Egyptian Mathematics - The Great Pyramid of Giza

Egyptian Mathematics - The Great Pyramid of Giza

The Egyptians built The Great Pyramid at Giza as an amazing burial monument for Pharaoh Khufu. But they did more. They also used it as a showcase for their mathematical and engineering skills.

Pi is probably the most famous number in mathematics. Draw any circle, then measure the length of the circumference and the length of the diameter. Divide the two numbers and you get pi. But a circle was not used in the design of The Great Pyramid of Giza, right? Wrong! Not only was a circle used it was totally fundamental to the design. Here's how..

The Khufu Pyramid (The Great Pyramid of Giza) had a design height of 280 royal cubits and a base length of 440 royal cubits. The Pyramid we see today is slightly different due to erosion and theft of stone. So let's stick with the original design size.

So the distance around the base of the pyramid is simply 4*440=1760

Let's divide this by twice the height, which is 560, so we get..

1760/560=22/7

The number 22/7 is the most famous approximation of pi, and it's a pretty good one, in fact..

22/7-pi=0.001

So the Khufu Pyramid was built on circular geometry. Which means the Egyptians knew pi by the time they built the Great Pyramid (2560 BC). For all we know they may have known it much earlier.

But it gets stranger, not only did they know pi, they used it to define the dimensions of their most sacred monument. What does this mean?

Content written and posted by Ken Abbott abbottsystems@gmail.com

Mobius Strip - A New Surprise

Mobius Strip - A New Surprise

The Mobius strip is such a simple object, yet full of surprises. Here's one you may not know about..

Take a strip of paper, give it 1 half twist before joining the ends. This is the famous Mobius Strip. Despite how it looks it only has 1 surface. Does a half twist clockwise give the same object as a half twist counter clockwise, or are these two different objects?

But here's the surprise..

Take a strip of paper, but give it 4 half twists before joining the ends. This has 2 surfaces. Now play around with it for a while. At some point it will suddenly "flip" into a double thickness band with 1 half twist. In other words if flips into a double thickness Mobius Strip. One surface has gone, and so have 3 half twists!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Prime Numbers in Base 2

Prime Numbers in Base 2

We're used to seeing numbers represented in base 10 "decimal" notation, and almost all prime number lists use base 10. But we can represent numbers in any base we please. In base 10 we use 10 symbols 0,1,2,3,...,9 and in base n we use n symbols 0,1,2,3,...,(n-1)

The simplest base is 2, because in that base we have only 2 symbols 0,1

Base 2 is also call "binary" and writing numbers in binary makes them look like computer data.

In binary the positions represent 1,2,4,8,16,32,.. so the general representation of a positive integer n is..

n=sum{ai*2^i} where the coefficients {ai} are all 0 or 1 and the sum is over i from 0 onward.

For example, the number 11 in binary is 1101, and this simply means..

11=1*(1)+1*(2)+0*(4)+1*(8)

Here's the first 20 prime numbers in binary..

2 01
3 11
5 101
7 111
11 1101
13 1011
17 10001
19 11001
23 11101
29 10111
31 11111
37 101001
41 100101
43 110101
47 111101
53 101011
59 110111
61 101111
67 1100001
71 1110001

Writing numbers in binary can help spot patterns we might not notice in other bases. For example..

In binary all prime numbers except 2 begin and end with 1.

The first 2 digits of the prime 71 is the prime 3 and the last 5 digits is the prime 17. So we could define a "+" operation and say that 3+17=71. Notice that the + operation depends on order, so 17+3=113 is different, but it's still a prime!

The prime 13 is just the prime 11 written backwards. The same is true for 23 and 29 and lots more. Many primes are just an earlier prime written backwards!

Some primes have all digits set to 1, so these primes are of the form (2^n)-1 where n is just the number of binary digits. Primes of this form are called Mersenne primes, named after Marin Mersenne, a French monk who studied them in the 17th century.

I wonder what we might discover if we used sophisticated computer pattern recognition on prime numbers in binary format?

Content written and posted by Ken Abbott abbottsystems@gmail.com