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The Mathematics of Design – Fibonacci, Fractals & Polyhedra

The Mathematics of Design – Fibonacci, Fractals & Polyhedra

It is common for people to group themselves into two categories: those who are good at art or design and those who are good at math or science.

The reason for this is that many people believe that the skills needed to be successful in creative services do not relate to the skills required to succeed in analytical.

Even though mathematics is on the list of the most challenging college classes, in reality, nothing could be further from the truth.

Many design concepts, such as symmetry, directly relate to mathematical concepts and discoveries.

So, if you are an artist or a designer, there is a good chance that you are already incorporating math into your work, you may not be doing it consciously.

Here are some very specific ways that the mathematics of design has not only influenced brands but has acted as a true game-changer.

The Fibonacci Sequence

Fibonacci In Design Golden Spiral Ratio

Consider this sequence numbers; 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55.

This particular sequence of numbers is what is known as the Fibonacci sequence.

If you look closer, you will notice that every number after the seed numbers 0 and the first 1, is the sum of the two numbers prior.

If you continued this pattern, you would add 55 and 34 to make 89.

At first glance, it may be difficult to see how any of this relates to art, but in fact, this sequence plays a significant role in art and design.

Imagine if each number is a shape. Let’s say the second number, 1 represents a square that is a 1-inch square. Then, let’s say that the 55 is a 55-inch square.

The combination of these squares can be used to form what is known as the golden rectangle.

You can find this rectangle in paintings such as the Mona Lisa and classical and modern architecture.

Fibonacci Architecture

Many web designers incorporate the golden rectangle in their designs by using the PHI calculator.

This calculator will help you to determine, for example, if you were to put the logo here (use your imagination), how the other elements of your page should be placed and sized.

Of course, this is not only limited to web design work. You could apply this to any design project.

Now, imagine the shapes representing circles. When arranged in specific ways, these can form the basis of starbursts, flowering patterns, branching, and more.

A perfectly formed spiral is based on the Fibonacci sequence. Not only do these things occur in art, but they also occur in nature.

The golden ratio, the reduction of numbers in the Fibonacci sequence that comes as close to zero as possible, can create the perfect spiral.

Many of Leonardo Da Vinci’s works incorporated the golden ratio in design mathematics. Even the human face closely follows the Fibonacci Sequence.

Other works that contain the golden ratio include the Great Pyramids and the Parthenon. Some people even believe the Apple computer logo uses the Fibonacci sequence, but experts dispute this claim.

Fractals

Design Mathematics Fractals

Fractals are repeating patterns that can be created via the mathematics of design but also appear in nature.

The most notable characteristic of fractals is that the repeating pattern can be noticed regardless of scale.

To better understand this, imagine using a microscope and turning up the magnification on an object.

No matter how closely you zoom in with fractals, you continue to see the same repeating patterns. This is known as self-similarity.

However, not all self-similarity is the same. For example, exact self-similarity means that the patterns within the fractal are perfectly identical.

On the other hand, quasi-self-similarity means that the patterns closely resemble one another but not perfectly.

Fractals can be created using computer software and mathematical formulas, but some of the most compelling examples of fractals come in nature. Here are just a few examples:

● Crystals
● Snowflakes
● Blood Cells
● Citrus Fruit
● Waves
● Mountain Ranges
● Fault Lines
● DNA
● Pineapple
● Animal Color Patterns

You can see an example of fractal patterns when you look at a window after a light frost. The similar patterns that repeat in the ice crystals are fractals.

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In art, some of the best-known examples of the use of fractals can be found in Jackson Pollock’s paintings.

Computer analysis has been done on his works that have determined the presence of fractals. This is even though they often appear quite random at first glance.

It is believed that his particular style of creating his paintings is the cause of this. Despite this, for the most part, fractals in the art are created digitally. Digital art and animation rely on fractals as their foundation.

Web designers and graphic artists frequently use fractal images. For example, if you have seen repeating patterns in background images on websites, these are often based on fractals.

Many believe the repeating patterns in fractals are both soothing and aesthetically pleasing.

If you would like to learn how to create fractals and use them in your designs, plenty of tools and examples are available to you on the internet.

Polykleitos Canon

Polykleitos was a famous Greek sculptor. He is widely regarded as one of the fathers of the classic Greek style of sculpture seen in many museums around the world.

His contributions do not stop there. He also wrote a Canon on symmetry regarding the sculpting of the male form. This canon represented his thoughts on aesthetics and artistic perfection.

He believed that all body parts appearing in a sculpture should be distinct from one another and proportioned using mathematics.

For example, the size and proportion of the first knuckle of the little finger would be the basis for the proportion of the rest of the finger; the finger would be the basis of the proportion of the palm of the hand, and so forth.

This was done by treating the tip of the finger as a square unit of measurement.

His work at determining the best ways to reach a perfect balance in sculpture is still influential today. Polykleitos, son and namesake, grew up to be a famous architect.

It is hard to imagine that his father’s work on proportion did not influence him greatly.

Polyhedra

Polyhedral Design Mathmatics Packaging

A polyhedron is a three-dimensional structure consisting of a collection of polygons joined along their edges.

Polyhedra have been incorporated into art and design for centuries. For example, Salvador Dali’s painting of The Last Supper depicts Jesus and his disciples within a dodecahedron, which is a type of polyhedron.

A polyhedral net is a net that has been unfolded for printing. Another of Salvador Dali’s works, Corpus Hypercubus, uses a polyhedral net as the cross.

Polyhedra was explicitly addressed in the book Education on Measurement by Albrecht Durer.

He was a German printmaker who wrote the book to educate others about perspective during the Renaissance. While some of his thoughts on that subject were a bit off base, his discussion of polyhedra and polyhedral nets was quite insightful.

These nets contain fascinating patterns often seen in mosaics and other art forms where geometric patterns are common.

Mathematics of Design And Other Art Forms

Even if artists and designers do not use mathematical formulas, the results often reflect using design mathematics.

If you notice symmetry, geometric patterns, balance, or proportion on a website, there is a mathematical principle behind those things.

Even the use of positive and negative space has mathematical origins. What’s even more interesting is that these principles do not only impact images.

Even the way that text is displayed on a screen can involve mathematics.

For example, the principle that text is more readable when there is plenty of white space primarily uses positive and negative space to create something more pleasing to the eye and more accessible to the process.

 

Logo Design Golden Ratio Mathematics

Conclusion

Mathematics goes hand in hand with art and design because the human brain appreciates and understands consistency.

For this reason, symmetry, balance, exactness, and proportion are vital parts of a design.

It is also why patterns frequently appear in art and design.

If you’ve ever visited a website or looked at a work of art that seemed off to you and was troubling, there is a good chance that the artist or designer failed in one of these areas.

It is one thing when this is done by the mathematics of design, for example, painting something that lacks symmetry for effect.

Unfortunately, when it is simply the result of poor execution, it becomes problematic.

Hopefully, some of the information here will help designers incorporate the mathematics of design principles into their work in more ways.

The results will likely be more aesthetically pleasing and easier to use.

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Stuart Crawford

Stuart Crawford is an award-winning creative director and brand strategist with over 15 years of experience building memorable and influential brands. As Creative Director at Inkbot Design, a leading branding agency, Stuart oversees all creative projects and ensures each client receives a customised brand strategy and visual identity.

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