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Drew,

The Eye of Horace is representative of all Ancient Egyptian Binary methods of additive multiplication and division.

Ancient Egyptians use of unit fractions and the Eye of Horace, producing is an infinite series.

Multiply the Perimeter of G1 of 1760 cubits times the Eye of Horace, 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 extending the series + 1/128 = 127/128 ect. Creating an infinite series never reaching one.

i.e.

1 –-1760 Base perimeter of G1.

1/2 –- 880

1/4 ---440

1/8 ---220

1/16---110

1/32 ---55

1/64 ---27 1/2

Adding the results fractional calcualtions totals 1732 1/2 cubits, this whole process is equal to our 1760 x (63/64) = 1732 1/2

Evidently I did not provide enough information since you seem not to understand when I said "The 'ro' is a Middle Kingdom creation that only distracts from what the missing 1/64 actually does!"

Here is some additional information for you to help:

[planetmath.org]

I did not say the Ancient Egyptians were not cognizant of rational values for phi, sqrt 2, sqrt 3, sqrt 5, in rational form, or their importance in calculations and constructions. The reality is there are fractional equivalents within 4 to 5 decimal places of accuracy for all our values listed above easily substituted for our current values. To the Ancient Egyptians our values simply did not exist.

You do know phi was first noted in rational form by Fibonacci in “Liber Abaci” in the 1299. originally expressed by a continued fractional series 1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 144/233 etc. The current incarnation of phi (1.618033989) is a modern irrational average based on the sqrt 5. That being said, the fact is the rise run of G1’s 5 1/2 seked 14/11 is within 0.000707623 of the current value for the sqrt phi 1.27201965. So it is very unlikely the Ancient Egyptians employed phi as we know and use it.

Sqrt 2: 1 + 1/9 times the rise run 1 + 1/4 * 1/44 of the 5 1/2 seked. Giving the value of 140/99 or a decimal value of 1.4141414… a difference of 0.000072148 from the sqrt 2, which is accurate enough for most calculations. i.e.140/99 = 1.414141414…minus the sqrt2 = 1.41213562 = 0.000072148

Several years ago Rob Miller found the Sqrt 3: in the form of 97/56 1 + 1/2 + 1/7 + 1/14 + 1/56 = 1 + 41/56, decimal value of 1.732142857…a difference of 0.00009205 compared to sqrt 3, again accurate enough for most calculations.

i.e. (97/56 = 1.732142857…) minus the (sqrt 3 = 1.732050808) = 0.00009205

As an example of what I told DavidK, understanding the difference between our current system and theirs will provide answers for many of the lingering questions accumulated over the years.

One interesting thing about 1 + 1/2 + 1/7 + 1/14 + 1/56 and just for fun because we know the Ancient Egyptians did not use angles as we do, but is: tan 97/56 = 60º 0’ 4.75” cotan 56/97 = 29º 59’ 55.25” compared to our measure for the latitude of G1 29ª 58’ 45.12“N having a difference of 0º 1’ 10.13” from our current noted location for G1.

Use or ignore the above information, It really doesn't matter to me it is your hypothesis.

Regards,

Jacob

The Eye of Horace is representative of all Ancient Egyptian Binary methods of additive multiplication and division.

Ancient Egyptians use of unit fractions and the Eye of Horace, producing is an infinite series.

Multiply the Perimeter of G1 of 1760 cubits times the Eye of Horace, 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 extending the series + 1/128 = 127/128 ect. Creating an infinite series never reaching one.

i.e.

1 –-1760 Base perimeter of G1.

1/2 –- 880

1/4 ---440

1/8 ---220

1/16---110

1/32 ---55

1/64 ---27 1/2

Adding the results fractional calcualtions totals 1732 1/2 cubits, this whole process is equal to our 1760 x (63/64) = 1732 1/2

Evidently I did not provide enough information since you seem not to understand when I said "The 'ro' is a Middle Kingdom creation that only distracts from what the missing 1/64 actually does!"

