46 Lessons in Early Geometry, part 7/10 / provisional version in freestyle English / a corrected version will follow in March, April or May (hopefully) / Franz Gnaedinger / February 2003 / www.seshat.ch

Lesson 33

The Egyptians knew a famous series they called Horus Eye:

1 = '2 '4 '8 '16 '32 '64 (...)

This series can be developed by means of a stairway of equations:

1 = '1

1 = '2 '2

1 = '2 '4 '4

1 = '2 '4 '8 '8

1 = '2 '4 '8 '16 '16

1 = '2 '4 '8 '16 '32 '32

1 = '2 '4 '8 '16 '32 '64 ('64)

................................

1 = '2 '4 '8 '16 '32 '64 ('128 '256 '512 ...)

1 = '2 '2x2 '2x2x2 '2x2x2x2 '2x2x2x2x2 '2x2x2x2x2x2 (...)

The principle of this series can be expanded as follows:

'2 = '2

'2 = '3 '6

'2 = '3 '9 '18

'2 = '3 '9 '27 '54

.......................

'2 = '3 '9 '27 '81 '243 '729 ...

'2 = '3 '3x3 '3x3x3 '3x3x3x3 '3x3x3x3x3 '3x3x3x3x3x3 ...

'3 = '4 '12

'3 = '4 '16 '48

'3 = '4 '16 '64 '192

........................

'3 = '4 '16 '64 '256 '1024 '4096 ...

'3 = '4 '4x4 '4x4x4 '4x4x4x4 '4x4x4x4x4 '4x4x4x4x4x4 ...

and so on

The powers required for these series are provided by additive number patterns:

1     1     1     1     1     1     1

2     2     2     2     2     2

4     4     4     4     4

8     8     8     8

16    16    16

32    32

64

1     2     4     8    16    32    64

3     6    12    24    48   96

9    18    36    72   144

27    54   108   216

81   162   324

243   486

729

and so on

The series of '3 is a subseries of the Horus Eye series:

1 = '2 '2x2 '2x2x2 '2x2x2x2 '2x2x2x2x2 '2x2x2x2x2x2 ...

'3 =    '4          '4x4                '4x4x4       ...

Hence we can establish a pair of subseries, whereby "3 = 2/3:

'3 =    '2x2        '2x2x2x2            '2x2x2x2x2x2 ...

"3 = '2      '2x2x2          '2x2x2x2x2              ...

And here is the stairway of the lower series:

"3 = '2 '6

"3 = '2 '8 '24

"3 = '2 '8 '32 '96

......................

Lesson 34

Having explained at length my basic number pattern for the calculation of the square I like to present the basic number pattern of down under algebra, using again my quasi hieratic notation of unit fractions:

1 = '2 '1x2 = '2 '2

'2 = '3 '2x3 = '3 '6

'3 = '4 '3x4 = '4 '12

'4 = '5 '4x5 = '5 '20

'5 = '6 '5x6 = '6 '30

'6 = '7 '6x7 = '7 '42

'7 = '8 '7x8 = '8 '56

'8 = '9 '8x9 = '9 '72

......................

This pattern allows the construction of a fascinating series:

1 = '1x2 '2

'2 = '2x3 '3

'3 = '3x4 '4

'4 = '4x5 '5

'5 = '5x6 '6

1 = '1x2      '2x3      '3x4      '4x5      '5x6  ...

Also this series can be developed by means of a stairway:

1 = '1

1 = '1x2 '2

1 = '1x2 '2x3 '3

1 = '1x2 '2x3 '3x4 '4

1 = '1x2 '2x3 '3x4 '4x5 '5

1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6

...................................

1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6x7 '7x8 '8x9 ...

Expanding the principle of the new series:

'2 = '2

'2 = '1x3 '6

'2 = '1x3 '3x5 '10

'2 = '1x3 '3x5 '5x7 '14

'2 = '1x3 '3x5 '5x7 '7x9 '18

..................................

The resulting series has a pretty subseries:

'2 = '1x3 '3x5 '5x7 '7x9 '9x11 '11x13 '13x15 '15x17 ...

'1x3      '5x7      '9x11        '13x15        ...

Now consider this pair of series which I derived from the above series and subseries:

'2  plus '1x3   plus '3x5  plus '5x7   plus '7x9 ...

'2  plus '1x3  minus '3x5  plus '5x7  minus '7x9 ...

The plus-plus series converges to 1, while the plus-minus series converges to a number close to 11/14:

0.5         average 0.6666...   0.8333...

0.7666...   average 0.7809...   0.7952...

0.7793...   average 0.7844...   0.7894...

0.7824...   average 0.7850...   0.7876...

.........   .................   .........

.........   11/14 = 0.7857...   .........

The area of the unit square measures 1 square unit, while the area of the inscribed circle measures practically 11/14 square unit. May it be that the above series oscillates around the area of the circle inscribed in the unit square?

