The first sexaxial figure of the Earth

ABSTRACT In this study, a sexaxial figure of the Earth was created using randomly distributed 6400 control points on the geoid, which was generated by the EGM2008 gravity model. The shape of the sexaxial ellipsoid is defined by its six orthogonal axes. To construct the new geometric figure, eight triaxial ellipsoids were fitted to the Cartesian coordinates of control points in each octant of the geoid using the least squares method. Its geometric origin is constrained to coincide with the centre of mass of the geoid, and the ellipsoid has no net rotation with respect to the WGS84 coordinate system. This study demonstrates that the new geometric figure provides a customised fit to the control points in each octant.


Introduction
Twenty-six centuries ago, Anaximander, who lived in Miletus, an Ionian city on the coast of what is now Turkey, postulated that the Earth was a stone floating in space without falling around twenty-six centuries ago (Rovelli, 2009).His idea was followed by Pythagoras, who suggested that the Earth was a sphere around 600 BC.Later, in 240 BC, Eratosthenes measured its circumference.Over the past four centuries, many notable scientists have studied the theoretical shape of the Earth, including Newton in 1689, who proposed its shape as an oblate spheroid.Other scientists who contributed to this field of study include Huygens in 1690, Cassini in 1701, Maupertuis in 1732, Clairaut in 1733, Euler in 1740, and Maclaurin in 1742.Since 1753, the Earth's shape has been determined and studied as an oblate ellipsoid by various scientists, including D'Alembert in 1756, Lagrange in 1759, Laplace in 1772, Legendre in 1784, Monge in 1787, Poisson in 1811, Gauss in 1813, Cauchy in 1815, Jacobi in 1834, Dirichlet in 1857, Dedekind in 1860, Riemann in 1860, Poincaré in 1885, Darwin (son of Charles Darwin) in 1906, Jeans in 1917, Cartan in 1924, Chandrasekhar in 1960, Heiskanen in 1962, and others listed in Ghys (2006).For a detailed account of these studies with references, please refer to Boccaletti (2018).
In recent times, estimating biaxial and triaxial ellipsoidal models of the Earth from the geoid has become routine thanks to the widespread use of the spherical harmonic representation of the Earth's gravity field.Many investigations have been carried out on this subject between 1850 and 1960, with numerous researchers involved, such as Schubert in 1859, Clarke in 1860,, 1866, 1878, Hill in 1884, Helmert in1915, Berroth in1916, Heiskanen in 1924, 1928, 1929, and 1938, Krasovskiy in 1942, Lambert in1945, Heiskanen in 1945, Subbotin in 1949, Uotila in 1957, Jongolovich in1957, Jeffreys in1959, Kaula in 1959, Izsak in 1961, and Kozai in1961.For a detailled account of the aformentioned publications with references please refer to Heiskanen (1962).
This study proposes the possibility of a sexaxial ellipsoidal representation of the Earth's geometric shape, which is motivated by the long list of previous investigations.The new representation would have six axes shared by eight triaxial ellipsoids representing each of the eight octants.The term 'Sexaxial Ellipsoid' is coined for the first time in this study.The name originates from Latin and conforms to the naming convention of biaxial and triaxial ellipsoids.A Greek counterpart, 'Hexaaxial Ellipsoid', could also be used.The proposed sexaxial model provides a tailor-made representation of the Earth's geometric figure octantby octant better fitting to the underlying geoid.
The following sections derive a set of randomly distributed Cartesian coordinates of control points on the geoid from the World Geodetic System 1984 (WGS84).A mathematical model for the triaxial ellipsoid and its solution is presented to be used in modelling sexaxial ellipsoidal representation of the Earth.The triaxial ellipsoid's estimated shape parameters also serve as a baseline for comparison.The parameters of the sexaxial ellipsoid are then estimated.Least Squares (LS) solutions use formulations based on the method of condition equations with unknown parameters.Finally, the conclusion section discusses the implications of the solution statistics for the new model in comparison to the biaxiality and triaxiality of the Earth's geometric representation.

