If you are a high school math student, you have heard of and studied equivalent fractions, differential equations, algebra, geometry, trigonometry (trig.), probability, Pythagorean theorem and other important mathematical concepts in the maths curriculum.

But what you may not realise is that this knowledge is built upon the maths basics that you have been taught since pre-school and primary school. Addition and subtraction, multiplication and division are the pillars for understanding all mathematics education and your ability to grasp and solve these math problems is what leads you now on the road to math mastery.

If you are serious about your math practice, you have studied sequences, fractions and algebraic math concepts. You will also recognise that math is not just made up of typical whole numbers but also symbols and letters which represent different parts of the number system.

Some of these special numbers are seen with pi, the distinctive Euler's e, the special numbers i, and of course the golden ratio.

Once you have left the primary grade math classroom where the curriculum was times tables, adding and subtracting. Maths begins to open up and with so does your understanding of the world. Rounding out your knowledge in the subject is a great skill as maths is one of the common core standards for many parts of our society today.

To learn math and more about its number patterns and mathematical models lets take a look at the golden ratio.

The Parthenon is a Greek temple
Plato used the golden ratio to design The Parthenon. Photo Source: Unsplash

What Is The Golden Ratio?

The golden ratio, divine proportion or golden proportion is said to be the perfect proportion that naturally occurs in nature. The term describes the relationship between two figures where the numbers of those figures are in a complementary ratio.

a+b/a = a/b = 1.618034

The ratio of the sum a + b of the two lengths to the largest (a ) is equal to that of the largest ( a ) on the smallest ( b )

For example, a rectangle which measures 12.94cm (a) by 8 cm (b) is in the golden ratio. How do we know? Let's look at an equation.

12.94/8 = 1.6175

As you can see the mathematical model has nothing to do with a square root or area and volume. This is classed as the golden ratio simply because we are dividing the long edge (12.94) by the short edge (8) which gives us our answer and the answer matches the golden ratio sequence. It is also important to know that the golden ration can’t be made into a fraction and its decimals go on infinitely, which makes it an irrational number.

It is the irrational number
All throughout the twentieth century, and even today the golden ratio continues to fascinate mathematicians, artists and architects. Photo Source: Unsplash

Mathematicians And The History Of The Golden Ratio

The golden ratio dates back to Egypt c.2600 BC, it is very old, used initially in geometry rather than arithmetic. As well as the Egyptians it may also have been used by the Pythagoreans, who may have used it to build pentagons using isosceles triangles.

  1. The first mathematical text really highlighting the golden ratio was written by Euclid c.300 BC.
  2. Plato is one of the first mathematicians to study the golden proportion exclusively as a mathematical concept. He used it to design The Parthenon
  3. Mathematician Al-Khawarizmi shines a new light on the golden ratio in the eighth century by proposing several problems of dividing a length of ten units into two parts. The solution of one of them is the initial size divided by the golden ratio.
  4. It is Fibonacci who talks about the equations along with his famous Fibonacci sequence. He found a link between the Fibonacci sequence and the golden ratio. By dividing a number in the series by the previous entry, the result comes very close to the golden ratio. This estimating becomes very accurate the further along in the sequence or, the higher in the sequence that you go.
  5. The irrationality of the golden number is taken on by Campanus through to who explores the golden ratio in geometry. He creates the golden spiral which is a logarithmic spiral. The growth factor of the spiral is equal to the golden ratio.
  6. Written about by Pacioli in his book divine proportion which was illustrated by Da Vinci.
  7. The German philosopher Adolf Zeising thinks that the golden ratio makes it possible to understand both scientific and artistic fields.

All throughout the twentieth century, and even today the golden ratio continues to fascinate mathematicians, artists and architects. It is the irrational number that gives rational meaning to beauty.

A Closer Look At Mathematician Fibonacci

Fibonacci, thanks to whom the golden ratio all began (well, not exactly, the concept was there always but it was he who first discovered it and revealed it to the world).

Fibonacci is one of the most famous names in this subject, but you may be interested to know that the mathematician we know as Fibonacci wasn't actually called Fibonacci! Leonardo Pisano was the man who immortalised the famous golden ratio sequence – 0, 1, 1, 2, 3, 5, 8, 13, ...

Pisano was born late in the 12th century in Pisa, Italy, and it was because of an innocent mix up that he became known as Fibonacci. Centuries after handwritten copies of his book Liber Abaci were published, scholars misread part of the title, declaring it to be"filius Bonacci" meaning "son of Bonaccio" – as the man's surname. And this is how the name Fibonacci was born.

Fibonacci introduced the decimal number system to the Latin-speaking world, and his first chapter read:

"These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated."

