengths are equal. The hexagons will decrease in number in a stepwise fashion to fit the finite geometry of the sphere. The dodecahedron represents the strongest configuration for a spherical shell, with 12 large pentagonal faces and no intervening hexagons. The truncated icosahedral Gondwanan fracture tessellation represents the second strongest configuration, with 12 pentagons and 20 intervening hexagons for a complete sphere. Note that Gondwana only occupies a portion of a sphere.

LIPs and hot spots

Nine LIPs ranging in age from Early Jurassic to Early Tertiary erupted as Gondwana rifted apart and its daughter continents dispersed. When the continents are gathered into their Gondwana configuration, the LIPs are congruent with vertices or edges of the fracture tessellation. This suggests that the fracture tessellation prepared the ascent routes for the eruptive sites, but that LIP outbreaks depended on later effects such as decompression melting as plate tectonic movements widened the fractures and opened conduits for LIP eruptions.

When Gondwana is reconstructed, the fracture tessellation may be superimposed on several major hot spots associated with Late Jurassic or Early Cretaceous rifts and LIPs. Heard, Marion, Bouvet, Gough, Tristan, St. Helena, Ascencion, and Fernando plot within a few degrees of the tessellation, mostly near vertices. However, neither older hot spots associated with opening of the Central Atlantic (Cape Verde, Canary, Azores), nor younger hotspots associated with the Deccan or Ethiopian LIPs are congruent with this position of the tessellation. The congruent hot spots may record a time of drift stagnation of Gondwana. Independent paleomagnetic data indicates that Gondwana moved in Early Jurassic time, stalled during Late Jurassic and Early Cretaceous time, and then rapidly broke apart in Late Cretaceous time (DeWit et al., 1988; Golonka et al., 1994).

Eruption of LIPs may have resulted from decompression melting upon opening of rifts along the fracture tessellation. Lingering hot spot activity may have been localized by alteration of chimneys in the upper mantle beneath the original sites of the LIP eruptions. Alternatively, Fairhead & Wilson (2006) suggest that some hot spot tracks may be fractures that propagated due to stress instabilities in the widening plates. Perhaps some hot spots are due to standing stress waves caused by adaptation of the symmetry of the fracture tessellation to the symmetry of sea-floor spreading. These considerations favor a lithospheric, rather than deep mantle, control for Gondwana LIPs and hot spots.

African geoid anomaly

Anderson (1982) proposed that the Atlantic-African geoid anomaly marks the Permian footprint of Pangaea, the decaying remnant of thermally expanded mantle that had been insulated beneath the supercontinent. He proposed further that LIP and hot spot activity in the region demonstrates its increased thermal content.

Figure 3 superimposes a contour map of the African part of the geoid anomaly on Gondwana in the mantle position it may have occupied during Triassic time (Golonka et al., 1994). If the geoid bulge was centered on Gondwana as shown, then the contours either paralleled or were orthogonal to the fracture and stress tessellations.

Figure 3. African geoid anomaly superimposed on Gondwana in its Triassic position. Note that contours of the geoid anomaly are generally orthogonal or parallel to stress tessellation edges, and that anomaly is centered on Gondwana. This is consistent with the hypothesis of Anderson (1982) that the geoid anomaly marks the paleoposition of Gondwana, and that Gondwana spread outward from the anomaly as it broke apart. Click here or on figure to enlarge.

Anderson (1982) suggested that Gondwana spread radially outward from the geoid high, consistent with the radial components of the stress tessellation. The outward spreading would also result in hoop stress parallel to the geoid contours, consistent with the non-radial components of the stress tessellation.

The increased radius of curvature that Gondwanan lithosphere attained above the geoid anomaly would have further stretched Gondwana outward, leading to additional radial and hoop tension. When the combination of these stresses was large enough to overcome the strength of the Gondwanan lithosphere, it cracked into the pattern that required the shortest total crack length.

The icosahedron in nature

Comparison of the Gondwanan tessellation with other natural examples of icosahedral arrangements provides insight into the Gondwanan fracture process. In nature, collections of particles commonly surface a sphere in icosahedral patterns. These include blastocysts, colloids, quasi-crystals, florets, and numerous icosahedral viruses, including wart, herpes, polio, and HIV (see Anderson, 2002).

Viral capsids are made up of collections of cells which are required by Euler’s theorem for convex polyhedrons to assume icosahedral configurations like the Gondwana fractures. Twelve cells have 5-fold coordination (pentamers) at the vertices of an icosahedron, and the remaining (n-12) cells have 6-fold coordination (hexamers). Because the pentamers form shorter, stronger bonds, bursting viral capsids crack along hexamers. The cracks zig-zag along polygonal boundaries much like those on fragmenting supercontinents.

Icosahedral configurations solve the classic Thomson problem of minimizing the energy of an array of mutually repulsive coulombic charges on a sphere (Altschuler et al., 1997). Twelve charges have five nearest neighbors and (n-12) have 6 nearest neighbors. The lowest energy configurations produce truncated icosahedral strain gradients closely similar to the rift patterns seen on Gondwana. For the supercontinent, the strain may have resulted from particles stretching away from one another to permit the spherical shell to adapt to an increasing radius of curvature as the sub-lithospheric mantle thermally expanded.

Conclusions

The self-organized Gondwana fracture tessellation is consistent with the hypothesis of Anderson (1982) that the supercontinent drove its own breakup by insulating the underlying mantle. The thermally-expanded mantle lifted Gondwana, placing it under uniform tension. When the tension exceeded the strength of the Gondwana lithosphere, it fractured into a symmetrical polygonal pattern commiserate with its strength and conforming to the geometric restrictions of a sphere and to the boundary conditions of the supercontinent. Likely, the fractures propagated in zig-zag fashion across the supercontinent, bending and branching at critical lengths. The fractures relieved the tension and separated the supercontinent into tiles that could move independently under the influence of plate tectonic processes. Separation of tiles opened rift valleys and ocean basins and drove decompression melting of the thermally expanded mantle, leading to outbreaks of LIPs and injection of dike swarms. These secondary effects were diachronous as global plate tectonics exploited the icosahedral fractures. Stationary hot spots that mark the eruptive sites of LIPs may represent long-lasting alterations of LIP ascent chimneys through the upper mantle, or tips of fractures fixed by standing stress waves in growing plates that convert icosahedral symmetry to plate tectonic symmetry.

References

last updated 26th April, 2006
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