The accompanying illustration compares the proposed system of lower-mantle convection rolls with the three global tectonic rings. According to the Convection Rolls Model, the lower-mantle rolls occupy two superimposed layers extending from approximately 670 km to 2,900 km depth, with their projected patterns partly overlapping. The general concept of mantle convection is familiar from introductory geology textbooks, which commonly illustrate the circulation of mantle material by means of paired convection cells. Similarly, the concept of a tectonic ring is familiar through the widely recognized Pacific Ring of Fire. A comparison of these two features suggests that they may represent related expressions of recurring geometrical patterns within the Earth.

At first glance, the mirrored arrangement of the green lines may appear confusing. This is because the illustration represents a three-dimensional convection system in two dimensions, with the patterns of two vertically superimposed layers projected onto a single geographical plane. Each lower-mantle convection roll extends approximately 15° from east to west, and the rolls are arranged side by side within their respective layers. The smaller convection rolls of the upper mantle, extending between approximately 120 and 670 km depth, are considerably more numerous. They are represented in the model by four layers of rolls, each approximately 1.5° wide. These smaller rolls are not included in the present illustration.

The Mid-Atlantic Ridge can be traced through the central region of the Atlantic tectonic ring, while the mid-ocean ridge system of the Indian Ocean occupies a corresponding position within the Indian Ocean ring. The Pacific Ocean requires particular consideration because of its extensive convergent plate boundaries along both its eastern and western margins, together with the divergent boundary represented by the East Pacific Rise.

The two-sided nature of the Pacific tectonic system is particularly relevant when considering the geometrical relationships of Antarctica. The Antarctic elliptical structure has two principal axes, each with two endpoints. These four directions appear to correspond to four major branches of the proposed global tectonic framework. One endpoint of the minor axis extends towards New Zealand, while the opposite endpoint points towards the Mid-Atlantic Ridge. Similarly, one endpoint of the major axis is associated with the East Pacific Rise, whereas the other corresponds to the region of the Indian Ocean ridge system and its triple junction.

In the proposed model, the three tectonic rings are represented by elliptical structures. The inner ellipse spans approximately 120° of longitude along the equator and has a major-to-minor axis ratio of 3:2. Its major axis is oriented NW–SE, at an angle of 45° to the north–south and east–west directions. On an equirectangular map, where each degree of longitude and latitude is represented by an equal unit of length, this orientation corresponds to a slope of −1.

The model proposes that the stresses generated by mantle convection provide the mechanical conditions necessary for the formation of these regularly arranged elliptical tectonic structures. Their dimensions are related to the 15° spacing of the lower-mantle convection rolls, which influence the smaller convection systems within the transition zone and asthenosphere. The width of the tectonically active zones may exceed the 15° width of an individual lower-mantle roll because of the additional effects of the smaller convection rolls in the overlying mantle.

The present distribution of geological activity and the dynamics of plate motion can be compared with this geometrical framework. Several relevant relationships have already been identified in connection with Antarctica. The Gulf of California and the Red Sea, for example, provide an interesting comparison because both occur at similar latitudes and are associated, in this interpretation, with the margins of elliptical tectonic structures.

Other examples can be identified throughout the world. New Zealand, Japan and Kamchatka are associated with the western Pacific tectonic system. The East African Rift, the volcanically active parts of the Mediterranean region and Iceland can be examined in relation to different sections of the Atlantic tectonic ring. Similarly, the western margin of South America follows a major convergent boundary associated with the Pacific tectonic ring.

Many of these locations have already been examined in previous articles. Comparing their geological characteristics with the proposed geometrical framework may help explain relationships between divergent boundaries, convergent boundaries, volcanic activity and the underlying mantle convection system.

It is important, however, to distinguish between the geometrical relationships demonstrated by the model and the dynamic processes that may produce them. The geometrical arrangement provides a framework within which present-day geological structures can be compared, while plate motions continuously modify the position and character of tectonically active zones. The proposed system is therefore not intended to represent a static Earth, but rather a recurring geometrical organization within an evolving tectonic environment.