Fibonacci wrote Liber Abaci to inform merchants who, especially in Italy which was at the time made up of small towns and regions with different money and weight systems, had many practical problems when it came to their commercial dealings. He provided them with examples which demonstrated how simply calculations could be done with this new number system compared to the existing Roman numerals.

Fibonacci is most famous, however, for a certain sequence of numbers that appeared as an example in Liber Abaci - the Golden Ratio.

Coolmath: The Uses Of The Golden Ratio

  • Plato used it to design The Parthenon
  • The Egyptians used it to design and build the pyramids
  • Architecture: Notre Dam, the Taj Mahal and the Great Mosque of Kairouan all have golden ratio elements
  • Artists: Da Vinci, Dali and other used and continuous to use the golden ratio in the layout of their artwork.
  • Music: The ratio can be detected in the music of Bach, Debussy, Beethoven and Chopin
  • Books: the bible talks about the golden ratio number system and things being built following the ratio
  • Nature: Flowers, plants, foods and more grow in accordance with the golden ratio. Even hurricanes and its chaos follow the number lines of the golden ratios.
  • Life: Insects, animals and humans have a connection to the golden ratio. Looking at Da Vincis The Vitruvian Man drawing, you can see that the human body is created based on the golden ratio. In the modern day, psychologists have noted that attractiveness may be judged by those who have features that are in line with the golden ratio. The golden ratio is even shown in our DNA

What do you think of that for some cool math? I wonder what the statistic is that a golden ratio is just a regular number? I am guessing that the golden ratio is a true interactive math sequence that is guiding the order of operations within the divine universe.

now learn other important mathematical concepts
If you are a high school math student, you have heard of and studied equivalent fractions. Photo Source: Unsplash

Why Use The Golden Ratio?

The Golden Ratio is a truly fascinating number as it finds itself everywhere, even the building blocks of our DNA follow this principle. I have no interest in the golden ratio mathematically, but it is a personal interest of mine. Don’t you find it fascinating? This exponential number has somehow permeated every corner of the universe. From galaxies to bees, from flowers to the weather.

The golden principle is interesting to me because of its logic, despite being an irrational number, it is a math fact that this sequence has a lot to offer us. You can use it when solving a math problem. You can use it to help you to design something that is aesthetically pleasing. You can even use it for some math fun and discover the golden ratio in your home or even on your own face.

You won't find this taught in the kindergarten or a primary level classroom, but perhaps the high school math classroom was designed using the golden ratio. This is fun math that exists on the borders of math, science, philosophy, reality and discovery. This is not like linear equations, differential equations, Polynomial triangles, or the mastery of other mathematical concepts.

Indeed, high school can be fun and exciting, even with math learning. This theorem is still being discovered and explored so if you want to put your name in the history books. The golden ratio could be your opportunity to win a noble prize, get your math worksheets ready!.

As a learner of maths knowledge about the golden ratio should be added to your list of math skills. However make sure you have your foundation in arithmetic, subtracting, multiplication problems or average grade math (basic math). Because it gives you a strong foundation to understand and work from.

The Golden Ratio And Horticulture

The website Maths Is Fun states that "Plants can grow new cells in spirals, such as the pattern of seeds in this beautiful sunflower.

The spiral happens naturally because each new cell is formed after a turn.

New cell, then turn,
then another cell, then turn, ..."

Many gardeners like to mimic this naturally occurring pattern in garden design, using layouts that add to the harmony and beauty of nature at its finest. Admirers of this sequence believe that, even if your garden is an awkward shape, you can still achieve a garden landscaped around the golden ratio.

Gardeners should keep in mind an imaginary golden rectangle, divided into a square and then another rectangle. The beauty of the design sequence is that it can be repeated again and again by drawing new golden rectangles within golden rectangles. Once a gardener has grasped this concept in the design phase, then they can begin to recognise this pattern in the plants that flourish in their garden.

Architecture Designed Using The Golden Ratio

While the golden ratio occurs naturally in plants and flowers, what's interesting is that some famous architects were so influenced by the mathematical concept that they structured the string of numbers into their design work. Let's take a look at some examples.

It is considered one of the most appealing of all the forms in geometry, with some calling it the Divine Proportion. Some are of the belief that the Egyptian pyramids were built using the golden ratio.

  • The Parthenon, Greece

The west façade of the Parthenon, part of Greece's famous Acropolis of Athens (468-430 BC), is said to have many proportions that resemble the sequence of the golden ratio. However, many scholars doubt that it can be possible that the Greeks knew about this set of golden numbers as a principle of aesthetic proportion. Work on the Acropolis was thought to have begun around 600 BC but Fibonacci's work on the golden ratio didn't get unearthed until between 468-430 BC.