Randomly distributed data on the Geoid
WGS is a consistent set of constants and Earth Gravitational Model (EGM) parameters that describe the size, shape, and gravity of the Earth.It is realised by an Earth-centred, Earth-fixed 3-dimensional terrestrial reference frame and geodetic datum.The latest realisations are called WGS84 and are based on EGM2008 (Pavlis et al., 2012) To generate uniformly distributed 6400 points on a unit sphere based on standard normal variates, the algorithm proposed by Mervin (1959) was used.These points were then converted to geodetic coordinates in the WGS84 system.Each randomly generated control point represents a grid size of approximately 280 square km.The online calculator by UNAVCO (2022) was used to generate geoid undulations at these locations.The WGS84 ellipsoid geodetic coordinates, along with their geoid undulations, were used to calculate the Cartesian coordinates of each control point on the geoid.Figure 1 shows the 6400 control points on the Plate Carrée projection, while Figure 2 displays the histogram of the geoid undulations whose magnitudes are shown on Figure 3.Note that the sample size of 6400 is markedly less than the 165,000 control points used in recent studies (Panou et al., 2020) but sufficient to estimate statistically significant model parameters.Fewer control points also reduces the effect of the omission of unknown positive correlations among the nearby controls in overestimating the uncertainties of the model parameters.
The following section discusses the mathematical model for a triaxial ellipsoidal figure of the Earth approximating the EGM2008 geoid and the estimation of its shape parameters.The representation and its estimated parameters will serve as a baseline for assessing the sexaxial model and its solution.

Modeling and estimation of the best fitting triaxial ellipsoid parameters
A geometric representation of the Earth's shape by a triaxial ellipsoid, shown in Figure 3, is given by the following expression, The principal axis a of the triaxial ellipsoid is the equatorial axis along the x-axis of the Mean Earth/ Polar axis reference system, b is the other equatorial  The method of condition equations with unknown parameters, as outlined in Appendix A, was utilised to estimate the parameters of the mathematical model given by Equation (1).The weight matrix P is associated with a total of n ¼ 6400 x 3 ¼ 19200 observations, i.e. the x, y, z coordinates of the control points on the geoid.The errors of the coordinates are assumed to be uniformly and identically distributed with a prior variance of unit weight equal to 1, i.Note that because the ellipsoidal latitudes and longitudes are not observed quantities, the errors are attributed to the departures of the geoid undulations from the underlying triaxial ellipsoid.
The estimated model parameters, along with their respective uncertainties, are presented in Table 3, alongside the corresponding parameters of one of the triaxial ellipsoid by Panou et al. (2020), the biaxial ellipsoid of WGS84, and the spherical radius of EGM2008 for comparative purposes.
The estimated biaxial parameters of the equatorial and polar radii of EGM2008 and WGS84 were unchanged using the triaxial ellipsoidal model in this solution.However, these parameters differed significantly from the triaxial estimates reported by Panou et al. (2020).It is worth mentioning that the G-T6 solution model of Panau et al. ( 2020) incorporates EGM2008 data and introduces an additional parameter to account for the net rotation angle of the triaxial ellipsoid's z-axis relative to the mean Earth/Polar axis of the WGS84 coordinate system.The estimated rotation angle of 14.9 degrees is significant in magnitude and plays the key role in explaining the differences between the ellipsoid's shape parameters compared to the other models.
The standard errors (SE) of the estimated shape parameters listed in Table 3 are  The triaxial model discussed in this section is to serve as a baseline for the sexaxial model elucidated in the following section.