Here is some additional information for you to help:

[planetmath.org]

Bold emphasis in above quote mine.Quote

Prior to 2050 BCE Old Kingdom Egyptian scribes rounded off rational numbers to six-terms binary representations for 1,000 years.The binary notation was stated in 1/2n units with a 1/64 unit thrown away. The [planetmath.org] Horus-Eye recorded rational numbers in the cursive pattern:

1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + (1/64).

Note that a potential 7th term (1/64) was rounded-off and thrown way.

The new Middle Kingdom math system ”healed” rounded off binary series by several finite methods. Two weights and measures finite systems can be reported by:

1. 1 hekat (a volume unit) used a unity (64/64)such that (32 + 16 + 8 + 4 + 2 + 1/64)hekat+ 5 ro

and (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64)hekat + 5 ro

meant (64/64)/n = Q/64 + (5R/n)ro

Note that the hekat unity was generally divided by rational number n. To divide by 3 scribal long-hand would have written out

(64/64)/3 = 21/64 hekat + 5/192 = (16 + 4 + 1)/64 hekat + 5/3 ro =

(1/4 + 1/16 + 1/64)hekat + ( 1 + 2/3)ro

2. (100-hekat)/70 = (6400/64)/70 = 91/64 hekat + 30/4480 = (64 + 16 + 8+ 2 + 1)/64 hekat + 150/70 ro = (1 + 1/4 + 1/8 + 1/32 + 1/64)hekat + (2 + 1/7)ro

meant (6400/64)/n = Q/64 + (5R/n)ro was applied for almost any hekat division problem.

The hieratic word ro meant 1/320 of a hekat in a grain weights and measures system. Note that 5 ro meant 5/320 = 1/64.

Generally, scribal shorthand recorded duplation aspects of mental calculations and fully recorded two-part hekat quotients and ro remainders.

At other times 2/64 was scaled to 10/320 such that (8 + 2)/320 = 1/40 + 2 ro

I did not say the Ancient Egyptians were not cognizant of rational values for phi, sqrt 2, sqrt 3, sqrt 5, in rational form, or their importance in calculations and constructions. The reality is there are fractional equivalents within 4 to 5 decimal places of accuracy for all our values listed above easily substituted for our current values. To the Ancient Egyptians our values simply did not exist.

You do know phi was first noted in rational form by Fibonacci in “Liber Abaci” in the 1299. originally expressed by a continued fractional series 1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 144/233 etc. The current incarnation of phi (1.618033989) is a modern irrational average based on the sqrt 5. That being said, the fact is the rise run of G1’s 5 1/2 seked 14/11 is within 0.000707623 of the current value for the sqrt phi 1.27201965. So it is very unlikely the Ancient Egyptians employed phi as we know and use it.

Sqrt 2: 1 + 1/9 times the rise run 1 + 1/4 * 1/44 of the 5 1/2 seked. Giving the value of 140/99 or a decimal value of 1.4141414… a difference of 0.000072148 from the sqrt 2, which is accurate enough for most calculations. i.e.140/99 = 1.414141414…minus the sqrt2 = 1.41213562 = 0.000072148

Several years ago Rob Miller found the Sqrt 3: in the form of 97/56 1 + 1/2 + 1/7 + 1/14 + 1/56 = 1 + 41/56, decimal value of 1.732142857…a difference of 0.00009205 compared to sqrt 3, again accurate enough for most calculations.

i.e. (97/56 = 1.732142857…) minus the (sqrt 3 = 1.732050808) = 0.00009205

As an example of what I told DavidK, understanding the difference between our current system and theirs will provide answers for many of the lingering questions accumulated over the years.

One interesting thing about 1 + 1/2 + 1/7 + 1/14 + 1/56 and just for fun because we know the Ancient Egyptians did not use angles as we do, but is: tan 97/56 = 60º 0’ 4.75” cotan 56/97 = 29º 59’ 55.25” compared to our measure for the latitude of G1 29ª 58’ 45.12“N having a difference of 0º 1’ 10.13” from our current noted location for G1.

Use or ignore the above information, It really doesn't matter to me it is your hypothesis.

Regards,

Jacob

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