The most immediately recognizable feature of the illustration is the large-scale division of the Earth’s tectonic framework into three principal systems associated with the Atlantic, Pacific and Indian Oceans. These are represented by the three global tectonic rings, whose geometrical relationships can be examined alongside the proposed system of mantle convection rolls.

Convection and tectonic rings relationship:

The Three Global Tectonic Rings: Geometrical Relationships with Mantle Convection Rolls and Major Plate Boundaries

Abstract

The proposed model of three global tectonic rings provides a geometrical framework for comparing the distribution of divergent, convergent and transform plate boundaries with the underlying system of mantle convection rolls. The rings are represented by elliptical structures associated with the Atlantic, Pacific and Indian Oceans. Their centers are separated by 120° of longitude, and their major axes are oriented NW–SE on an equirectangular geographical grid.

A particularly interesting geometrical relationship is found along the minor axis of the inner Pacific tectonic ring. Its northeastern endpoint lies near the Gulf of California, while its southwestern endpoint is situated near New Zealand. At both locations, the inner Pacific ellipse intersects the outer ellipse of an adjacent tectonic ring. The northern intersection, involving the Atlantic ring, occurs at approximately 31.017°N, 113.063°W. The southern intersection, involving the Indian Ocean ring, occurs at 43.017°S, 174.463°E.

Both intersections are remarkably close to the corresponding endpoints of the minor axis. These relationships, together with the orientation of major fault systems and the proposed convection-roll geometry, provide a basis for investigating whether the elliptical structures represent a recurring organization of tectonic deformation.

1. The geometrical framework of the tectonic rings

The general distribution of active plate boundaries reveals three large oceanic systems associated with the Atlantic, Pacific and Indian Oceans. These systems are interconnected, but their tectonic characteristics differ considerably.

The Atlantic Ocean is dominated by divergent boundaries along the Mid-Atlantic Ridge, whereas the Pacific Ocean is characterized by extensive subduction zones, transform faults and the divergent East Pacific Rise. The Indian Ocean contains an extensive network of spreading ridges and plate-boundary intersections.

The proposed tectonic-ring model attempts to place these different structures within a common geometrical framework.

Each ring consists of an inner and an outer elliptical boundary. Their dimensions, expressed in degrees on an equirectangular map, are:

Parameter	Inner ellipse	Outer ellipse
Semi-major axis	77.7°	96.5°
Semi-minor axis	51.8°	71.0°
Major/minor axis ratio	1.500	1.359
Major-axis orientation	−45°	−45°

The elliptical structures share a common central latitude of 6°S.

Their central longitudes are:

• Atlantic tectonic ring: 29.3°W.
• Indian Ocean tectonic ring: 90.7°E.
• Pacific tectonic ring: 149.3°W.

The centers are separated by 120°, corresponding to one-third of the Earth’s circumference along the equator.

The general equation of an ellipse rotated through an angle (\theta) is:

[
\frac{x’^2}{a^2}+\frac{y’^2}{b^2}=1
]

where

[
x’=(x-x_0)\cos\theta+(y-y_0)\sin\theta
]

[
y’=-(x-x_0)\sin\theta+(y-y_0)\cos\theta
]

and (\theta=-45^\circ).

The major axis therefore follows a line with a slope of −1, while the minor axis is perpendicular to it, with a slope of +1.

All calculations use equal numerical scales for longitude and latitude. Consequently, the elliptical structures are exact within the chosen rectangular coordinate system, although their corresponding dimensions and orientations on the spherical Earth require geographical corrections.

2. The equator and the dimensions of the tectonic rings

The equator provides a fundamental reference for the geometrical arrangement of the tectonic rings.

The three centers are positioned at 120° intervals in longitude, while the entire system is displaced 6° southward.

The dimensions of the inner ellipses provide an equatorial span of approximately 120°, equivalent to one-third of the Earth’s circumference in longitude.

The relationship between the inner and outer elliptical boundaries is particularly interesting.

