46 Lessons in Early Geometry, part 8/10 / provisional version in freestyle English / a corrected version will follow in March, April or May (hopefully) / Franz Gnaedinger / February 2003 / www.seshat.ch


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Lesson 37


Let us imagine a lesson held at Akhmim some 4,000 years ago. The teacher reads the following numbers from a piece of wood and comments on them:


1 '3

double this line

2 "3

double this line

4 1 '3

a new line

'64 '256 "3

double this line

'32 '128 '3

a new line

'16 '64 '256 "3

double this line

'8 '32 '128 '3

a new line

'4 '16 '64 '256 "3

double this line

'2 '8 '32 '128 '3

a new line

1 '4 '16 '64 "3

double this line

2 '2 '8 '32 '3


Now I ask you: are my multiplications correct?


Pupil A: I see a pattern, the last numbers are repeating.


Teacher: What do you mean?


Pupil A: '3 "3 '3 "3 '3 "3 '3 "3 '3 "3 '3


Teacher: So you assume that my calculations are correct?


Pupil A: You always tell us that we should look out for patterns.


Pupil B: However, there is a mistake.


Teacher: Yes?


Pupil B: When you double a fraction like '64 you can half the number and will obtain '32, but when you multiply "3 you can't simply half '3 '3 and obtain '3. No, the correct multiplication yields 1 '3, as explained in the first multiplication.


Teacher: Quite so. You pointed out a very common mistake among the aspiring scribes. I can only tell you: be very careful. Not every pattern is promising, some can be misleading. Looking out for patterns is very fine, but checking a pattern and checking all calculations is no less important for a successful scribe.



This lesson was held to young pupils. Advanced learners will be given a more demanding task.


Teacher: Transform my above numbers into a table of exact multiplications and fractions of 3. Begin with the last pair of lines:


'3 '3

'4 '12

'4 '16 '48

'4 '16 '64 '196

'4 '16 '64 '256 '768 and so on


"3 '2 '6

'2 '8 '24

'2 '8 '32 '96

'2 '8 '32 '128 '384 and so on



Now for another game. Consider the given lines:


'4 '16 '64 '256 "3


'2 '8 '32 '128 '3


Complete them in the above sense and you obtain 1:


'4 '16 '64 '256 '1024 '4048 '16384 ... "3 makes 1


'2 '8 '32 '128 '512 '2048 '8192 ... '3 makes 1


Proceed in the same way with all the lines, and you obtain the following sums:


'192 "3 makes '2 '8 '32 '64

'96 '3 makes '4 '16 '32

'48 "3 makes '2 '8 '16

'24 '3 makes '4 '8

'12 "3 makes '2 '4

'6 '3 makes '2

'3 "3 makes 1

"3 '3 makes 1




Lesson 38


Another lesson held at Akhmin. The teacher reads the following numbers from another wood tablet, and the pupils copy them on clay tablets:


1 10 1

10 100 10

20 200 20

2 20 2

4 40 4

8 80 8

11 1

1 '16 64 '96 '11

'2 '8 '32 '64 '128 '384 '68 (sic)

'4 '16 '32 '64 '256 '33

'8 '2 '8 '16 '16 '32 '192 "3 '22 '66


These are multiples and fractions of eleven. Now please check my lines and correct them if you can.


All pupils find easily that the multiplications are correct, while the fractions must be wrong. Some pupils try hard in correcting the wrong lines, while the most gifted one comes up with an easy and elegant solution:


1 '2 '2

'2 '3 '6

'3 '4 '12

'11 '12 '132



'11 of 1 '12 '132

'11 of 2 '6 '66

'11 of 4 '3 '33

'11 of 8 "3 '22 '33



'12 '12

'12 '16 '48

'12 '16 '64 '192


'11 of 2 makes '16 '64 '192 '132



'6 '6

'6 '8 '24

'6 '8 '32 '96

'6 '8 '32 '128 '384


'11 of 4 makes '8 '32 '128 '384 '66



'3 '3

'3 '4 '12

'3 '4 '16 '48

'3 '4 '16 '64 '192

'3 '4 '16 '64 '256 '768


'11 of 4 makes '4 '16 '64 '256 '768 '33



"3 '2 '6

"3 '2 '8 '24

"3 '2 '8 '32 '96


'11 of 8 makes '2 '8 '32 '96 '22 '66 or "3 '22 '66






(part 1)


I developed my lessons on early geometry in sci.math.symbolic, believing that this online forum would allow the discussion of all kinds of relations between symbols and numbers, but I was wrong, the forum is meant for symbolic computing, and so I had to defend my case:


Professor F. asked me via e-mail whether I can prove that mathematics depends on symbols? whether I believe that anybody cares about old discarded methods? and whether I can propose something useful for symbolic computing?