Modeling and estimation of the best-fitting sexaxial ellipsoid parameters
This study presents the concept of a sexaxial ellipsoid as a geometric representation of the Earth (refer to Figure 4).The sexaxial figure comprises six axes that divide the Earth into eight octants whose boundaries are highlighted in red in Figure 1.The boundaries of each octant, as well as the number of control points allocated within it, are listed in Table 1.A triaxial ellipsoid, which is discussed in the previous section, is assigned to each octant, and the neighbouring octants share common axes as shown in Figure 5. Figure 6 illustrates the distribution of approximately 800 geoid undulations in each octant, which were generated from the previously generated 6400 control points.The plots reveal that the distribution of geoid undulations in each octant varies in magnitude and frequency, as indicated by their statistics presented in Table 2.
The composite mathematical/geometric model that unifies all eight triaxial ellipsoids of each octant in a single representation is expressed using the following set of condition equations with unknown parameters: The model is nonlinear and involves six unknown shape parameters: a, b, c, a', b', and c', as shown in Figure 4.These parameters define the principal axes of each triaxial ellipsoid, which are aligned with those of the neighbouring octants.The values of these parameters are determined from the Cartesian coordinates of the control points on the geoid allocated to each octant, which are the assumed observables subject to error.
The solution was performed using the LS method, as described in Appendix A. The corresponding condition equations for each octant and their partitioned normal equations are provided in Appendix B. The solution was obtained iteratively, starting with the estimated triaxial ellipsoid parameters as initial approximate values.It should be noted that this approach is necessary to ensure convergence to an optimal minimum, as the composite model may converge to different local minima depending on the initial approximate values.This is one of the reasons why some of the earlier triaxial studies entertained additional statistically significant parameters of different magnitudes, which also included additional parameters such as translation of geocenter and rigid body motion of the triaxial ellipsoid.
The estimates and their corresponding uncertainties are presented in Table 3, alongside the parameters of the triaxial ellipsoid for comparison.The normal equations of the estimated model parameters are wellconditioned, as evidenced by the correlations not exceeding 0.4 among the shape parameters.
The estimates for the semi-axes, a, b, and c, for both the triaxial and sexaxial models are nearly identical (Table 3).In other words, both models do not distort the properties of the underlying biaxial properties of the gravitational representation.However, the differences between a vs. a', b vs. b', and c vs. c' are statistically significant at α ¼ 0:05 level.
The impact of the sexaxial figure can further be analysed through Octant-by-Octant residual differences of the triaxial and sexaxial solutions, as shown in Figure 6.The residuals indicate that the sexaxial figure has a significant impact on the geoid undulations at Octants II, VI, VII, and VIII as shown in Figure 7.

Conclusion
It is easy to have ideas; it is difficult to pick out good ideas and find the arguments to show that they are 'better' than the current options.C. Rovelli 2008, The First Scientist Anaximander and his legacy.Today, the Earth already has the most sophisticated figure, it is called the geoid.Its irregular geometric shape is conceived through its undulation values, which can be produced at any desired density.Its underlying biaxial ellipsoid is well-suited and well-studied for mapping and surveying.Investigating its equatorial ellipticity is therefore motivated by the possibility of a deviation of the rotating Earth from an isostatic equilibrium due to the distribution of topographic masses and the cluster of positive or negative gravity anomalies, which are demonstrated indirectly in Figure 5 through the histograms of the geoid undulations in each octant.Its triaxiality or sexaxiality may also be caused by the relatively deep disturbing masses extending even at the boundary between the earth's mantle and core as speculated by Heiskanen as early as (Heiskanen, 1962).
Over 60 years ago Professor Heiskanen stated: With the gravimetric method it has not been possible so far to obtain as high an accuracy as we wanted.The reason lies not in the method, but in the lack of gravity material.This problem, in addition to several others, can be solved with very high accuracy as soon as the gaps in the gravity anomaly field are filled.
He then added: You have read of the Roman Senator Cato who lived in the second century B.C.He was an orator and gave addresses in the Forum Romanum.He was a member of the Roman peace commission after the second Punic War in Carthage and saw how rich and powerful Carthage was.Whatever his topic had been in his orations, his last words were always, 'Ceterum censeo Cartaginem esse delendam', which means, 'I, however, am of the opinion that Carthage must be destroyed'.I would like to end my paper similarly by saying, 'I am, however, of the opinion that the big gaps in the gravity anomaly field must be filled without delay ' (Heiskanen, 1962).
I would like to report here that we successfully destroyed Carthage after 60 years of progress in satellite geodesy.Since then, several studies demonstrated the presence of a statistically significant triaxial figure of the Earth, and even more, this study conceived and calculated the first sexaxial ellipsoid.Although we were able to refute your conjecture: I personally doubt the possibility of obtaining the triaxiality from satellite tracking (Heiskanen, 1962), the magnitudes of the statistically significant departures of their semi-axes in the equatorial plane are small, on the order of a hundred metre or less, as in the case of previous studies for its triaxiality since 1860 (Heiskanen, 1962).But unfortunately, for now, there are no demonstrated raisons d'être for the Earth's triaxiality as well as its sexaxiality investigated in this study, other than being just an interesting geodetic exercise of the day.New studies are therefore needed to determine their impact, if there is any, for understanding the kinematics and dynamics of the Earth, such as, density variations in its interior, rotational stability of its orientation and rotation, in association with temporal changes in mean sea level, etc.Having said that, a sexaxial ellipsoid is better suited to represent and study amorphic objects such as astronomical bodies, geologic and geophysical formations, civil engineering, tracking malignant growths in living bodies, 3D printing, etc.

