The Triangular Analytical Module
Figure 1.
The Triangular Analytical Module, ABC, illustrates the proportional and angular relationships among the largest annual or monthly earthquakes between consecutive regional mainshocks at G and C, where GB represents aftershock sequences and the general decline in magnitudes following the first mainshock at G and AC represents foreshock sequences and the build-up in magnitudes preceding the next mainshock at C which has also arisen suddenly and perpendicularly from B. The mid-term at D is frequently characterized by either large peaks or troughs during this change-over period between declining aftershocks and rising foreshocks. DB represents the usual decline in the size of events just prior to a mainshock, whilst EC and DC often contain significant foreshocks to the mainshock at C.
The emphasis here is on the geometry of the build-up phase AC and surrounding events leading up to a forthcoming mainshock at apex C. Figure 1 shows the 30-60-90-degree Triangular Analytical Module made up of the internal isosceles triangle ABD with angles of 30-30-120 degrees and the equilateral triangle DBC with angles of 60 degrees. The sides of these triangles designated as ‘a’ are identical in length.
Triangulations at B, C, and X are powerful simultaneous geometric statements about the magnitude and timing of a mainshock. The Triangular Analytical Module is a conceptual model expressed geometrically. Theoretically, and in practice, it has been found to apply to all numerical analyses of earthquakes: whether monthly or yearly, and whether applying to the size or the timing of events. But it does not stand alone.
The Triangular Analytical Module operates in conjunction with other modules such as results from the author’s yearly and monthly Timing Analytical Modules, ensuring cross-validation and consistency. These interdependent modelling approaches, together with the Mainshock Analytical Module, are employed to facilitate practical forecasting outcomes in all examples featured on this website, which aims to assess and advance prospective forecasting methodologies.
Importantly, if the Triangular Module fits a data set it does not definitively confirm that a mainshock will occur at its apex C, which indicates both a specific magnitude and an anticipated time of occurrence. Conversely, if the model fails to fit the dataset, the forecast is incorrect. This conclusion applies to retrospectively tested mainshocks by the author of this website who is now developing and testing prospective application
The following points refer to
figures on the updated Japan page
Key features of the new Triangular Analytical Module are given below and show the potential ability to forecast the timing and magnitude of a forthcoming mainshock from two simultaneous triangulations in one geometric module, and with potential confirmation from annual and monthly modules.
Mainshock’s Date & Magnitude given simultaneously by two triangulations:
The Date of a Mainshock is given by triangulation at B on the baseline.
a) The date B for the next mainshock is given by where DB intersects base AB.
b) The line FXB (theoretical or not) must intersect the baseline at B.
c) At the intersection of the perpendicular CB which forms the 60-deg angle ACB subtended from C.
The Magnitude of a Mainshock is given by triangulation at C on the apex.
a) On a 30-deg rise from baseline AB to C and often including a major foreshock.
b) At the intersection of AC & EC, the latter being either theoretical or actual and arising at an angle of 60 degrees from AB to join C. Major foreshocks may also occur on EC.
c) At the intersection with the perpendicular BC from the appropriate date B.
Axes are adjusted so that AC & DB, by triangular theory, form 30-deg angles with AB and thus 120 degrees at D; then, the intersection of DB and AB gives the date (day number) of the mainshock at C which subtends the 60-deg angle ACB. When the axes are being adjusted, DX is important as it should intersect where EC and FB cross over, whether these lines are theoretical or not.
The 60-deg angle ACB is divided by EC which crosses the important line DB at an angle of 90 degrees (for example, as shown in Figures 1-2, on the Northern Japan forecasts page), EC arises from baseline AB where it is intersected at 90 degrees by GE. Horizontal lines HF and GX highlight key intersections through which theoretical lines can be drawn if data are lacking.
Page updated: 21 June, 2026
Acknowledgement
The author, Reg. Roberts, wishes to acknowledge the profound contribution of his son Wayne Roberts’ book, ‘Principles of Nature, towards a new visual language’, Canberra, 2003, ISBN 0 9750903 5 6, which drew attention to the highly resonant angles of 30, 60, 90, and 120 degrees. These angles are recognised as significant across various disciplines in the natural and physical world.