Let me begin with a few general remarks.


As a boy I was attracted to mathematics because our teacher told us that in geometry and mathematics all is logic, from base to top, and all belongs together. Following my teacher I see more connections than fences between the various areas and sub-disciplines. The wonder of mathematics are patterns that emerge everywhere, on each level, with no end; and a great pleasure are the seemingly distant areas of research that can suddenly and unexpectedly meet and join.


I told you repeatedly what impresses me about early methods: they are simple and robust on the one hand, clever and complex on the other hand. People who develop computers and programs may well profit from having a pleasure walk along the old ways.


My course on early geometry is an experiment: I try to say it ever simpler, giving a few lessons instead of writing a book. When I use many words I can always sneak around some critical points, but when I make it real short I consider each word and line. I have to reconsider my arguments, and then I often find a new and delightful aspect which I have missed before. As I make it simpler it grows more complex. My brother Steve, a programmer, makes the very same experience in his job. Our advice: make it ever simpler, and you shall be rewarded by seeing your work growing more complex and more effective.


Now let me answer the question about old discarded methods:


With the rise of Renaissance, medieval art had been discarded as being gothic, meaning barbarous. With our modern knowledge of fractal geometry we gain new pleasure from looking e.g. at the cathedral of Strasbourg with towers that grow smaller towers that grow again smaller towers ... Or think of the magnificent frontispiece of the Irish Book of Kells.


Similarly, there are no really discarded mathematical methods, I believe, only methods no longer in use. If all is logic from base to top, each really working method is a valuable approach.


The unit fraction series of Ancient Egypt came to life again with Medhavan, Nilakantha, Gregory, Leibniz, Newton, and others.

I found indirect evidence for the use of many pi values in the Rhind Mathematical Papyrus, among them 19/6 or 3 1/6 or 3 '6. We can easily discard such a value. However, a couple of years ago one K. Lange found an elegant continued fraction starting from 3 1/6 (published in the American Mathematical Monthly, here given in my ASCII notation):


pi = 3 plus 1x1/ 6 + 3x3/ 6 + 5x5/ 6 + 7x7/ 6 + 9x9/ 6 + ...


Who can exclude the possibility that some old discarded methods may one day reveal themselves being first steps into new areas of research?


This allows me to answer the third question:


Number patterns are prior to formulas, and pretty effective. Is there a way of rendering such patterns in symbolic computing?


Consider a simple case


1 3 5 7 9 ...

4 8 12 16

12 20 28

32 48



which provides the Medhavan series


pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 ...


and a natural logarithm


ln4 = 1 + 1/4 + 1/12 + 1/32 + 1/80 ...


What about other series?


1 '3x4 '5x8x12 '7x12x20x32 '9x16x28x48x80 ...


'3exp1 '5exp3 '7exp5 '9exp7 ...


I calculated many such series, and perhaps occasionally ran across a useful one, which, however, I was not able to recognize, so I would like to have a program allowing me to check all kinds of series derived from simple number patterns. The program should calculate a synthetic series and automatically compare the result with important numbers. There may be some meaningful superfast series including pi series waiting around the corner (a hope of mine).


Furthermore I would welcome a program that allows me to handle number patterns in 3 dimensions and explore further possibilities of discrete mathematics - not sure if there will be any rewarding results, but with mathematics one never knows.


Having said enough for today I shall answer the first question on another occasion. What I can offer is basically a better understanding of mathematics itself, including how the mind processes mathematical tasks. More in my next reply or message (economy of the mind / symbols used in mathematics perhaps mapping a mental landscape).




