But there is something more complete : see page 32 of the user manual for the TK-Yupana tool (Tk-Yupana r07, by Kunturweb) : http://kunturweb.altervista.org/tk-yupana/doc/tk-yupana-EN.pdf.
[ this version is no longer available as of July 21 2014, but the Spanish and Italian versions are still there : http://kunturweb.altervista.org/tk-yupana/doc/tk-yupana-ES.pdf and http://kunturweb.altervista.org/tk-yupana/doc/tk-yupana-IT.pdf ; and the english version is also on http://es.sourceforge.jp/projects/sfnet_tk-yupana/downloads/doc/tk-yupana-EN.pdf/ ]
Use :
- the board is used in the vertical position
- each row represents a power of ten, from 0 (bottom) to n-1 (n=number of rows)
- each circle in each row of the board represents a different value (from 1 to 11) from right to left
- seeds are placed in the circles only [ 0 or 1 seed in each circle, probably ]
Value of each circle in a row is as follows :
Comments
At first glance the system doesn't seem of the most practical kind :
- requires a very precise placing of seeds,
- multiplies the opportunities of mistakes : from mispositioning, from movements of the board, and because there is no intuitive hints to the value of each circle in a cell, it is difficult [at least for slightly dyslexic people] to remember the value of each circle, ...
- actually, this last point is not so important - see below some easily memorized configurations - when you memorize the configurations of simple numbers, you don't really need to memorize the value of the cells (sort of).
- collaborative calculus is less easy (several people seated around a board, each with a differing view of it, will have more difficulty - than with various other systems - to know the value of the seeds),
- requires a rather carefully crafted board : using an improvised earthy or sandy surface as a board is almost impossible,
- performing a multiplication with one board only and without knowing the tables doesn't seem allowed easily.
[ update 2014-03-10 : I tried with a big multiplication : rather interesting http://youtu.be/m3n6QhZslLc
]
Advantages :
- like some other systems, it allows most numbers to be reprensented as a variety of seeds configurations, which is interesting for the agility in calculations,
- allows the use of a smaller number of seeds for numbers greater that 5,
- each row can represent a number from 0 to 66 : usefull in calculations, before the simpification that will keep in each row only a number from 0 to 9 [ for easy recording in a khipu or simply for being able to enunciate the number verbally ]
- some configurations of seed are really easily visually memorized and represents interesting numbers : 45, 15, 5, 20, 10, 50, 60, 30, 16. For instance, take a look at those fascinating configurations :
10
10
10
15
15
15
14
16
cool !
20
20
a bit confusing with the similar 15
20
cool !
20
mirrored-image of the "horizontal" 14
25
30
cool ! compare with 20
30
30
satisfactory
30
compare with the mirrored-image for 25
40
45
cool !
45
50
50
50
60
cool !
60
- and it really explains one aspect of Guaman Poma's drawing where seeds seem to be placed carefully in some of the circles in the cells and not anywhere in the cells.
from Guaman Poma Website (see below)
- But on the other hand, how would we easily translate on the khipu, an accumulation of seeds that needs to be placed on other rows, in order to easily give a number expressed in base 10. OK, the drawing may represent an intermediary stage in a calculation : before the expression is simplified.
For those who want a "true" (?) source for the design of Guaman Poma's Yupana, see page 360 [362], "Drawing 143. Chief accountant and treasurer, Tawantin Suyu khipuq kuraka, authority in charge of the knotted strings, or khipu, of the kingdom" http://www.kb.dk/permalink/2006/poma/362/en/text/?open=id2974973
on the Guaman Poma Website (National Library of Denmark and Copenhagen University Library)
Larger image : http://www.kb.dk/permalink/2006/poma/362/en/text/?open=id2974973&imagesize=XL
I corrected the labels of some of the images (images of numbers that can be represented on one row) :
ReplyDelete- three dots on the lower line = 14
to be compared to three dots on the center line = 16
and three dots on the upper line = 25
- two couples of dots at the upper left (like this : \ \ ) = 30
to be compared to the mirrored image (like this : / / ) = 25
Caution : if you download the images, pay attention to their name : the number in the name is generally the number represented on the image - according to Chirino's theory - but I made 2 or 3 mistakes.
I tried my usual big multiplication (438 * 367) with Chirinos' system : it works.
ReplyDeleteActually, I don't know how Chirinos describes its way to perform multiplications. But it can be done, at least by using the same technique than with (my version of) the "yupana dinamica".
That is :
- use different colors or shapes of beads / seeds for the multiplicand, the multiplicator and the result
- put all this on the same yupana
- split the "big" numbers (9, 8, 7, 6, for which, supposedly, as beginner quipucamayoc, you don't yet know the multiplication tables) in a sum of smaller numbers (5+3+1, 5+3, 5+2, ...)
- then you have nothing more to perform than multiplications from 1*1 to 5*5
The problem with this way of doing is that you must adapt the system :
- you'll have to put, rather often, several seeds in the same circle ; and not only zero or one seed per circle
- and if your seeds are big, you'll have to put some of them outside the circles, which brings more risks of error
I tried a simpler and easier way for computing a big multiplication mindlessly (without any knowledge of the multiplication tables, and following a process like would do a robot) with the use of Chirinos' Yupana.
ReplyDeleteSee http://youtu.be/m3n6QhZslLc
Maybe it is not what Chirinos prescribes for big multiplications... But as I couldn't find documentation, I was obliged to guess... Which was almost as fun as trying to interpret / reinvent the real thing from the very few information that survived the past 500 years.