Eye of Horus and Unit Fractions

 The ancient Egyptian symbol “eye of Horus” (from the sky god Horus who was usually depicted as a falcon) was considered as a protective amulet to the Egyptian. It has been commonly painted on the bows of boats both protected the vessels and "saw" the way ahead. Moreover, the eye is constructed in six fractional parts, representing the shattering of the eye of Horus into six pieces. According to historical documents, these six parts also represent six senses of human. The inner corner of the eye indicates one half, the iris is one fourth, the eyebrow is one eighth, the outer corner of the eye is one sixteenth, and the decorations below the eye are one thirty-second and one sixty-fourth respectively. 

Note: The infinite geometric series with ratio less than one is:

The unit fraction numbers in the Eye of Horus are the first six terms of the geometric series:

Hence, it looks like Egyptian scribes would approximate one using the first six terms of this series because the sum: 1/2 + 1/4 +...+ 1/64 = 63/64.

In Milo Gardner paper (The Arithmetic used to Solve of an Ancient Horus-Eye Problem, 2006), explanations were given on how Egyptian scribes divide one (64/64) by number. In ancient Egypt hekat was a volume unit. One hekat can be considered 64/64 hekat. There are also other volume units such as hin, dja and ro where

1 hin = 1/10  hekat

1 dja = 1/64 hekat

1 ro = 1/320 hekat

Scribes would express the division of a hekat in terms of Horus Eye unit fractions and these smaller units of hekat. Here are some examples from Milo Gardner paper(2006).

In addition to Horus Eye unit fractions, Egyptians also used other unit fractions. All fractions can be represented as a sum of unit fractions (1/n, n is any natural number). According to (Katz, 2008, pg.5) the Egyptians expanded the fraction with non-unit numerators into the sum of unit fractions. Whenever a number of objects need to be equally divided into a number of receivers, this expansion method can be useful. For example, suppose we need to equally divide five loaves of bread to seven people. Then one can expand 5/7 into sum of unit fractions as follows: 

5/7  = 1/2 + 1/7 + 1/14 

This expansion also gives the plan of how to cut the bread. That is each person will get half of a bread loaf plus one seventh of a bread loaf plus one fourteenth of a bread loaf. So it is very interesting to know that some ancient peoples made good use of unit fractions in their daily business when they did not have the luxury of modern mathematics.

Many numbers have special properties, meaning or applications. Cultures are also connected to some numbers. In this Wichita State University website (http://www.math.wichita.edu/history/Topics/snumbers.html), there is some information on special numbers such as perfect numbers (a positive integers that is equal to sum of its divisors excluding itself such as 6 = 1+2+3). 

Amicable numbers are the pairs of numbers with the following properties:

1. Sum of divisors of first number (excluding the first number) equals the second number

2. Sum of divisors of second number (excluding the second number) equals the first number

An example is 220 and 284. Sum of divisors of 220 is:

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.

Sum of divisors of 284 is:

1 + 2 + 4 + 71 + 142 = 220.

There are also special numbers in cultures. Ancient Greeks are said to have believed in four elements (earth, water, air, and fire). Native American culture also talked about four directions (east, west, north, south) from where the wind came, and each of these winds had accompanying stories.