At the western intersection with the equator, the separation between the boundaries is calculated to be approximately:

[
\Delta x=19.451^\circ
]

This is remarkably close to 19.5°, which can be expressed in terms of the dimensions of the proposed convection-roll system:

[
19.5^\circ=15^\circ+3(1.5^\circ)
]

In the model, the lower-mantle convection rolls are arranged at 15° intervals, whereas smaller rolls in the upper-mantle system occur at 1.5° intervals.

One possible interpretation is that the width of the tectonically active region reflects the combined influence of one major lower-mantle unit and three smaller upper-mantle units.

However, this correspondence is geometrical. A mechanical calculation is required to determine whether the stresses transmitted between the different convection systems could actually produce the proposed width.

The 6° southward displacement introduces another important feature. The tectonic rings are not symmetrical about the equator, although their internal geometrical arrangement remains symmetrical about their common central latitude.

This displacement may be significant when considering the interaction between the northern and southern parts of the mantle convection system.

3. The northeastern endpoint of the Pacific minor axis: California

The inner Pacific tectonic ellipse has its center at:

[
(x_0,y_0)=(-149.3^\circ,-6^\circ)
]

Its semi-minor axis measures 51.8°.

Because the minor axis is inclined at 45° to the coordinate axes, its northeastern endpoint can be calculated directly:

[
x=x_0+\frac{51.8^\circ}{\sqrt2}
]

[
y=y_0+\frac{51.8^\circ}{\sqrt2}
]

The resulting coordinates are:

[
\boxed{30.628^\circ N,\ 112.672^\circ W}
]

This location is near the northern Gulf of California.

The geographical relationship is particularly interesting because the Gulf of California and the San Andreas Fault System represent connected sections of the Pacific–North American plate boundary.

The Gulf of California is characterized by oblique divergence and transform motion, whereas the San Andreas Fault is predominantly a right-lateral transform boundary.

The inner Pacific ellipse follows a NW–SE orientation near California, broadly corresponding to the orientation of the Gulf of California and major sections of the San Andreas Fault System.

At the northeastern endpoint of the minor axis, the tangent to the ellipse is parallel to its major axis. This provides a direct geometrical explanation for the NW–SE orientation of the elliptical boundary in this region.

A comparison with the San Andreas Fault also reveals an interesting relationship between the distance of the fault system from the inner ellipse at its northern and southern extremities.

Using representative locations near the Salton Sea and Cape Mendocino, the calculated distances to the ellipse are approximately 32 km and 44 km, respectively.

Although the intermediate parts of the fault system do not follow a constant distance from the ellipse, the similarity of these two endpoint distances deserves closer examination.

The orientation of the relative motion between the Pacific and North American plates also corresponds broadly to the tangent of the ellipse in southern California. The agreement becomes less precise farther north.

Thus, the geometrical relationship concerns not only the location of the fault system but also its orientation and the direction of relative plate motion.

4. The intersection of the Pacific and Atlantic tectonic rings

Another important geometrical relationship occurs near the northeastern endpoint of the Pacific minor axis.

The inner Pacific ellipse intersects the outer Atlantic ellipse at:

[
\boxed{31.017^\circ N,\ 113.063^\circ W}
]

This intersection is located close to the northern Gulf of California, approximately 57 km from the calculated endpoint of the Pacific minor axis.

The two ellipses intersect slightly south of 32°N.

This latitude is important within the proposed convection-roll system because the centers of the northern hemispheric convection arcs are positioned along the 32nd parallel.

The corresponding equation is:

[
(x-C_n)^2+(y-32)^2=35.341^2
]

At 32°N, the tangents to these circular convection boundaries are oriented north–south on the rectangular geographical grid.

Therefore, the northern Gulf of California occupies an interesting position in the model: it is situated near the intersection of two tectonic rings, near the endpoint of a principal elliptical axis and close to the latitude at which the convection-roll boundaries have a north–south orientation.

This combination suggests a possible location where different components of the proposed mantle-driven stress system might interact.

Whether such an interaction has mechanical significance remains to be established through calculations of mantle flow, transmitted traction and lithospheric deformation.

