To understand the design of the lower shafts it is necessary to look back at the principal ellipse which was formed from the base length of the pyramid. This construction was done using Ramanujan's ellipse circumference formula in reverse, to determine the ellipse's axes from its circumference. If you re-arrange Ramanujan's formula and substitute a=fb as was shown in the algebraic solution for the upper shaft length, you obtain the following formula. Click here to show the formula.
Even though Ramanujan clearly indicates that there is a minus sign in the formula, it is an inescapable mathematical fact that when this formula is used it must produce two results, because there is a square root sign in it. The positive square root gives the correct result, and the negative square root gives the incorrect result.
To get an idea of the relationship between the two results by doing some quick mental arithmetic, set the value of 'f ' to 1 (which it very nearly is) and you can work out that the bracketed part of the denominator of the fraction will give a result of either 2 or 10. In other words when the primary ellipse was created, a secondary ellipse which is approximately five times smaller must have been created at the same time from the negative square root.
It is this ellipse that is used to create the lower shafts of the building, and is highlighted on the drawing with its center at the base center of the pyramid and its half circumference line also displayed.
This ellipse is another Earth cross section, with a scale approximately 5 times that of the primary ellipse. The value at this stage of the work is 1 : 434210.