**Figure 2**. A cube is dissected, using Liu Hui's approach, into three solids that are \({\frac{1}{2}}, {\frac{1}{3}},\) and \({\frac{1}{6}}\) of its volume. To see the dissection from different perspectives, click then drag the image below.

Liu Hui (刘徽, ca. 220 AD – ca. 280 AD) was a third-century Chinese mathematician, well known as the “Euclid of China” for his dominant contributions to Chinese mathematics (Dauben, 2007). Although little is known today about his personal life, Liu Hui is widely recognized for his original commentaries on the ancient Chinese mathematical text, *Jiu Zhang Suan Shu* or the *Nine Chapters on the Mathematical Art*, which laid the foundation for the development of Chinese mathematics (Li & Du, 1987). Over time, Liu Hui's commentaries were further edited and annotated by successive mathematicians such as Li Chunfeng (李淳风, 602 – 672 AD) and Li Huang (李潢, ca.1746 – 1812). Thus, there are in existence various editions of this classic of Chinese mathematics (e.g, Liu & Li, n.d.; Li, 1820). In his commentaries, Liu Hui elaborated on 246 problems in nine chapters, covering a wide variety of mathematical ideas from arithmetic to matrix algebra, along with geometry in various real-world applications characteristic of his era, such as farming and earthworks.

In Chapter Five of *Nine Chapters on the Mathematical Art,* entitled *Shang Gong* (商功, construction consultations), Liu Hui commented on a series of 28 problems concerning the volumes of solids in the context of civil engineering and grain measurement. Here, Liu Hui may not have been as much interested in designing a puzzle as finding algorithms to calculate the volumes of various polyhedral solids. Problem 15 of Chapter Five is of special significance in our present case. The original problem, answer, and the corresponding rule in the *Nine Chapters on the Mathematical Art *are as follows (trans. Shen, Crossley & Lun, 1999, p. 269):

Now given a *yangma*, with a breadth of 5 *chi*, a length of 7 *chi* and an altitude of 8 *chi*. Tell: what is the volume? Answer: \(93{\frac{1}{3}}\) [cubic] *chi*. The Rule [for a *yangma*]: Multiply the breadth by the length, then multiply by the altitude, divide by 3.

The term *yangma* (阳马), which makes little sense nowadays in modern Chinese, is an ancient Chinese term for a special kind of rectangular pyramid. The term *chi* is a Chinese linear unit, still in limited use today with a modern definition. In ancient Chinese, *chi* was also used as a square unit and a cubic unit (as in the problem above), subject to the specific context. In his commentary on the problem above, Liu Hui detailed his method of cube dissection for finding the volume of a *yangma*. He started with a cube, which has three equal dimensions, and dissected it along two face diagonals, as described below (trans. Shen, Crossley, & Lun, 1999, p. 269):

Take a cube 1 *chi* in each of breadth, length and altitude. The product is 1 [cubic] *chi*. Cutting the cube on a diagonal gives two right triangular prisms (*qiandu*). Dissecting a right triangular prism (*qiandu*) on a diagonal gives a *yangma* and a *bie'nao*. The *yangma *occupies 2 parts and the *bie'nao* 1 part. These are fixed rates. Two *bie'nao* constitute 1 *yangma*, and 3 *yangma* constitute 1 cube.

Liu Hui's approach is clear once we get hold of a cube and perform the dissecting operations (see **Figure 2**, above). His use of terms, however, calls for some explanation.

- A
*qiandu* (堑堵), meaning literally an *embankment beside a trench*, is a metaphoric term for a *right triangular prism* such as the green piece in **Figure 2**.
- A
*yangma* (阳马) is the corner of a rectangular pyramid such as the red piece in **Figure 2**; it probably has its origin in architecture. Interestingly, in its Japanese explanation, *yangma* is called a “sunshine-carrying horse,” which in fact conveys part of its literal meaning in Chinese (四角錐, n.d.).
- A
*bie'nao* (鳖臑), meaning literally a *turtle's foreleg bone*, is a metaphor for a *tetrahedron with a right triangular base* or, in short, a *triangular pyramid* such as the purple piece in **Figure 2**. However, in the context of the problem above, a *bie'nao* is taken to be *a tetrahedron with four right triangular faces*.