5. The southwestern endpoint of the Pacific minor axis: New Zealand

The opposite endpoint of the same minor axis is found by subtracting the horizontal and vertical components from the center of the Pacific ellipse.

The calculated coordinates are:

[
\boxed{42.628^\circ S,\ 174.072^\circ E}
]

This point lies near the northeastern part of New Zealand’s South Island, in the broader Kaikōura tectonic region.

The geological setting differs considerably from California.

New Zealand is situated along the boundary between the Pacific and Australian plates. The Alpine Fault is a major oblique-slip plate-boundary fault extending along the western side of the South Island. Farther northeast, the Marlborough Fault System connects the Alpine Fault to the Hikurangi Subduction Zone.

The endpoint of the Pacific minor axis is not located directly on the principal trace of the Alpine Fault. Nevertheless, it is situated within the broader region where the geometry and character of the Pacific–Australian plate boundary change significantly.

The most important geometrical relationship involves the intersection between the inner Pacific ellipse and the outer Indian Ocean ellipse.

Their southern intersection occurs at:

[
\boxed{43.017^\circ S,\ 174.463^\circ E}
]

The intersection is approximately 54 km from the southwestern endpoint of the Pacific minor axis.

This result closely resembles the corresponding relationship in California, where the Atlantic and Pacific ellipses intersect approximately 57 km from the northeastern endpoint.

Consequently, the two ends of the Pacific minor axis display a similar geometrical arrangement involving the boundary of an adjacent tectonic ring.

6. The symmetrical relationship between California and New Zealand

The results can be summarized as follows:

Geometrical feature	California	New Zealand
Pacific minor-axis endpoint	30.628°N, 112.672°W	42.628°S, 174.072°E
Adjacent tectonic ring	Atlantic	Indian Ocean
Intersection with Pacific inner ellipse	31.017°N, 113.063°W	43.017°S, 174.463°E
Distance from minor-axis endpoint	~57 km	~54 km
Principal tectonic setting	Transform and oblique divergence	Oblique collision, transform faulting and subduction

The similarity is particularly striking because the two intersection points lie on opposite sides of the Pacific Ocean.

The northern intersection occurs at approximately 31°N, while the southern intersection occurs at approximately 43°S.

Their latitudes are symmetrical around 6°S:

[
\frac{31.017^\circ+(-43.017^\circ)}{2}
=-6^\circ
]

This relationship follows directly from the common latitude of the ellipse centers and the symmetrical mathematical construction.

It is therefore an exact prediction of the chosen geometry rather than an independent geological confirmation.

Nevertheless, its geological relevance can be tested by comparing the locations and orientations of active faults, plate movements and deformation zones with the predicted elliptical boundaries.

The California and New Zealand regions provide two important opportunities for doing so.

7. Mantle convection rolls and the origin of elliptical stress fields

The proposed relationship between the tectonic rings and mantle convection involves both vertical and horizontal mechanical effects.

Vertical movements within the mantle convection system may influence the distribution of buoyancy forces, pressure and lithospheric stresses.

At the same time, horizontal mantle flow can transmit shear traction to the overlying lithosphere.

These forces interact with plate motion, including spreading at divergent boundaries, convergence at subduction zones and lateral displacement along transform faults.

In the proposed model, the lower mantle contains large convection rolls arranged at characteristic intervals of 15°. Their geometrical organization may influence smaller convection systems in the overlying mantle.

The mathematical approximation of the convection-roll boundaries employs circular arcs with a radius of:

[
R=35.341^\circ
]

and centers along the latitudes:

[
y=\pm32^\circ,\qquad y=\pm96^\circ
]

An important question is whether the stresses generated by these systems could organize lithospheric deformation into large elliptical structures.

The orientation of the tectonic rings provides one possible constraint.

A shear-dominated horizontal stress field can have principal directions inclined at 45° to the geographical coordinate axes, provided the horizontal normal stress components are equal.

This offers a possible mechanical route towards explaining the orientation of the ellipses.