(part 2)


The early methods of geometry and mathematics also fascinate me because they are very simple if regarded by themselves, yet solving highly demanding problems when working together.


A biologist from Zurich university and his teams are doing research on the desert ant cataglyphis since nearly 30 years now. Cataglyphis lives in small cavities and goes out looking for food in the hottest hours of the day, when no predator is underway. As soon as it has found a piece of meal, or if there is a danger occurring, it runs home in a straight line, rarely ever missing the entrance of its cavity. How does the minute brain of cataglyphis solve the problem of finding home directly? By considering the heavenly patterns of polarized light, by remembering some prominent features of the ground, and by summing up the vectors of all steps made since leaving home. Each ability alone is weak and prone to errors, but working together they perform a wonder.


How are we processing mathematical tasks?


Neurology found two mathematical areas in the human brain. One carries out exact calculations (on the left hemisphere, near the Broca center, in 10 or 11 o'clock position, if my memory serves me right), while a more intuitive area judges whether a given amount is bigger or smaller than another one (near a main visual area, in 7 position, if my memory serves me well).


Vision is processed by over thirty areas occupying one third of the human brain. We may assume that mathematical tasks are also processed in more than two areas. Telling from my own personal experience I can say that an economic way of calculating requires a feeling for the complexity of a given problem. I have such a sense, and it was a great help during my schooldays. I could most always 'see' how complex the solution can be, and had to check only a few possible results. Thus I often solved a problem correctly within a minute, while others used half an hour or more and came up with a wrong result.


How can a sense of complexity possibly be implemented in computing? I think it has to do with reasoning on more than one level, which again might have to do with kind of a mental landscape. Let me explain this.


A pair of autistic twins are known for solving highly demanding calculations by just wandering around in a landscape of numbers, as they explain it. When they are asked to solve a problem they know exactly where to go, and there lies the answer ready to be grasped. From Oliver Sacks we know that neurological injuries or defects reveal how the regular brain is working. It may well be that we all have access to an inner landscape, and if there is any map of it, I (relying on my sense of complexity) can make out only one possible candidate, namely the symbols used in mathematics (numbers, operation signs, and their composites).


The first plus and minus signs that came down to us are found in the Rhind Mathematical Papyrus: a pair of legs walking in one direction (gaining way), or walking in the opposite direction (loosing way). The Egyptian sign for equal had the meaning of make: "3 '5 '10 '20 er (makes) 1, and showed an eye. The equal sign we are using today was introduced as a pair of parallels === by Robert Recorde (whetstone of witte, London, 1557). Look up my favorite History of Mathematics by David Eugene Smith (Gynn and Company 1923, Dover New York 1958), and you will see how the notations evolve, and how the process of shaping the mathematical symbols involved some of the greatest minds.


Ahmes had a beautiful handwriting. The Rhind Mathematical Papyrus is pleasing to look at. In my opinion it has an almost musical quality, which confirms my vague idea of mathematical notations being some kind of a map of the mind. Music and mathematics are related. Musical elements may play a key role in coordinating the various areas of the brain involved in solving a problem. There is actually a frequency of 41 Hertz (if my memory serves me well) coordinating the otherwise autonomous areas.


My advice: look at your work from many angles of view, consider more than one level, and allow that the symbols themselves can be a topic of this forum. I see the same playful mind at work at every level of mathematical evolution, from early geometry to so-called symbolic algebra. I see the same trust in symbols and symbolic operations everywhere. Proceeding from ONE apple, TWO loaves of bread and THREE jugs of oil to the bare numbers 1 2 3; from storing goods to adding numbers; from actual trading to symbolical trading by adding and subtracting numbers; from distributing goods among a group of people to handling unit fractions and unit fraction series, which were also and most successfully used for solving geometrical problems, and which, moreover, opened a door into number theory --- all that was no less revolutionary an emancipation than are the works of Peacock, De Morgan, Hamilton et al. in our time (freeing the numbers from objects, proceeding from counting to calculating, was no less radical than freeing the algebraic operations from the restrictions of conventional numbers). And I see an all embracing love for patterns that emerge everywhere, superimpose, interweave, and may well be the fabric of that mysterious landscape those twins are speaking about.







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