Note
1.The method of conditions equation with unknown parameters was formulated by Friedrich Robert Helmert (1843-1917), a geodesist.The details of the following summary narrative can be found in Uotila (1988). ) where k = I; II; . . .; VIII.Hence, and Similarly, it can be shown that,

Figure 1 .
Figure1.Shown on the Plate Carrée projection are the 6400 randomly generated control points on the geoid.Each control point represents the location of an average geoid undulation for a grid size of ~280 square km.
axis parallel to the y-axis, and z is the polar axis coincides with the z-axis.The position of the geometric centre of the triaxial figure coincides with the Center of Mass, CM, of the Earth, which is the origin of the Mean Earth/Polar axis reference system: The unknown shape parameters, namely, a, b, and c can be estimated using the known Cartesian coordinates of the control points generated in the previous section.
larger than the ones reported byPanou et al. (2020) using EGM2008 (D 2.1 model).The differences are due to the large number of ~ 165,000 control points used in their study compared to the 6400 control points deployed by this study.Nonetheless, lower number of control points prevent overestimating the uncertainties due to the omission of the high positive correlation caused by the use of dense data.The a posteriori variance of unit weight of the solution is σ0 ¼ 30:7.It measures the goodness of fit of the model summarising the discrepancy between observed values and the values expected under the model in question but not reported byPanou et al. (2020), which is replaced by the statistics of the adjusted geoid undulations.

Figure 4 .
Figure 4. Triaxial representation of the Earth's geometric shape.O is the geometric center of the ellipsoid and coincides with the CM of the Earth.

Figure 6 .
Figure 6.Histograms of the geoid undulations in each octant.Vertical and horizontal scales are the same for all plots to enable direct visual comparisons.

Figure 7 .
Figure 7. Histograms of the residuals in each octant from the triaxial model solution (left) and the sexaxial model solution.Horizontal scales have the same range for visual assessment.

Table 1 .
The boundaries of the eight octants of the sexaxial ellipsoid.

Table 2 .
Descriptive statistics of geoid undulations for each octant in metres.Standard error is denoted by SE.

Table 3 .
The estimated ellipsoidal parameters and their uncertainties are in metres.N/A:Not Applicable.N/R:Not Reported.The number of control points and the a posteriori variance of the unit weight are denoted by n and σ0 respectively.Note that the solution model, G-T6 by Panau et al. (2020) using EGM2008 data includes another parameter, a net rotation of the triaxial ellipsoid about its rotation axis with respect to WGS84 coordinate system.