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
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 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.
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