A Comparison of the Organization and Use of Chinese and Mesoamerican Abaci
by David B. Kelley
The remains of a 3/4 vigesimal^{1} (base twenty number system) abacus were purportedly found in Mexico. Whether or not this is true, the fact remains that the Mesoamerican baranddot number signs appear to fit very systematically into such an arrangement, as demonstrated below. Additionally, this arrangement, involving seven rows and thirteen columns, also matches the most common arrangement for the Chinese decimal/hexadecimal (base sixteen) abacus is something that cannot be ignored. The evident similarities in the design of the Chinese solidandbrokenbar and the Mesoamerican baranddot number symbols is suggestive of some sort of relationship between the two systems. What that relationship may be is not clear, but it is hoped that further research will reveal its nature.
Terminology
The Comparisons
In the following set of comparisons, the same number is represented in the left image using the Chinese solidandbrokenbar system and in the right image using the Mesoamerican baranddot system.
Chinese Representation  Mesoamerican Representation 
3/4 Abacus: simple column values with Chinese solidandbrokenbar signs and number values below. This is NOT a functional Duodecimal (base 12) Abacus, and although each lower deck bead has a value of "1", each Broken Bar sign below, actually has a numeric value of "2". 
3/4 Vigesimal Abacus: simple column values with Mesoamerican baranddot signs and number values below. This is a fully functional Vigesimal Abacus, but owing to a 13column limitation, only 11 baranddot signs and values, and two uses of the "zero" sign are shown. 
3/2 Duodecimal (base 12) Abacus: simple column values with Chinese solidandbrokenbar signs and number values below. This is a fully functional Duodecimal Abacus, and to make it functional, the value of the Broken Bar sign has been changed to "1", and the signs and values rearranged. 
3/2 Abacus: simple column values with Mesoamerican baranddot signs and number values below. This is NOT a vigesimal abacus. 
Examples of PlaceValue Use
3/2 Duodecimal Abacus: placevalue column values with Chinese solidandbrokenbar signs and number values below showing the decimal number 1,999.

3/4 Vigesimal Abacus: Regular placevalue column values with Mesoamerican baranddot signs and number values below showing the decimal number 1,999.


3/2 Duodecimal Abacus: placevalue column values with Chinese solidandbrokenbar signs and number values below.
The solidandbrokenbar signs are markedly similar to the Mesoamerican baranddot signs, especially those associated with the Calendric use of the Mesoamerican signs. In the case of the example above, we see a total of five solidbar signs and four brokenbar signs each consisting of two subparts, are very similar to the five Mesoamerican barsigns and eight dotsigns. 
3/2 Vigesimal Abacus: Calendric placevalue column values with Mesoamerican baranddot signs and number values below.
The abacus shown above has been modified to reflect the Calendric use of the Mesoamerican vigesimal number system. This means that the third column has a limit of three upperdeck beads (with each bead = "5") and two lowerdeck beads ((with each bead = "1"). Accordingly, all columns to the left of the second column have different values from those derived from the Regular vigesimal abacus presented earlier. 
Final Comments
The solidandbrokenbar (IChing) number signs represent one of the several variant Chinese numbersymbol systems. The Chinese IChing number symbols are based on a combination of the two Yao () 'Elementary Forms'. And, the two Yao form the basis of the PaKua () 'Eight Diagrams' found in the IChing () 'Book of Divination'. On the two Yao, the IChing specifically states that, "The number 3 was assigned to heaven, 2 to earth, and from these came the (other) numbers."; the original line in Chinese is, ""^{2}. This has been interpreted to mean that the SolidBar had a numeric value of "3" and the BrokenBar a value of "2". However, as the above analyses demonstrate, with those values the two Yao cannot generate a complete series of numbers, because there is no sign with a value of "1".
And so, the pertinent line from the IChing was perhaps misinterpreted; the line appears to describe an actual 3/2 duodecimal abacus, and was referring to its two decks: the upper one ("Heaven") with three beads per column, and the lower one ("Earth") with two beads per column. Finally, in order for the duodecimal abacus to be fully functional, each Upper Deck bead and associated Solid Bar sign would have necessitated a value of "3", and each Lower Deck bead and associated Broken Bar sign, a value of "1".