The ‘牟合方蓋’ (‘mouhefanggai’ – double box-lid) model was first thought of by Liu Hui, the great ancient Chinese mathematician in China, though, nowadays in the western society this solid is better known as ‘Steinmetz Solid’ or a ‘bicylinder’ (since the solid is the intersection of two cylinders), for the reason that the Prussian-American mathematician Charles.P.Steinmetz (1865-1923) used to study them, but Liu came up with the model of the solid over 1500 years earlier than Steinmetz. The solid was ‘created’ by Liu mainly to solve the formula of the volume of sphere (although one of the greatest Greek scientist and philosopher Archimedes had successfully concluded the formula around 500 years before Liu was born), also to remain a response to the incorrect formula that was indicated in the ‘Nine Chapters‘ below is a gif that clarifies how we get the ‘bicylinder’ model:
We could see from the image above how this perfect-looking solid could be got by intersecting two congruent cylinders. Liu came up with this model hoping so solve for the formula of sphere. The formula of volume of sphere indicated in ‘Nine Chapters‘ was that V = 9d^3/16 (with d being the diameter of the sphere), and of course Liu doubted the formula while correctly pointed out that this formula tended to underestimate the volume of a sphere. Coincidentally, the incorrect formula later turned out to be the formula of volume of the ‘bicylinder’ model, which Liu failed to figure out. Despite the fact, Liu stated that the ratio of the ‘bicylinder’ and its inscribed sphere is 4:pi, which was proved to be correct and is significant and monumental about the Chinese way of exploring the formula of the volume of sphere. Liu felt pitiful that he was unable to solve the equation and wrote that ‘let us leave this problem to whom that is able to tackle it’. The conclusion that Liu has drawn, however, can be proved in a very unexpected way, since the bicylinder model could be also thought of as in this following way:
First, think the sphere as composed (or rather, piled up) by a bunch of circles whose center points (origins) is in the same straight line (that forms one of the diameters of the sphere), but with different radiuses. (As figure 1 above)
Then replace every circle by their own circumscribed square (with exactly the same method of piling) to get a figure 2, which coincides to be a bicylinder. A more direct and viewable figure is shown as in figure 3 how this convention turns a sphere into a bicylinder:
The reason why the intersecting surfaces of two orthogonally intersected cylinders are all squares is because that the two cylinders has exactly the same radius and height, and the two parameters are also equal. Thus, the cross section tetragon happens to have equal sides, which could be seen in figure 4 above.
Therefore, think every circle/square that composed the corresponding sphere/bicylinder as a ‘layer’, and abstractly the bicylinder and its inscribed sphere has exactly the same number of ‘layers’. Now that we know for every single of their ‘layer’ pair, the ratio of the area of a square and its inscribed circle is 4:pi. Hence, holistically the volume of the bicylinder is 4/pi times of the volume of the sphere. That is why we would know the formula of volume of sphere once we discovered the formula of volume of bicylinder.
It took long even for me, who lives in this highly-developed society to illustrate the fact, thus Liu Hui was really intelligent to have discovered this. Several hundred years later after Liu Hui found this out, Zu Chongzhi（祖沖之）and his son Zu Geng（祖暅）figured a way out to solve for the volume of a bicylinder by using the Cavalieri’s Principle (though they used this method around 1200 years before the Italian mathematician Cavalieri was born, and in Chinese this principle is called ‘Zu Geng’s Principle’). First, they considered one eighth of a bicylinder in its circumscribed cube and another upside-down pyramid with its base surface being a square with side r (radius of the ball inscribed in the bicylinder) and height r as well. (Please refer to the figure below)
In ancient China, the solid on the right is called a Yangma（’陽馬’）. It has its own name because of its special attributes, that is: We could assembly three Yangmas in a way to make a full cube as shown below in figure 5:
Then back to figure 4, the two solids (1/8 of the bicylinder and the Yangma) have the same height r. If we consider the transverse section of the Yangma and the SPACE BETWEEN THE CUBE AND THE 1/8 BICYLINDER at same height h, we know intuitively that the transverse of the Yangma is a square, while the transverse of the space is the difference of two squares.
Numerically, for the left area
S = r^2 – x^2 where x^2 = r^2 – h^2, so S = h^2
For the right area we instantly know that the area of the cross section is h^2 (since the base of the Yangma is r^2 at height r)
Thus the volume of the space and the volume of the Yangma are equal based on Cavalieri’s Principle. Also, we know that the volume of the Yangma is 1/3 of its corresponding cube. Thus, the area of the 1/8 bicylinder is 2/3 of the cube. So for a full bicylinder we have:
1/8V (bi) = (2/3)r^3 thus V (bicylinder) = (16/3)r^3
Finally we figured out a way to solve for the volume of sphere which is
V (sphere) = pi/4 * V (bicylinder) = pi/4 * (16/3)r^3 = (4pi/3)r^3
Though Archimedes got this formula way earlier than Liu Hui, Zu Chongzhi and Zu Geng, the way of thinking of the three great ancient Chinese mathematicians shall be referred and studied by us in order to better explore the unknown field of mathematics.