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

early geometry 1 / early geometry 2
/ early geometry 3 / early
geometry 4 / early geometry 5 / early geometry 6 / early geometry 7
/ early geometry 8 / early geometry 9 / early geometry 10

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

Intermezzo

(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

80

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.

early geometry 1 / early geometry 2
/ early geometry 3 / early
geometry 4 / early geometry 5 / early geometry 6 / early geometry 7
/ early geometry 8 / early geometry 9 / early geometry 10