All three terms were essential in Liu Hui's efforts to find the volumes of polyhedral solids, although they do not make much sense in modern Chinese without further explanation. Therefore, they ought to be used with respect to Liu Hui's specific geometric dissections. The context will help readers understand the underlying mathematical processes and the resulting objects. About the terminology, Martzloff (1987/1997) suggested, "Whatever the exact etymologies, there is no doubt that here these terms have no more to do with architecture than terms like 'pyramid' used in a mathematical context" (p. 283). Indeed, Liu Hui may have just used these pieces as *qi* (棋, chess pieces) to make sense of other solids or to play games, hence the puzzle nature of Liu Hui's dissections (Martzloff, 1987/1999). As a further linguistic note, Chinese nouns do not have plural forms. Accordingly, in literal translations, their plural forms are the same as their singular forms, such as in the case of “3 *yangma* constitute 1 cube.”

It is noteworthy that Liu Hui's approach and his use of terms such as *qiandu*, *yangma*, and *bie'nao* are not restricted to cubes only but are applicable to any rectangular prism or rectangular parallelepiped. In fact, Liu Hui used *lifang* (立方, a rectangular prism) in the Chinese text. When he needed to specify a cube, he would give the *lifang* three equal dimensions. Therefore, it is important that we make sense of Liu Hui's approach in the context of arbitrary rectangular prisms and treat a cube as a special case (Wagner, 1979; Ying, 2011; Li, 2002). In English translations, an arbitrary rectangular prism (or rectangular parallelepiped) is sometimes called a (rectangular) box or cuboid. For a mathematical puzzle, a cube is more appropriate in light of its rich symmetries. When a cube is used, Liu Hui's approach leads to one type of *qiandu*, one type of *yangma*, and two types of *bie'nao,* depending on the choice of the second diagonal cut. The two types of *bie'nao* are mirror images of each other. It is visually evident that two congruent *qiandu* make a cube (see **Figure 3**) and three congruent *yangma* also make a cube (see **Figure 4**). In the case of a cube, therefore, it is fairly easy to see that a *yangma* is one third of the cube volume. Given six *bie’nao*, if three of them are mirror images of the rest, one can also make a cube (see **Figure 5**). Thus, a *bie’nao* is one sixth of the cube volume. These relationships can also be explored visually using dynamic models.

**Figure 3**. Two *qiandu* (triangular prisms) make a cube. To see the dissection from different perspectives, click then drag the image below.

**Figure 4**. Three *yangma* (triangular prisms) make a cube. To see the dissection from different perspectives, click then drag the image below.

**Figure 5**. Six *bie'nao* (triangular pyramids) make a cube after mirroring three of them. To see the dissection from different perspectives, click then drag the image below.

By contrast, an arbitrary rectangular prism is more complicated because it does not have all the symmetries of a cube. Consequently, Liu Hui's dissection leads to three types of *qiandu*, six types of *yangma*, and twelve types of *bie'nao* (with four right triangular faces), depending on the choice of both diagonals during the dissecting process. Three of the *yangma* are mirror images of the rest; six of the twelve *bie'nao *are mirror images of the rest (Shen, Crossley, & Lun, 1999, pp. 272-273). **Figure** **6** shows Liu Hui's dissection of an arbitrary rectangular prism, where the green piece is a *qiandu* (1/2 of the rectangular prism), the red piece is a *yangma* (1/3 of the rectangular prism), and the purple piece is a *bie'nao* (1/6 of the rectangular prism). **Figure** **7 **further shows three ways to dissect the same rectangular prism, each way yielding a distinct *qiandu*, *yangma*, and *bie'nao*. The three *yangma* in **Figure 7 **have distinct bases, and they can be assembled to form the original rectangular prism (**Figure 8**). Their mirror images also form the original rectangular prism in a different orientation. As can be seen, Liu Hui's approach seems to apply to all arbitrary rectangular prisms as well as to a cube.

**Figure 6.** Liu Hui's dissection of an arbitrary rectangular prism into a *quiandu* (green piece), a *yangma* (red piece), and a *bie'nao* (purple piece).

**Figure 7.** Three different dissections of an arbitrary rectangular prism that yield three distinct *quiandu,* *yangma,* and *bie'nao*. The blue pieces are *quiandu,* the red pieces *yangma,* and the green pieces *bie'nao*.

**Figure 8**. Three distinct *yangma* can be assembled into the original rectangular prism.

But, in the general case, does each *yangma* (rectangular pyramid) have the same volume? If so, what is that volume? Liu Hui did not answer these questions directly using the visual method in **Figure**** 8**, but instead set out to show that there is a 2-to-1 ratio between the volumes of a *yangma* and a *bie'nao*. Given that the *qiandu* has half the volume of the rectangular prism, this 2-to-1 ratio of the volume of the *yangma* to that of the *bie'nao* would then guarantee volumes of \({\frac{1}{2},} {\frac{1}{3},} \) and \({\frac{1}{6}}\) of the volume of the rectangular prism for, respectively, the *qiandu,* *yangma,* and *bie'nao.*