written by Luca Pacioli and illustrated by Leonardo da Vinci. Printed in 1509.
Lógos, a ratio in mathematics, is a value that indicates the relation between two quantities. a/b = x
Example for a = 6 and b = 3: 6/3 = 2.
A proportion refers to the equality of two ratios. a/b = c/d = x.
Example: The ratios 6/3 and 4/2 form a proportion because 6/3 = 4/2 = 2.
Proportion is also the comparison of one quantity with the sum of two quantities.
x = a/(a+b) and y = b/(a+b), which can also be expressed as a/(a+b) = X% and b/(a+b) = Y%.
Example for a = 6 and b = 2: x = 6/(6+2) = 6/8 = 3/4 and y = 2/(6+2) = 2/8 = 1/4.
also expressed as x = 3/4 = 75% and y = 1/4 = 25%.
Golden Ratio
The Golden ratio between two quantities, a > b, exists if their sum divided by the larger quantity is equal to the ratio of the larger quantity to the smaller quantity. In other words, if the following equation holds true: (a + b) / a = a / b = φ (where φ -phi- is the golden ratio).
For a = 1, then b = √5 / 2. Therefore, the value of φ is: φ = (1+√5)/2 = 1.618… (an infinitely long decimal, an irrational number)
The golden ratio is also referred to as the Golden rule and the divine proportion. When applied to the division of a straight segment, it is also called the Golden Ratio or Extreme and Mean Ratio.
The inverse of the ratio is referred to as the Golden ratio conjugate (or Silver ratio) and is symbolized by the capital letter Φ:
Φ = 1 / φ = 0.618… However, Φ can also be expressed as Φ = φ – 1 = 1.618… – 1 = 0.618…
This demonstrates the unique property of the golden ratio among positive numbers, 1/φ = φ – 1 but also: 1/Φ = Φ + 1.
Arithmetic Proportion
If among three given numbers a > b > c, the equality of differences a – b = b – c is valid, we say that an Arithmetic proportion is formed. The middle number b is called the arithmetic mean and is expressed in terms of the two extreme numbers as:
Arithmetic proportion is also known as a difference proportion because it does not involve equality of ratios between its terms but equality of their differences.
Geometric Proportion
If among three given numbers a > b > c, and the equality of ratios a/b = b/c holds, we say that a Geometric proportion is formed.
The middle number b is called the geometric mean and is expressed in terms of the two extreme numbers as:
An example of a geometric proportion is formed by the numbers 4, 2, 1, because 4/2 = 2/1 and 2 = √(1 · 4).
Harmonic Proportion
If among three given numbers a > b > c, and the ratio of the maximum term to the minimum term is equal to the ratio of the difference of the mean from the maximum term to the difference of the minimum term from the mean
The middle number b is called the harmonic mean and is expressed in terms of the two extreme numbers as:
An example of a harmonic proportion is formed by the numbers 6, 4, 3, because
