Monday, August 31, 2009
Sunday, August 30, 2009
Similarly, one might guess that the pyramidal numbers are proportional to the triangular numbers and divide a pyramidal number by its corresponding triangular number. One again gets a linear sequence of numbers and the factor can be easily determined.
Saturday, August 29, 2009
The sum of n 1s is n. The sum of 1 through n is n(n+1)/2. So it would seem likely that the sum we are looking for is some cubic expression and by substituting k-1 for k we get the prior sum and the difference.
We can equate the coefficients of k on both sides of the second equals sign to obtain equations for the coefficients.
Note that only unit fractions are needed for the sum. Finally we deduce the formula for the total number of blocks in a pyramid of n rows.
If we assume that the blocks are square in shape with width w and height h the volume of the pyramid is
V = n(n+1)(2n+1)w²h/6.
If we ignore the additive constants and set W = nw and H = nh we get the usual formula for the volume of a pyramid,
V = W²H/3.
A method for computing the volume of a truncated pyramid if found in the Moscow Mathematical Papyrus.
It is not necessary that all the scribes knew this rudimentary level of geometry and algebra. It would take only one master of sufficient skill to deduce the methods of calculation. The builder of the first Step Pyramid was known as Imhotep. He was immortalized by the ancient Egyptians.
It could be that the pyramids contained the mathematical knowledge of the ancient Egyptians. It is not unlikely that they would set this knowledge down in stone. At least the pyramids provided an opportunity to do so.
Tuesday, August 25, 2009
The procedure for computing the seked was to take half the base of the pyramid and divide by the height. This makes computing the slope of the Great Pyramid of Giza especially simple since the ratio of the base to height was 11/7. The seked is just 11/2 or 5½ (palms) since the unit length, the cubit, is 7 palms.
Sunday, August 23, 2009
But our main focus is the royal foot (remen) whose use can be seen in the design of the pyramids of the Old Kingdom. Let's take a look at the Great Pyramid of Giza whose original dimensions are estimated to have a base of 230.5 m and a height of 146.6 m. What does this tell us? Well, the ratio of the base to the height is 1.57231 which doesn't say much by itself but we can look at its continued fraction.
which shows that one gets a best fit for 11/7 with a scale factor of 40 cubits or 70 feet. This implies that in Khufu's time,
1 royal foot = 299.35 mm
1 royal cubit = 523.86 mm
which is very close to the values for the cubit stick from the 18th Dynasty.
An interesting coincidence is that,
11/7 = 1.5714
π/2 = 1.5708
Saturday, August 22, 2009
Another aspect of the story is the secrecy associated with the Book of Thoth. It was hidden away and protected by a series of defenses. But no matter how well something is protected the defenses can always be defeated.
Thoth appears in the story to have a dark side. While Ra, the Sun god, is the source of truth and light Thoth, the Moon, is more obscure. To obtain vengeance on the thief he first gets power over him by taking him and his family to the land of the dead. Thoth is jealous of his power and it is not easily taken.
Friday, August 21, 2009
Thursday, August 20, 2009
One of the duties of the scribes was to distribute goods. When precision was required the royal measures were beyond question.
Wednesday, August 19, 2009
Tuesday, August 18, 2009
Monday, August 17, 2009
Sunday, August 16, 2009
Saturday, August 15, 2009
The fingers of the royal cubit were marked as follows: 1finger, 2 fingers, 3 fingers, 4 fingered palm, five fingered palm, fist, and so on up to 8 fingers. The glyph for the remen is placed above the 15th finger on the scale. This appears to be what Wikipedia calls the bw (foot) which was 15 or 16 fingers. The term remen was also used to represent half the diagonal of a cubit square which was about 20 fingers. And a remen square would have a diagonal about equal to a cubit. The remen marked on the cubit seems to be associated with the forearm and not a foot. Perhaps this is an older usage that of the diagonal of a square came later.
It you look at the side of a royal cubit you will see that there are fingers which are subdivided into parts varying from 2 to 16. The Egyptian scribes needed all these divisions since they used unit fractions. But the scale shows that with 16 subdivisions they were capable of millimeter accuracy. One can compare the royal cubit with the modern engineer's scale or architect's scale.
Friday, August 14, 2009
We can compare various feet which have been used and the nsec,
1 pes (Roman foot) = 296 mm
1 pous (Ionian foot) = 296 mm
1 English foot = 305 mm
1 Egyptian bd = 300 mm
1 nsec = 300 mm
The Egyptian bd is 4 "hand breadths" of 4 fingers and there are 7 hand breadths to a cubit so the bd was 16 fingers and the cubit was 28 fingers. If the bd was the primary unit the cubit would be 7/4 of it. Is there any significance to this?
7 = 1 + 2 + 4 → 7/4 = 1 + 1/2 + 1/4
These are Eye of Horus fractions and the representation of integers as a sum of powers of 2 was used for multiplication.
The Egyptians made the conversion 2/7 → 1/4 +1/28 so 4/7 = 1/2 + 1/14 could also be easily represented and calculations involving 4/7 would not pose any difficulty.
Eye of Horus fractions would have been of greater significance to the Egyptians. 7/4 was more fundamental.