However, an additional explanation is required for their axis ratios, geographical positions and 6° southward displacement.

The geometry of the Earth must also be included. Longitude intervals become progressively shorter towards the poles, changing the relationship between geographical coordinates, distances and local flow directions.

The curvature of the Earth therefore enters the mathematical description of mantle flow and deformation, although it does not, by itself, establish the proposed circular convection trajectories or elliptical tectonic boundaries.

The physical origin of the complete system remains an important subject for further investigation.

8. The role of Antarctica in the global framework

Antarctica provides another important reference for the proposed global organization.

The Antarctic Plate is surrounded by extensive divergent plate boundaries connecting the southern Atlantic, Indian and Pacific oceanic ridge systems.

The geometry of these spreading systems can be compared with the major and minor axes of the proposed Antarctic elliptical structure.

The two principal axes define four directions. These can be examined in relation to New Zealand, the Mid-Atlantic Ridge, the East Pacific Rise and the Indian Ocean ridge system.

The development of oceanic lithosphere around Antarctica represents a major component of the horizontal plate-motion system.

Accordingly, the tectonic-ring model proposes that the observed distribution of active plate boundaries reflects the combined effects of horizontal plate movement and stresses associated with the underlying mantle.

A quantitative examination of this relationship would require reconstruction of plate motions through geological time, rather than comparison with present-day geography alone.

9. Geological implications and further investigation

The geometrical relationships identified in California and New Zealand are sufficiently specific to permit quantitative testing.

The locations of ellipse intersections and minor-axis endpoints can be calculated without reference to the positions of individual faults.

Their relationship to observed tectonic structures can then be evaluated independently.

A particularly important test concerns the orientations of the ellipse tangents, convection-roll boundaries and measured plate-motion vectors.

The San Andreas Fault System provides one opportunity for such an analysis. The Alpine Fault, Marlborough Fault System and Hikurangi plate boundary provide another.

Further investigation should examine whether the observed relationships are more systematic than would be expected from alternative elliptical geometries or other possible arrangements of the global tectonic system.

The mechanical interpretation also requires an explicit model of the stresses transmitted from mantle convection into the lithosphere.

This should include the effects of mantle viscosity, velocity gradients, vertical circulation, lithospheric resistance and existing plate-boundary forces.

The ultimate question is whether a mechanically self-consistent convection model can generate the proposed elliptical boundaries without prescribing their shapes in advance.

10. Conclusions

The proposed three-ring model provides a common geometrical framework for examining the relationship between major oceanic tectonic systems and mantle convection.

The most significant results presented here concern the minor axis of the inner Pacific tectonic ellipse.

Its northeastern endpoint lies near the Gulf of California, while its southwestern endpoint lies near New Zealand.

At both ends, the inner Pacific ellipse intersects the outer ellipse of an adjacent tectonic ring at a distance of approximately 54–57 km from the corresponding minor-axis endpoint.

In California, the relevant intersection is between the Pacific and Atlantic rings. In New Zealand, it is between the Pacific and Indian Ocean rings.

These relationships follow from the proposed mathematical geometry and provide two geographically distinct locations for testing its geological significance.

The correspondence between the elliptical boundaries, plate-boundary orientations and regional tectonic structures is particularly interesting in the context of the San Andreas Fault System and the complex Pacific–Australian boundary in New Zealand.

The equator, the 6° southward displacement of the ellipse centers, the 15° spacing of the proposed lower-mantle convection rolls and the 45° orientation of the ellipse axes provide additional geometrical constraints.

Together, these properties form a coherent mathematical description of the proposed global tectonic-ring system.

Their physical significance, however, depends on establishing a mechanical relationship between mantle convection, lithospheric stress and plate motion.

The present analysis should therefore be regarded as a geometrical hypothesis supported by specific, testable correspondences, rather than a completed dynamical explanation of the Earth’s plate-boundary system.

The central objective of further research is to determine whether the large-scale organization represented by the three tectonic rings can emerge from the interaction between mantle convection and the moving lithospheric plates.