Golden ratio

Some shapes have a special balance.

You can make a special rectangle. If you cut a square from it, a smaller rectangle stays. This new rectangle has the same shape.

This pattern is in many places. You can see it in a five-sided star. It is also in the way leaves grow on plants.

Some artists use it too. They think these shapes look very pretty. It is a math rule found in nature.

Imagine you have two lengths. If you add them together, they make a new, longer line. The golden ratio happens when the whole line and the long part share the same balance as the two parts do.

This special balance is a number called phi. It is an irrational number. This means you cannot write it as a simple fraction.

Math experts have studied this for a long time. Ancient Greeks found it in shapes like the five-sided star.

It also helps build shapes like the dodecahedron. A mathematician named Hippasus found its strange nature. He showed it was not a whole number.

Later, people found links to the Fibonacci numbers. These are numbers where you add the last two to get the next one.

You can see these patterns in nature. They show up in how leaves grow on plants. Some artists and builders use it too. They think these shapes look very pleasing to the eye.

Imagine you have two different lengths. If you add them together, they make one long line. The golden ratio is a special balance between these parts. This balance happens when the whole line and the long part share the same ratio as the two original parts.

Mathematicians use the Greek letter phi to represent this number. It is an irrational number, which means it cannot be written as a simple fraction. This number is approximately 1.618. It is also the solution to a specific math puzzle called a quadratic equation.

This ratio creates very interesting shapes in geometry. A golden rectangle has sides that follow this special balance. If you cut a square out of a golden rectangle, the piece left over is another golden rectangle.

This pattern can repeat forever in a smaller and smaller way. The ratio also appears in a five-sided star called a pentagram. It is used to build shapes like the dodecahedron and the icosahedron.

These shapes are part of a group called the Platonic solids. Even patterns like Penrose tiling use this ratio to work.

People have studied this ratio for thousands of years. Ancient Greek mathematicians first noticed it in their geometry studies. A mathematician named Hippasus lived in the 5th century BC. He discovered that the ratio was irrational, which surprised his fellow thinkers.

Later, Euclid wrote about it in his famous book, Elements. Around the year 850, a mathematician named Abu Kamil used it for his work. In 1509, Luca Pacioli wrote a book called Divina proportione. Leonardo da Vinci even drew pictures for that book. He called the ratio the golden section.

Many famous thinkers found new ways to look at phi. In 1597, Michael Maestlin wrote down a decimal version of the ratio. Johannes Kepler later studied the Kepler triangle, which uses this ratio.

In the 1800s, a man named Jacques Philippe Marie Binet found a special formula. This formula uses the golden ratio to find any Fibonacci number. Even the inventor Mark Barr helped by choosing the symbol phi in 1910. These discoveries helped math grow much deeper over many centuries.

We can see the golden ratio in the world around us. It shows up in nature, like the spiral of leaves on a plant.

Some artists and architects use it to make their work look beautiful. They believe these proportions are very pleasing to the eye.

It is also closely linked to the Fibonacci sequence. In this sequence, you get the next number by adding the two before it. As the numbers in the sequence get larger, their ratio gets closer to phi. This connects simple counting to the deep patterns of the universe.

The golden ratio is a unique mathematical constant that describes a specific relationship between two quantities. Two quantities, often called $a$ and $b$, are in the golden ratio if the ratio of their sum to the larger quantity is equal to the ratio of the larger quantity to the smaller one. This relationship is expressed by the Greek letter phi ($\phi$). Mathematically, it is defined by the equation $\frac{a+b}{a} = \frac{a}{b} = \phi$. This constant is an irrational number, meaning it cannot be written as a simple fraction of two integers. Its approximate value is 1.618.

To understand the mechanism of this ratio, one can look at a golden rectangle. This is a rectangle where the side lengths are in the golden ratio. If you cut a square from a golden rectangle, the remaining piece is another, smaller golden rectangle.

This process can be repeated indefinitely, creating a sequence of smaller and smaller rectangles. This geometric property is closely linked to the quadratic equation $x^2 - x - 1 = 0$. The positive root of this equation provides the exact value for $\phi$. Because it is a root of this polynomial, $\phi$ is considered an algebraic number.

Geometry reveals many distinct types of shapes defined by this ratio. In a regular pentagon, the ratio of a diagonal to its side is exactly the golden ratio. This property is essential for constructing complex shapes like the dodecahedron and the icosahedron.

Another fascinating application is found in Penrose tiling. Developed by Roger Penrose between 1973 and 1974, these patterns use two different rhombic tiles. The ratio of the areas of these tiles, and their frequency within the pattern, is related to the golden ratio.

This mathematical structure was later connected to the discovery of quasicrystals by Dan Shechtman.

The history of the golden ratio spans thousands of years of human discovery. Ancient Greek mathematicians first studied it through geometry, specifically regarding pentagrams. A mathematician named Hippasus, living in the 5th century BC, reportedly discovered its irrationality. This was a shocking revelation to the Pythagoreans of that time.

Later, Euclid provided the first known definition of the ratio in his work, *Elements*. During the Islamic Golden Age, the mathematician Abu Kamil used the ratio in calculations involving pentagons and decagons around the year 850.

In the Renaissance, the ratio gained new cultural significance. Luca Pacioli published *Divina proportione* in 1509, which explored the ratio's properties in Platonic solids. Leonardo da Vinci illustrated this book and referred to the ratio as the *sectio aurea*, or golden section. While some believe Pacioli promoted the ratio for aesthetic beauty, historians note this may be a later interpretation. By the late 16th century, mathematicians like Rafael Bombelli were solving complex geometric problems using these proportions. In 1597, Michael Maestlin provided one of the first decimal approximations of the ratio.

One of the most significant connections is between the golden ratio and the Fibonacci sequence. The Fibonacci sequence starts with 0 and 1, where each subsequent number is the sum of the two preceding ones. As you move further into the sequence, the ratio of any two consecutive numbers converges to $\phi$. This relationship was noted by mathematicians like Johannes Kepler in 1608. Later, in 1843, Jacques Philippe Marie Binet discovered a formula that uses the golden ratio to calculate any Fibonacci number directly. This is now known as Binet's formula.

Today, the golden ratio is studied across many different scientific and artistic fields. In nature, it appears in the spiral arrangements of leaves and other vegetation.

Some 20th-century architects, such as Le Corbusier, and artists, such as Salvador Dalí, used these proportions in their work. They believed the golden rectangle was aesthetically pleasing. Additionally, the ratio is used to analyze artificial systems, including financial markets, though some of these applications are considered dubious. Whether in the structure of a crystal or the spiral of a plant, the golden ratio remains a fundamental concept in understanding patterns.