Yes, it does, and similar ways of reasoning may have led the Indian mathematician Medhavan (ca.1340-1425) to his discovery of the famous series

pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 ...

which is an equivalent of this series

pi/8 = '1x3 '5x7 '9x11 '13x15 '17x19 '21x23 '25x27 ...

that is approached from above by another kind of stairway

'1x3 '16

'1x3 '5x7 '32

'1x3 '5x7 '9x11 '48

'1x3 '5x7 '9x11 '13x15 '64

'1x3 '5x7 '9x11 '13x15 '17x19 '80

.......................................

Lesson 35

The following stairway approximates pi from above:

8 times '1x3 '16

8 times '1x3 '5x7 '32

8 times '1x3 '5x7 '9x11 '48

.................................

The ever longer lines can be fine-tuned by adding 1/1 to 16, 1/2 to 32, 1/3 to 48, and so on:

8 times '1x3 '(16 '1)

8 times '1x3 '5x7 '(32 '2)

8 times '1x3 '5x7 '9x11 '(48 '3)   and so on

8 times '3 '17

8 times '3 '35 '32.5

8 times '3 '35 '99 '48.333...   and so on

Play with the numbers and you see a new pattern emerging:

8 times '2x2-1 '4x4+1

8 times '2x2-1 '6x6-1 '8x8+1 '8x8+1

8 times '2x2-1 '6x6-1 '10x10-1 '12x12+1 '12x12+1 '12x12+1

..............................................................

Let me calculate some lines:

8 times 1/3 1/17

8 times 1/3 1/35 2/65

8 times 1/3 1/35 1/99 3/145

8 times 1/3 1/35 1/99 1/195 4/257

8 times 1/3 1/35 1/99 1/195 1/323 5/401

8 times 1/3 1/35 1/99 1/195 1/323 1/483 6/577   and so on

line 1  3.137...  line 2  3.14139...  line 3  3.141563...

line 15  3.14159264...

pi  3.14159265...

The Indian mathematician Nilakantha, who flourished in the 15th century, may have known the fine-tuned lines. We owe him many pi-series, for example this product:

square root of 12   times

1 minus '3x3 plus '5x3x3 minus '7x3x3x3 plus '9x3x3x3x3 ...

=  sr12  times  1 - 1/9 + 1/45 - 1/189 + 1/729 - 1/2673 ...

Lesson 36

I see the same playful mind at work at every level of mathematical evolution, from early geometry to so-called symbolic algebra, from number patterns to formulas and theorems, and every pattern, insight and law found a proper application. Finally even the octonions, which, by the way, are calculated by means of a special triangular number pattern, come to life in string theory. Sooner or later each Sleeping Beauty is getting kissed by her prince daring and charming ...

1     1     2

2     3     4

5     7    10

12    17    20

29    41    58

70    99   140

Divide 99 by 70 and give the result in unit fraction series:

99 : 70 = 198 : 140 = 140+28+20+10 : 140 = 1 '5 '7 '14

140 : 99 = 297 : 210 = 210+70+14+3 : 210 = 1 '3 '15 '70

Let the side of a square measure 411 royal cubits or 2,877 palms. How long is the diagonal? Multiply the numbers by a series and round all results:

411 times  1   '5   '7   '14

411  82   59   29  sum 581

The diagonal measures 581 royal cubits (mistake 12.659... cm)

2877 times  1    '5    '7    '14

2877  575   411   ???

Here we have a rounding problem, so let us use the other series:

2877 times  1    '3    '15   '70

2877  959   192   41   sum 4069

The diagonal measures 4069 palms or 581 cubits 2 palms (mistake only 23 millimeters).

In the Rhind Mathematical Papyrus are found many conversions, for example this one: "101 = '101 '202 '303 '606. Let me use that series for a calculation of interest. Vizier Milo saved 68,954 Egyptian Dollars and brings the money to the First City Bank of Amarna, which offers interest at '101 '202 '303 '606 (a little more than 2 percent). How will Vizier Milo's fortune rise in the coming years? Proceed like above:

Year 1   fortune  68,954   interest  683 341 228 114  =  1,366

year 2   fortune  70,320   interest  696 348 232 116  =  1,392

year 3   fortune  71,712   interest  710 355 237 118  =  1'420

year 4   fortune  73,132   interest  724 362 241 121  =  1,448

year 5   fortune  74,580   interest  738 369 246 123  =  1,476

year 6   fortune  76,056   interest  753 377 251 126  =  1,507

year 7   fortune  77,563   interest  768 384 256 128  =  1,536

year 8   fortune  79,099   interest  783 392 261 131  =  1,567

year 9   fortune  80,666   interest  799 399 266 133  =  1,597

year 10  fortune  82,263   interest  814 407 271 136  =  1,628

year 11  fortune  83,891   interest  831 415 277 138  =  1,661

year 12  fortune  85,552   interest  847 424 282 141  =  1,694

year 13  fortune  87,246   interest  864 432 288 144  =  1,728

year 14  fortune  88,974   interest  881 440 294 147  =  1,762

year 15  fortune 90,736

And the exact value? 68,954 x (1 + 2/101) exp 14 = 90736.365...

Although we have rounded all numbers, the margin of error is not even 40 cents. And if we begin with 6,895,400 cents instead of 68,954 dollars, the mistake would be less than 4 (four) cents.