Wikipedia states that:

"The Parthenon (447–432 BC), was a temple of the Greek goddess Athena. The Parthenon's facade as well as elements of its facade and elsewhere are claimed to be circumscribed by a progression of golden rectangles. Some more recent studies dispute the view that the golden ratio was employed in the design.

Hemenway claims that the Greek sculptor Phidias (c. 480–c. 430 BC) used the divine proportion in some of his sculptures. He created Athena Parthenos in Athens and Statue of Zeus (one of the Seven Wonders of the Ancient World) in the Temple of Zeus at Olympia. He is believed to have been in charge of other Parthenon sculptures, although they may have been executed by his alumni or peers. In the early 20th century, American mathematician Mark Barr proposed the Greek letter phi (φ), the first letter of Phidias's name, to denote the golden ratio.

Lothar Haselberger claims that the temple of Apollo in Didyma (c. 334 BC), designed by Daphnis of Mileto and Paionios of Ephesus, has golden proportions."

  • Notre-Dame Cathedral, Paris

The cathedral, prior to its devastating fire earlier this year, displayed golden proportions recognised by many mathematicians.

Wikipedia once again confirms that:

"In his 1919 book Ad Quadratum, Frederik Macody Lund, a historian who studied the geometry of several gothic structures, claims that the Cathedral of Chartres (begun in the 12th century), the Notre-Dame of Laon (1157–1205), and the Notre Dame de Paris (1160) are designed according to the golden ratio. Other scholars argue that until Luca Pacioli's 1509 De Divina Proportione (see next section), the golden ratio was unknown to artists and architects, although this is not likely the case since the ratio was explicitly defined by Euclid."

  • The Great Pyramid, Egypt

Goldennumber.net states that the Great Pyramid of Egypt "closely embodies Golden Ratio proportions."

They say:

"Great Pyramid of Giza, Egypt with golden ratio proportions There is debate as to the geometry used in the design of the Great Pyramid of Giza in Egypt. Built around 2560 BC, its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with absolute certainty. The outer shell remains though at the cone, so this does help to establish the original dimensions.

There is evidence, however, that the design of the pyramid may embody these foundations of mathematics and geometry:

Phi, the Golden Ratio that appears throughout nature.
Pi, the circumference of a circle in relation to its diameter.
The Pythagorean Theorem – Credited by tradition to mathematician Pythagoras (about 570 – 495 BC), which can be expressed as a² + b² = c²."

How many other buildings in our towns have been designed and constructed using this intellectual system do you think?

Fun Ways To Help You Learn The Golden Ratio

When you are enjoying yourself, you retain knowledge longer and thus learn things much easier. This is why teachers of primary school aged pupils and younger will introduce games and puzzles to teach the young students about complicated subjects like Maths. When you were younger, did your math teacher use games to help you remember your times tables, calculus, adding and subtracting or maybe for solving problems?

Learning the fun way with quizzes, puzzles, perhaps a number jigsaw are interactive and engaging ways to study math.

It's all very well keeping things fun when you're teaching yourself (because who wants to do the boring bits?) but it's actually not impossible to teach yourself about maths theories and concepts on your own.

If your personal study doesn’t follow the math curriculum or core standards it's ok, as long as you are learning. Reinforce what you need to know and play with the rest, don’t limit your mind. As the golden ratio has shown us there really is no limit.

Even if you are dedicated and motivated as a student and teacher, don't forget that it is possible to find a tutor who can continue with your informal training and take the pressure off of you to find resources to use.

Learn Maths With A Teacher

Math's isn't an easy subject and so, if you come across an area that you just can't work through, it is very hard to work through it yourself.

A teacher can help you by explaining those concepts you find more difficult so that you don't get stuck and ht a brick wall (because much of maths depends on you understanding one concept to be able to move forward with another).

Get a private maths tutor for someone to one training. You can look for tutors advertising in local shops and newspapers or use an online tuition for maths platform such as Superprof to find a registered teacher near you.

Superprof Tutors

The advantage of having a private maths tutor is that you have the full, undivided attention of your teacher, and they can tailor the work to your needs.

Superprof, a leading platform for tutors and students to connect and form working partnerships, offers a user-friendly website on which you can instantly locate tutors offering maths tutoring services in your area, as well as those who are able to offer remote online tuition.

There are close to 60,000 tutors listed online who can help you electronically or via video call with your maths concerns. With prices starting from just £5 per hour, you can find someone who can meet your needs and get you moving forward with your math lessons. Some are mathematicians, some are qualified teachers, whilst others are individuals who are talented with numbers and want to pass on their knowledge and skills.

So, if you are still finding it hard getting to grips with the perfect numbers, or you want to educate yourself enough to challenge the experts and find the 52nd perfect number before anyone else does, then go ahead and get learning!

If you have an interest in special numbers read our blog about the special number pi, the important prime number series and the rarity that is the perfect numbers.

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